Projective Varieties With Unexpected Properties

Projective Varieties with Unexpected Properties Projective Varieties with Unexpected Properties A Volume in Memory of...

Author: Ciro Ciliberto; Antony V. Geramita

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Projective Varieties with Unexpected Properties

Projective Varieties with Unexpected Properties A Volume in Memory of Giuseppe Veronese Proceedings of the International Conference ‘Varieties with Unexpected Properties’ Siena, Italy, June 8⫺13, 2004

Editors C. Ciliberto A.V. Geramita B. Harbourne R. M. Miro´-Roig K. Ranestad

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Walter de Gruyter · Berlin · New York

Editors Ciro Ciliberto Dipartimento di Matematica Universita` degli Studi di Roma “Tor Vergata” Via della Ricerca Scientifica 00133 Roma, Italy. e-mail: [email protected] Antony V. Geramita Department of Mathematics & Statistics Queen’s University Jeffery Hall, University Ave. Kingston, ON K7L 3N6, Canada Dipartimento de Matematica Universita` degli Studi di Genova Via Dodecaneso, 35 16146 Genova, Italy e-mail: [email protected]

Brian Harbourne Department of Mathematics University of Nebraska ⫺ Lincoln 331 Avery Hall Lincoln, NE 68588-0130, USA e-mail: [email protected] Rosa Maria Miro´-Roig Faculty of Mathematics University of Barcelona Gran Via 585 08007 Barcelona, Spain e-mail: [email protected] Kristian Ranestad Mathematical Institute University of Oslo POB 1053 ⫺ Blindern 0316 Oslo, Norway e-mail: [email protected]

Mathematics Subject Classification 2000: 14M07, 14M15, 14M17, 14M20, 14N05, 14E08, 14J10

앝 Printed on acid-free paper which falls within the guidelines of the 앪 ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data Projective varieties with unexpected properties : a volume in memory of Giuseppe Veronese : proceedings of the international conference “Varieties with Unexpected Properties,” Siena, Italy, June 8⫺13, 2004 / edited by Ciro Ciliberto ... [et al.]. p. cm. ISBN-13: 978-3-11-018160-9 (hardcover : acid-free paper) ISBN-10: 3-11-018160-6 (hardcover : acid-free paper) 1. Algebraic varieties ⫺ Congresses. 2. Geometry, Projective ⫺ Congresses. I. Veronese, Giuseppe, 1854⫺1917. II. Ciliberto, C. (Ciro), 1950⫺ QA564.P787 2005 516.3153⫺dc22 2005025650

ISBN-13: 978-3-11-018160-9 ISBN-10: 3-11-018160-6 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at ⬍http://dnb.ddb.de⬎. ” Copyright 2005 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Cover design: Thomas Bonnie, Hamburg. Typeset using the authors’ TEX files: M. Pfizenmaier, Berlin. Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen.

Preface The Conference “Projective Varieties with Unexpected Properties” was held in Siena (Italy) during the week June 8–13, 2004. The conference was dedicated to Giuseppe Veronese, who was born on May 7, 1854, on the occasion of his 150th birthday. As is well known, Veronese was one of the initiators of modern algebraic geometry, in particular of the study of special projective varieties, which was the main topic of the conference. This volume contains refereed papers related to the lectures and talks given during the conference as well as papers that grew out of discussions among the participants and their collaborators. All the papers are research papers, but some of them do also contain expository sections which aim at updating the state of the art of the subject of special projective varieties and their applications. We hope they will be of value both as reference papers for people who work in the ﬁeld, as well as for non-experts. The Scientiﬁc and Organizing Committee of the conference consisted of: Cristiano Bocci (Milano, Italy), Luca Chiantini (Siena, Italy), Ciro Ciliberto (Rome, Italy), Anthony V. Geramita (Kingston, Canada and Genova, Italy), Brian Harbourne (Nebraska, USA), Antonio Lanteri (Milano, Italy), Rosa Maria Mir´ o–Roig (Barcelona, Spain), Kristian Ranestad (Oslo, Norway). All the members of the Scientiﬁc and Organizing Committee actively collaborated in the preparation of the present volume. Special thanks go to Cristiano Bocci and Luca Chiantini for their perfect organization of the conference and for the friendly atmosphere they were able to create and which was essential for the success of the event. We are extremely grateful to all participants and to all contributors to this volume. We also thank: • the research project Geometria delle Variet` a Algebriche; • the Gruppo Nazionale Strutture Algebriche, Geometriche e le loro Applicazioni - Istituto Nazionale di Alta Matematica “F. Severi”; • the european research project EAGER; • the University of Siena; • the Banca Monte dei Paschi di Siena; • the Azienda Provinciale per il Turismo of Siena; which all contributed funds for the conference. The Editors: Ciro Ciliberto Antony V. Geramita Brian Harbourne Rosa Maria Mir´ o–Roig Kristian Ranestad

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v E. Arrondo, R. Paoletti Characterization of Veronese varieties via projections in Grassmannians . . . . . . . 1 E. Ballico, C. Fontanari Birational geometry of defective varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 M. Beltrametti, M.L. Fania Fano Threefolds as hyperplane sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 C. Bocci, L. Chiantini Triple points imposing triple divisors and the defective hierarchy . . . . . . . . . . . . . 35 A. Bruno, A. Verra M15 is rationally connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 E. Carlini Codimension one decompositions and Chow varieties . . . . . . . . . . . . . . . . . . . . . . . . . 67 M.V. Catalisano, A.V. Geramita, A. Gimigliano Higher secant varieties of Segre-Veronese varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 A. Van Tuyl Appendix to a Paper of Catalisano, Geramita, Gimigliano: The Hilbert function of generic sets of 2-fat points in P1 × P1 . . . . . . . . . . . . . . . 109 F. Catanese, B. Wajnryb The 3-cuspidal quartic and braid monodromy of degree 4 coverings . . . . . . . . . 113 L. Chiantini, C. Ciliberto On the classiﬁcation of defective threefolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131 C. Ciliberto, R. Miranda Matching conditions for degenerate plane curves and applications . . . . . . . . . . . 177 T. De Fernex Negative curves on very general blow ups of P2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 P. De Poi, E. Mezzetti Linear congruences and hyperbolic systems of conservation law . . . . . . . . . . . . . 209 C. De Volder, A. Laface A note on very ampleness of complete linear systems on blowing ups of P3 . . 231 N. Eriksson, K. Ranestad, B. Sturmfels, S. Sullivan Phylogenetic algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

viii

Contents

E. Guardo, A. Van Tuyl Some results on fat points whose support is a complete intersection minus a point 257 B. Harbourne The (unexpected) importance of knowing α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 A. Iarrobino Hilbert functions of Gorenstein algebras associated to a pencil of forms . . . . . 273 A. Iliev, L. Manivel Varieties of reductions for gln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 P. Ionescu Birational geometry of rationally connected manifolds via quasi-lines . . . . . . . . 317 A. Lanteri, R. Mu˜ noz On the discriminant of spanned line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 A. Noma Multisecant lines to projective varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 Z. Ran Cycle map on the Hilbert scheme of nodal curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 Schedule of the conference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 List of participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 List of contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

Characterization of Veronese varieties via projection in Grassmannians Enrique Arrondo and Raﬀaella Paoletti

A Giacomo Paoletti, amico e padre Abstract. We characterize for any d the d-uple Veronese embedding of Pn as the only variety that, under certain general conditions, can be projected from the Grassmannian of (d − 1)-planes in Pnd+d−1 to the Grassmannian of (d − 1)-planes in Pn+2d−3 in such a way that two (d − 1)-planes meet at most in one point. We also study the relation of this problem with the Steiner bundles over Pn . 2000 Mathematics Subject Classiﬁcation: 14N05, 14N20

Introduction In [2], the ﬁrst author characterized, under certain conditions, the embedding of Pn in G(1, 2n + 1) via OPn (1)⊕2 as the only n-dimensional subvariety of G(1, 2n + 1) that is isomorphically projectable into a subvariety of G(1, n + 1) (under a projection induced by a linear projection from P2n+1 to Pn+1 ). In this paper we deal with the analogous problem for the embedding of Pn in G(d − 1, nd + d − 1) via OPn (1)⊕d . One ﬁrst problem is that the naive generalization to arbitrary d of the case d = 2 does not work (see Remark 2.3). As we will see in Section 2, the right notion for a subvariety of a Grassmann variety of (d − 1)-planes to be special under projection is to be projectable into a subvariety of G(d − 1, n + 2d − 3) in such a way that any two (d − 1)-planes meet at most in one point. With this notion in mind, in Section 3 we prove our main projectability result (Theorem 3.1) by essentially repeating most of the steps in the proof of [2]. The only step we cannot reproduce is the result that a projectable variety has positive defect, so that we will need to add this as a hypothesis. Finally, in Section 4 we observe that the projectability of Pn in Grassmannians is closely related to Steiner bundles. However, not all of them appear. We will see for instance that, if n ≥ 3, we never obtain the most particular case, namely the Schwarzenberger bundles.

2

Enrique Arrondo and Raﬀaella Paoletti

Both authors acknowledge the ﬁnancial support of the Italy-Spain project “Acci´on Integrada” HI00-128. The ﬁrst author has also been supported by the Spanish Ministry of Science and Technology project BFM2000-0621. The second author has been partially supported by Italian MURST and GNSAGA-INdAM. Both authors belong to EAGER (European contract PRN-CT-2000-00099).

1. Preliminaries and projectability in Grassmannians of lines We will work over an algebraically closed ﬁeld of characteristic zero. We will denote by G(r, m) the Grassmann variety of r-linear spaces in Pm . A linear projection G(r, m) G(r, m ) will mean the natural (rational) map induced by the corresponding linear projection Pm Pm . The main example we are going to consider is the following. Example 1.1. Consider the natural embedding of Pn in G(d − 1, nd + d − 1) deﬁned by OPn (1)⊕d and call V its image. In coordinates, it can be described by associating to each (x0 : . . . : xn ) ∈ Pn the (d − 1)-plane spanned by the rows of the matrix ⎛

x0 . . . xn ⎜ 0...0 ⎜ ⎜ .. ⎝ . 0...0

0...0 x0 . . . xn

... ... .. .

0...0 0...0 .. .

0...0

...

x0 . . . xn

⎞ ⎟ ⎟ ⎟. ⎠

We consider now the linear projection Pnd+d−1 Pn+d−1 deﬁned by (z10 : z11 : . . . : z1n : z20 : . . . : zd,n−1 : zdn ) → (z10 : z11 + z20 : . . . : z1,d−1 + . . . + zd0 : . . . . . . : z1n + z2,n−1 + . . . + zd,n+1−d : . . . : zd−1,n + zd,n−1 : zdn ). This projection induces a projection from G(d − 1, nd + d − 1) to G(d − 1, n + d − 1) under which the image of the above Pn consists of the (d − 1)-planes spanned by the rows of the matrix ⎞ ⎛ x0 x1 . . . xn 0 . . . 0 ⎜ 0 x0 x1 . . . xn . . . 0 ⎟ ⎟ ⎜ (1) ⎜ .. . ⎟. .. .. .. ⎝. . . . .. ⎠ 0

...

0

x0

x1

...

xn

This is still an embedding of Pn in G(d − 1, n + d − 1), since the maximal minors ucker embedding of of the above matrix (which give the image of Pn after the Pl¨

Characterization of Veronese varieties via projection in Grassmannians

3

n+d

G(d − 1, n + d − 1) in P( d )−1 ) deﬁne the d-uple Veronese embedding of Pn in n+d P( d )−1 . By this reason we will call this subvariety V ⊂ G(d − 1, nd + d − 1) (or any of its isomorphic projections) the n-dimensional Veronese variety. Following [2] we will use the following deﬁnitions: Deﬁnition 1.2. We will say that a subvariety X ⊂ G(d − 1, N ) of dimension n is nondegenerate if the union of all the (d − 1)-planes parametrized by X is not contained in a hyperplane of PN . We will also say that X is uncompressed if the union in PN of all of the (d − 1)-planes parametrized by X has the expected dimension n + d − 1. Otherwise, if the dimension is smaller, we will say that X is compressed. The main result of [2] is the following: Theorem 1.3. The only nondegenerate uncompressed n-dimensional subvariety of G(1, 2n + 1) that can be isomorphically projected to G(1, n + 1) is the Veronese variety (i.e. the one of Example 1.1 for d = 2). The main tool to prove that result is the use of the generalization to the Grassmannians context of the notions of secant variety and secant defect. The precise deﬁnitions are as follows: Deﬁnition 1.4. The k-secant variety of a variety X ⊂ G(1, N ) is the variety S k X ⊂ G(rk , N ) consisting of the closure of the set of linear spans of k + 1 general lines of X (observe that, although one should expect rk to be 2k + 1, its value could be actually smaller). The k-secant defect of X is the dimension δk of the set of lines of X contained in a general space of S k X. It follows that S k X has dimension (k + 1)(n − δk ). The two main ingredients in the proof of Theorem 1.3 are the fact that a projectable variety has positive ﬁrst defect δ1 (actually in this case one can prove that δ1 = 1) and the fact that more precisely the set of lines contained in a general 3-plane of S 1 X is a conic (speciﬁcally, the set of lines meeting two skew lines). It is thus easy to prove that a projectable variety in G(1, 2n + 1) has defects δk = k for k = 1, . . . , n and that the divisors of X obtained for k = n−1 yield an isomorphism of X to Pn whose inverse is the double Veronese embedding.

2. Projectability in arbitrary Grassmannians The obvious natural generalizations of the notions of k-secant varieties and ksecant defects in arbitrary Grassmannians are the following: Deﬁnition 2.1. The k-secant variety of X ⊂ G(d − 1, N ) is the closure S k X in G(rk , N ) of the set of linear spans of k+1 general (d−1)-planes of X. The k-secant

4

Enrique Arrondo and Raﬀaella Paoletti

defect of X is the dimension δk of the set of (d − 1)-planes of X contained in a general space of S k X. It follows that S k X has dimension (k + 1)(n − δk ). In the case k = 1, we will just speak about secant variety SX and secant defect δ. In general, one should expect r1 = 2d − 1. Example 2.2. The Veronese variety V deﬁned in Example 1.1 has ﬁrst secant variety of dimension 2n−2, i.e. less than expected. Indeed, let (z10 : z11 : . . . : z1n : z20 : . . . : zdn ) be homogeneous coordinates on Pnd+d−1 and for each k ∈ {1, . . . , d} consider the n-dimensional linear subspaces Πk deﬁned by the equations: zij = 0 for j = 1, . . . , d and i = k. Then each point of V can be represented as the span of d corresponding points P1 , . . . , Pd s.t. Pk ∈ Πk . Let ϕ : V ×V G(2d−1, nd+d−1) deﬁned by ϕ ((Λ1 , Λ2 )) = Λ1 , Λ2 . Then SV = Im ϕ. Let π ∈ Im ϕ, π = Λ1 , Λ2 with Λ1 = P1 , . . . , Pd and Λ2 = Q1 , . . . , Qd . Then π = P1 Q1 , . . . , Pd Qd where Pk Qk is the line through Pk and Qk . Thus

1 , . . . , Q d ) / P 1 = Q 1 ∈ P1 Q1 ϕ−1 (π) = (P 1 , . . . , P d , Q has dimension 2 and dim SV = 2n − 2. In the same way, it is easy to see that V has secant defects δk = k for k = 1, . . . , n. Remark 2.3. Let us see that the Veronese varieties cannot be characterized by means of the projectability to G(d − 1, n + d − 1) that we have seen in Example 1.1. In fact, let X ⊂ G(d − 1, N ) be an irreducible nondegenerate ndimensional variety and let us analyze when it is possible to project it isomorphically to G(d − 1, n + d − 1). A projection will be induced by a linear projection πA : PN → Pn+d−1 with center a linear space A of dimension N − n − d. The fact that πA induces an isomorphism between X and its image in G(d − 1, n + d − 1) is equivalent to the following property: for any Λ1 , Λ2 ∈ X (maybe inﬁnitely close), dim(Λ1 , Λ2 ∩ A) < dim(Λ1 , Λ2 ) − d. Actually this condition says that dim πA (Λ1 , Λ2 ) ≥ d, so that πA (Λ1 ), πA (Λ2 ) represent distinct points in G(d − 1, n + d − 1). In particular, if Λ1 , Λ2 are skew the above condition states that dim(Λ1 , Λ2 ∩ A) < d − 1. Consider the following incidence variety: I = {(A, Π) | dim(A ∩ Π) ≥ d − 1} ⊂ G(N − n − d, N ) × SX and let p, q be the corresponding projections. The elements of p(I) represent “bad centers” of projections. A dimensional count on the ﬁbers of q shows that dim(I) = dim(SX)+dim G(N −n−d, N )−nd. Since dim(SX) ≤ 2n (and in fact one expects to have an equality), it follows that dim(I) ≤ n(2 − d) + dim G(N − n − d, N ). Therefore, only if d = 2 one can expect p to be surjective, which means that all the possible centers of projection should be bad. On the contrary, if d ≥ 3, one should be able to always ﬁnd a good center of projection. Hence only for d = 2 the Veronese varieties would be special by means of its projectability to G(d − 1, n + d − 1) as in Example 1.1. This is in fact the content of Theorem 1.3.

Characterization of Veronese varieties via projection in Grassmannians

5

The right notion of projectability will be the following: Deﬁnition 2.4. Let X ⊂ G(d − 1, N ) be a smooth irreducible variety and let k be an integer such that 0 ≤ k ≤ d − 1. We will say that X is k-projectable to G(d − 1, M ) if there exists a projection from G(d − 1, N ) to G(d − 1, M ) such that any two (d − 1)-planes of the image of X (maybe inﬁnitely close) do not meet along a linear space of dimension greater than or equal to k. Remark 2.5. 1) If X is k-projectable, then any two (d − 1)-planes of X itself do not meet along a linear space of dimension greater than or equal to k. 2) If k = d − 1, then X is (d − 1)-projectable to G(d − 1, M ) if and only if X is isomorphically projectable to G(d − 1, M ). 3) If k = 0, then X is 0-projectable to G(d−1, M ) if and only if any two (d−1)planes of the image of X are skew. In other words, the union of the (d − 1)-planes of X is a smooth scroll and it is isomorphic to its image in PM . 4) If X is k-projectable to G(d − 1, M ), it is also (k + 1)-projectable to G(d − 1, M − 1). In particular, if X is 1-projectable to G(d − 1, M ) it is also isomorphically projectable to G(d − 1, M − d + 2). 5) If X is uncompressed, then it is certainly not (d − 1)-projectable to G(d − 1, n + d − 2). Therefore, X is not 0-projectable to G(d − 1, n + 2d − 3). Example 2.6. We have seen (Example 1.1) that the Veronese variety is isomorphically projectable to G(d−1, n+d−1), but this was not a suﬃciently special property to characterize it (Remark 2.3). By the above fourth remark, 1-projectability to G(d − 1, n + 2d − 3) is a stronger condition. Let us see that this is now a very restrictive property satisﬁed by the Veronese variety. Let us repeat ﬁrst the dimensional count of Remark 2.3 considering now 1projectability to G(d − 1, n + 2d − 3) instead of (d − 1)-projectability to G(d − 1, n + d − 1) (obviously the two notions coincide if d = 2). So let X ⊂ G(d − 1, N ) be an irreducible nondegenerate n-dimensional variety. A projection to G(d − 1, n + 2d − 3) will be induced by a linear projection πA : PN → Pn+2d−3 with center a linear space A of dimension N − n − 2d + 2. For any Λ1 , Λ2 ∈ X (maybe inﬁnitely close), the fact that πA (Λ1 ) and πA (Λ2 ) meet at most in one point is equivalent to dim(Λ1 , Λ2 ∩ A) < 1. Consider now the following incidence variety: I = {(A, Π) | dim(A ∩ Π) ≥ 1} ⊂ G(N − n − 2d + 2, N ) × SX and let p, q be the corresponding projections. The elements of p(I) represent “bad centers” for 1-projectability. A new dimensional count on the ﬁbers of q shows that this time dim(I) = dim(SX) + dim G(N − n − d, N ) − 2n. Therefore, dim(I) ≤ dim G(N − n − d, N ) and in general one expects to have an equality. This shows that 1-projectability to G(d − 1, n + 2d − 3) is the right property to study (observe also that the ﬁfth remark above shows that we can never have 0-projectability unless X is compressed).

6

Enrique Arrondo and Raﬀaella Paoletti

On the other hand, the Veronese varieties V are 1-projectable to G(d − 1, n + 2d − 3): this follows immediately from the above dimensional count and the fact that the dimension of SV is smaller than expected (Example 2.2). We thus strengthen the conjecture in [2] to the following: Conjecture 2.7. The only smooth irreducible nondegenerate n-dimensional variety of G(d − 1, nd + d − 1) that is 1-projectable to G(d − 1, n + 2d − 3) is the Veronese variety.

3. Characterization of Veronese varieties in arbitrary Grassmannians. In this section we will prove the following evidence of Conjecture 2.7: Theorem 3.1. Let X ⊂ G(d−1, nd+d−1) be a smooth irreducible nondegenerate n-dimensional variety such that any two (possibly inﬁnitely close) (d − 1)-planes of X do not meet. If X has positive defect and is 1-projectable to G(d − 1, n + 2d − 3), then X is the Veronese variety. The case n = 1 of Conjecture 2.7 is very easy to prove. Before proving it, we state without proof an easy technical lemma that we will need. Lemma 3.2. The set Ω ⊂ G(d − 1, 2d − 1) of (d − 1)-planes of P2d−1 meeting a ﬁxed (d − 1)-plane Λ is (after the Pl¨ ucker embedding) a hyperplane section of G(d − 1, 2d − 1) having a point of multiplicity d at the point represented by Λ. The intersection of G(d − 1, 2d − 1) with the tangent cone of Ω at this singular point consists of the set of (d − 1)-planes meeting Λ in a space of dimension at least d − 2. Proposition 3.3. The only smooth irreducible nondegenerate curve X in G(d − 1, 2d − 1) that is 1-projectable to G(d − 1, 2d − 2) is the embedding of P1 via the vector bundle OP1 (1)⊕d . Proof. Let m be the degree of X after the Pl¨ ucker embedding. For any Λ ∈ X, consider the set of (d − 1)-planes of X meeting Λ. From Lemma 3.2 there are other m − d of them besides Λ. Hence there are m − d hyperplanes of P2d−1 containing two (d − 1)-planes of X (one of them being Λ). Of course everything is counted with multiplicity, and in particular it could happen that some or all of the above (d − 1)-planes are inﬁnitely close to Λ. But using in this case the second part of Lemma 3.2 we also get hyperplanes containing two inﬁnitely close (d − 1)-planes of X (and in fact we get not just hyperplanes but even linear spaces of dimension d).

Characterization of Veronese varieties via projection in Grassmannians

7

As a consequence, if m = d we would get, varying Λ, an inﬁnite family of hyperplanes containing two (d − 1)-planes of X. Therefore, through any point p ∈ P2d−1 we ﬁnd a hyperplane containing two (maybe inﬁnitely close) (d − 1)planes of X. This implies that the projection of X from any point would produce in the image two (d−1)-planes meeting along a line, which contradicts our hypothesis on X. We thus get that X has degree d. We then have that the union of the (d − 1)-planes of X is a d-dimensional scroll in P2d−1 of degree d, and hence it is a rational normal scroll. Since two (d − 1)-planes of the scroll do not meet, the splitting type is necessarily (1, . . . , 1),

i.e. X is the embedding of P1 via the vector bundle OP1 (1)⊕d . The proof of Theorem 3.1 is based, following the steps in [2], on two main lemmas that we will state and prove ﬁrst. Lemma 3.4. Let X ⊂ G(d − 1, N ) be a smooth irreducible nondegenerate variety of dimension n with positive defect and such that any two (maybe inﬁnitely close) (d − 1)-planes are skew. Then, for general Λ1 , Λ2 ∈ X, the set YΠ of (d − 1)-planes of X contained in Π := Λ1 , Λ2 is the one-dimensional family (or one of the two families if d = 2) of the Pd−1 ’s of the Segre variety P1 × Pd−1 ⊂ Π. In particular, the secant defect δ of X is equal to 1. Proof. Since the defect is positive, YΠ has dimension at least one. In the Grassmann variety of (d − 1)-planes of Π, the set of all those meeting Λ1 is a hyperplane section (under the Pl¨ ucker embedding). Therefore, if YΠ had dimension bigger than or equal to two, then it would meet that hyperplane section in at least a curve. This implies that there would be inﬁnitely many (d − 1)-planes of X meeting Λ1 . This contradicts the assumption that any two (d − 1)-planes of X are skew. Hence YΠ is a curve, and the result follows now from [3], Theorem 5.1 or from (the proof of) Proposition 3.3.

Lemma 3.5. Let X ⊂ G(d − 1, N ) be a smooth irreducible nondegenerate variety of dimension n with positive defect and such that any two (maybe inﬁnitely close) (d − 1)-planes are skew. Then, for any integer k > 1, δk ≥ min {δk−1 + 1, n}. Proof. Since clearly δk ≥ δk−1 , there is nothing to prove if δk−1 = n. We hence assume δk−1 < n. Take k + 1 general (d − 1)-planes Λ0 , . . . , Λk of X and write Π = Λ0 , . . . , Λk−1 and Π = Λ0 , . . . , Λk = Π , Λk . By the generality of these (d − 1)-planes, YΠ has dimension δk−1 and Λk is not contained in Π . Consider the incidence variety I = {(Λ , Λ) | Λ ⊂ Λ , Λk } ⊂ YΠ × YΠ and let p, q be the corresponding projections. Consider a general element Λ ∈ YΠ ; then p−1 (Λ ) = G(d−1, Λ , Λk )∩X = YΛ ,Λk and this has dimension δ1 ≥ 1 by hypothesis. Therefore, I has dimension δk−1 (X) + δ1 , so it is enough to show that

8

Enrique Arrondo and Raﬀaella Paoletti

q is generically ﬁnite over its image. We thus take a general element Λ ∈ Im q. Then q −1 (Λ) = {Λ ∈ YΠ | Λ ⊂ Λ, Λk } = {Λ ∈ X | Λ ⊂ Π , Λ ⊂ Λ, Λk } = X ∩ G(d − 1, Π ∩ Λ, Λk ). Since Λ, Λk is not contained in Π (because Λk is not) then Π ∩ Λ, Λk has dimension at most 2d − 2. Therefore any two (d − 1)-planes inside Π ∩ Λ, Λk should meet. Since two (d − 1)-planes of X cannot meet, it follows that X ∩ G(d − 1, Π ∩ Λ, Λk ) consists of at most one element. This completes the proof.

Proof of Theorem 3.1: By Lemma 3.4, we have δ1 = 1 and iterating Lemma 3.5 we get δk ≥ min{k, n}. This implies in particular that δm = n for some m ≤ n. Therefore all the (d − 1)-planes of X are contained in the span of m + 1 general elements of X, which has dimension at most md + d − 1. Since X is nondegenerate in G(d − 1, nd + d − 1), it follows that m = n and the span of n + 1 general elements of X is the whole Pnd+d−1 , i.e. the elements are in general position. Moreover, we get that, for k = 1, . . . , n, it holds δk = k and k + 1 general elements of X span a linear space of dimension kd + d − 1. In the particular case k = n − 1 we have that n general elements of X span a linear space Π of codimension d in Pnd+d−1 and YΠ is a hypersurface of X. This hypersurface is contained in the hyperplane section HΠ of G(d − 1, nd + d − 1) consisting of the (d − 1)-planes meeting Π. By Lemma 3.2, this hyperplane section is singular with multiplicity d along the set of (d−1)-planes contained in Π. Hence, as a divisor on X, the hyperplane section HΠ can be written as aYΠ + EΠ , with a ≥ d. If now Π is the span of two general elements of X, we know from Lemma 3.4 that YΠ is the set of (d−1)-planes of the Segre embedding of P1 ×Pd−1 in Π . This has degree (after the Pl¨ ucker embedding) equal to d, i.e. the intersection product of HΠ and YΠ is d. This means, using the above identity, that d = a + EΠ · YΠ . Therefore a = d and EΠ · YΠ = 0. Since YΠ “moves”, we obtain as in [2] that EΠ = 0. Summing up, we got that d · YΠ is linearly equivalent to the hyperplane section of X. Now the same reasoning as in [2] shows that X ∼ = Pn and that X is the Veronese variety.

4. Relation with Steiner bundles In this section we will study the relation of the projection of Veronese varieties in Grassmannians with the so-called Steiner bundles on Pn . Remark 4.1. Let us analyze more closely Example 1.1. The matrix 1 represents a set of (d − 1)-planes in Pn+d−1 parametrized by Pn or, dually, a set of (n − 1)planes in Pn+d−1 . It is well-known that this dual representation corresponds to the set of n-secant spaces to the rational normal curve. Indeed we can take the standard rational normal curve in Pn+d−1 (u0 : u1 : . . . : un+d−1 ) = (λn+d−1 : λn+d−2 μ : . . . : μn+d−1 ).

Characterization of Veronese varieties via projection in Grassmannians

9

Then a set of (maybe inﬁnitely close) n points on the curve is given by the zeros of a homogeneous polynomial x0 λn + x1 λn−1 μ + . . . + xn μn in the variables λ, μ. On the other hand, the system of hyperplanes of Pn+d−1 containing the (n − 1)plane spanned by this set of points is generated by the independent hyperplanes of Pn+d−1 : x0 u0 + x1 u1 + . . . + xn un = 0 x0 u1 + x1 u2 + . . . + xn un+1 = 0 .. . x0 ud−1 + x1 ud + . . . + xn un+d−1 = 0 whose coeﬃcients yield the matrix (1) in Example 1.1. Of course this is not the most general situation, in the sense that only after a very particular projection we will get such a dual representation. In the language of vector bundles, the embedding of Pn in G(d−1, nd+d−1) is equivalent to the eval⊕(n+1)d uation epimorphism OPn → OPn (1)⊕d (with kernel ΩPn (1)⊕d ) while the pro⊕(n+d) → OPn (1)⊕d . jection to G(d−1, n+d−1) is equivalent to an epimorphism OPn The dual of its kernel is called a Steiner bundle. And the bundle corresponding to the n-secant spaces to a rational normal curve (called Schwarzenberger bundle) is the most particular case. We refer to [1] for a thorough study of these bundles. Deﬁnition 4.2. We will call Schwarzenberger-Veronese variety a Veronese variety in G(d − 1, n + d − 1) that corresponds to a Schwarzenberger bundle. A ﬁrst fact showing how special Schwarzenberger-Veronese varieties are is the following: Proposition 4.3. If n ≥ 3, the Schwarzenberger-Veronese variety with d ≥ 3 is never obtained as a projection of a Veronese variety V ⊂ G(d − 1, n + 2d − 3) such that any two (d − 1)-planes of V meet at most in one point. Proof. Following the notations introduced in Example 1.1, we want to show that if n ≥ 3 it is not possible to factorize the projection ϕ of the Veronese variety V Pn → G(d − 1, nd + d − 1) to G(d − 1, n + d − 1) by means of a projection ψ to V ⊂ G(d − 1, n + 2d − 3) such that any two (d − 1)-planes of V meet at most in one point. Assume for contradiction that it were possible. Then this projection would be induced by a linear map Pnd+d−1 Pn+2d−3 deﬁned by (z10 : z11 : . . . : z1n : z20 : . . . : zd,n−1 : zdn ) → (z10 : z11 + z20 : . . . : z1,d−1 + . . . + zd0 : . . . . . . : z1n + z2,n−1 + . . . + zd,n+1−d : . . . : zdn : L1 : . . . : Ld−2 ) where L1 , . . . , Ld−2 are linear forms in z10 , . . . , zdn . Then the image under ψ of a point (x0 : . . . : xn ) ∈ Pn V would be represented by the (d − 1)-plane spanned by the rows of a matrix of the following type:

10

Enrique Arrondo and Raﬀaella Paoletti

⎛ ⎜ ⎜ ⎜ ⎝

x0 0 .. .

x1 x0

0

...

... x1 .. .

xn ... .. .

0 xn

0

x0

x1

... ... .. .

0 0 .. .

l1,1 l2,1

... ...

...

xn

ld,1

... ...

⎞ l1,d−2 l2,d−2 ⎟ ⎟ ⎟ ⎠ ld,d−2

(2)

where l1,1 , . . . , ld,d−2 are linear forms in x0 , . . . , xn . We will ﬁnd a contradiction by showing that there exist two of these (d − 1)-planes meeting in more than one point. The span in Pn+2d−3 of the (d−1)-planes corresponding to the (clearly distinct) points (t0 : . . . : tn−1 : 0), (0 : t0 : . . . : tn−1 ) would be the linear space generated by the rows of the matrix ⎛ ⎞ t0 t1 . . . tn−1 0 0 ... 0 l1,1 . . . l1,d−2 ⎜ 0 t0 t1 ⎟ . . . tn−1 0 ... 0 l2,1 . . . l2,d−2 ⎜ ⎟ ⎜ . ⎟ .. .. .. .. .. ⎜ .. ⎟ . . . . ... . ⎜ ⎟ ⎜ 0 ... 0 t0 t1 . . . tn−1 0 ld,1 . . . ld,d−2 ⎟ ⎜ ⎟ ⎜ ⎟ . . . tn−1 0 ... 0 l1,1 . . . l1,d−2 ⎜ 0 t0 t1 ⎟ ⎜ ⎟ ⎜ 0 0 t0 t1 . . . tn−1 . . . 0 l2,1 . . . l2,d−2 ⎟ ⎜ ⎟ .. .. ⎜ .. ⎟ .. .. .. ⎝ . ⎠ . . . ... . . t1 . . . tn−1 ld,1 . . . ld,d−2 0 0 ... 0 t0 where li,j = li,j (t0 , . . . , tn−1 , 0) and li,j = lij (0, t0 , . . . , tn−1 ). Performing the obvious elementary transformations on the rows, the matrix can be reduced to the matrix: ⎞ ⎛ t0 . . . tn−1 0 0 ... 0 l1,1 ... l1,d−2 ⎟ ⎜ 0 t0 . . . tn−1 0 ... 0 l2,1 ... l2,d−2 ⎟ ⎜ ⎟ ⎜ . .. .. .. .. ⎜ .. ⎟ . . . ... . ⎟ ⎜ ⎟ ⎜ 0 ... . . . tn−1 0 ld,1 ... ld,d−2 0 t0 ⎟ ⎜ ⎟. ⎜ ... 0 t0 . . . tn−1 ld,1 ... ld,d−2 ⎟ ⎜ 0 0 ⎟ ⎜ ⎜ 0 ... ... 0 l1,1 − l2,1 ... l1,d−2 − l2,d−2 ⎟ ⎟ ⎜ .. .. .. ⎟ ⎜ .. ⎠ ⎝ . . . . − ld,1 . . . ld−1,d−2 − ld,d−2 0 ... ... 0 ld−1,1

Obviously, the upper-left block of the matrix has rank d + 1. The lower-right block can be regarded as a (d − 1) × (d − 2) matrix of linear forms in the Pn−1 of coordinates t0 , . . . , tn−1 . Thus this block has rank ≤ d − 3 on a subvariety of Pn−1 of codimension at most 2. Since n ≥ 3, it follows that there exist t0 , . . . , tn−1 such that the rank of matrix is at most 2d − 2. In other words, the (d − 1)-planes corresponding to the points (t0 : . . . : tn−1 : 0), (0 : t0 : . . . : tn−1 ) meet in more than one point. This yields the wanted contradiction.

Let us consider now the case n = 2.

Characterization of Veronese varieties via projection in Grassmannians

11

Denote by (a : b : c) a system of Pl¨ ucker coordinates on the space P2∗ of lines in 2 P : if l is the line passing through the points (x0 : x1 : x2 ) and (y0 : y1 : y2 ), then a = x0 y1 − x1 y0 , b = x0 y2 − x2 y0 , c = x1 y2 − x2 y1 .

(3)

Remark 4.4. Observe that the line trough the points (t0 : t1 : 0) and (0 : t0 : t1 ) has coordinates (t20 : t0 t1 : t21 ) which satisfy the equation b2 − ac = 0; viceversa, each line whose coordinates (a : b : c) satisfy the same equation is a line generated by two points of type (t0 : t1 : 0), (0 : t0 : t1 ). Proposition 4.5. If n = 2, the Schwarzenberger-Veronese variety with d ≥ 3 can be obtained as a projection of a Veronese variety V ⊂ G(d − 1, 2d − 1) such that any two (d − 1)-planes of V meet at most in one point. Proof. According to the proof of Proposition 4.3, we need to ﬁnd linear forms l1,1 , . . . , ld,d−2 in the matrix(2) such that for any two (a priori possibly inﬁnitely close) points (x0 : x1 : x2 ), (y0 : y1 : y2 ) in P2 the matrix ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

x0 0 .. . 0 0 y0 0 .. . 0 0

x1 x0 ... ... y1 y0 ... ...

x2 x1 .. . 0 0 y2 y1 .. . 0 0

0 x2 .. . x0 0 0 y2 .. . y0 0

0 0 x1 x0 0 0 y1 y0

... ... .. . x2 x1 ... ... .. . y2 y1

0 0 .. . 0 x2 0 0 .. . 0 y2

l1,1 l2,1 ld−1,1 ld,1 l1,1 l2,1 .. . ld−1,1 ld,1

... ... ... ... ... ... ... ... ...

l1,d−2 l2,d−2 ld−1,d−2 ld,d−2 l1,d−2 l2,d−2 .. . ld−1,d−2 ld,d−2

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(4)

where li,j = li,j (x0 , x1 , x2 ) and li,j = lij (y0 , y1 , y2 ) has rank at least 2d − 1. A long series of matrix manipulations shows that the choice ⎛ ⎞ t2 0 ... 0 ⎜t0 + 2t1 + t2 ⎟ t2 0 ⎟ ⎞ ⎜ ⎛ ⎜ ⎟ .. l1,1 . . . l1,d−2 ⎜ ⎟ t0 + 2t1 + t2 . −t0 ⎜ ⎟ ⎜l2,1 . . . l2,d−2 ⎟ ⎜ ⎟ .. ⎜ ⎟=⎜ ⎟ . 0 0 −t0 ⎝ ⎠ ⎜ ... ⎟ ⎜ ⎟ 0 0 t2 ld,1 . . . ld,d−2 ⎜ ⎟ ⎜ ⎟ .. .. ⎝ . . . . . t0 + 2t1 + t2 ⎠ 0 0 −t0

satisﬁes the required property. It is helpful for the computations to remark that the rank of the matrix (4) does not change if we substitute the points P = (x0 : x1 : x2 ) and Q = (y0 : y1 : y2 ) with any two distinct points of the line P, Q . In particular, we should not care about the possibility that P and Q become inﬁnitely close.

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Enrique Arrondo and Raﬀaella Paoletti

References [1] Ancona V., Ottaviani G., Unstable hyperplanes for Steiner bundles and multidimensional matrices, Adv. Geom. 1 (2001), 165-192. [2] Arrondo E., Projections of Grassmannians of lines and characterization of Veronese varieties, J. Algebraic Geom. 8 (1999), 85-101. [3] Lanteri A., Turrini C., Some formulas concerning nonsingular algebraic varieties embedded in some ambient variety, Atti. Accad. Sci. Torino 116 (1982), 463-474. [4] Schwarzenberger R.L.E., Vector bundles on the projective plane, Proc. London Math. Soc. 11 (1961), 623-640. Enrique Arrondo Departamento de Algebra, Facultad de Ciencias Matem´ aticas Universidad Complutense de Madrid 28040 Madrid, Spain Email: enrique [email protected] Raﬀaella Paoletti Dipartimento di Matematica e Applicazioni per l’Architettura Piazza Ghiberti 23 50123 Firenze, Italy Email: raﬀ[email protected]ﬁ.it

Birational geometry of defective varieties Edoardo Ballico and Claudio Fontanari

Abstract. Here we investigate the birational geometry of projective varieties of arbitrary dimension having defective higher secant varieties. We apply the classical tool of tangential projections and we determine natural conditions for uniruledness, rational connectedness, and rationality. 2000 Mathematics Subject Classiﬁcation: 14N05

1. Introduction We work over an algebraically closed ﬁeld K with char(K) = 0. Let X ⊂ Pr be an integral nondegenerate n-dimensional variety. Recall that for every integer k ≥ 0 the k-secant variety S k (X) is deﬁned as the Zariski closure of the set of the points in Pr lying in the span of k + 1 independent points of X. An easy parameter count shows that the expected dimension of S k (X) is exactly min{r, n(k + 1) + k}. However, there are natural examples of projective varieties having secant varieties of strictly lower dimension: for instance, the ﬁrst secant variety of the 2-Veronese embedding of P2 in P5 has dimension 4 instead of 5. More generally, one deﬁnes the k-defect δk (X) = min{r, n(k + 1) + k} − dim S k (X) and says that X is k-defective if δk (X) ≥ 1. It seems reasonable to regard defective varieties as exceptional and try to classify them. The ﬁrst result in this direction, stated by Del Pezzo in 1887 and proved by Severi in 1901 (see [5], [10], and also [4] for a modern proof), characterizes the 2-Veronese of P2 as the unique 1-defective surface which is not a cone. Along the same lines, subsequent contributions by Palatini ([6] and [7]), Scorza ([8] and [9]) and Terracini ([11] and [12]) set up the classiﬁcation of kdefective surfaces and of 1-defective varieties in dimension up to four. This classical work has been recently reconsidered and generalized by Chiantini and Ciliberto in [1] and [2]. It turns out that one of the main tools for understanding defective varieties is provided by the technique of tangential projections. Namely, assume that X is not (k − 1)-defective and let p1 , . . . , pk be general points on X ⊂ Pr . The general k-tangential projection τX,k is the projection of X from the linear span of its tangent spaces at p1 , . . . , pk . By the classical Lemma of Terracini (see [11] and [3] for a modern version), Xk := τX,k (X) is lower dimensional than X if and only if X is k-defective. Therefore, the classiﬁcation of defective varieties reduces to the

14

Edoardo Ballico and Claudio Fontanari

classiﬁcation of varieties which drop dimension in the general tangential projection. However, we believe that the only reasonable goal in arbitrary dimension is to determine some geometrical properties of defective varieties, and the present paper is indeed a ﬁrst attempt in this direction. In order to state our main results, we recall from [1] that the contact locus of a general hyperplane section H tangent at k + 1 general points p1 , . . . , pk+1 of X is the union Σ of the irreducible components of the singular locus of H containing p1 , . . . , pk+1 . One deﬁnes νk (X) := dim Σ and says that X is k-weakly defective whenever νk (X) ≥ 1. The reason for this terminology is simply that a k-defective variety is always weakly defective (indeed, we are going to show in Proposition 1 that νk (X) ≥ δk (X)), but the converse is not true (look at cones). We point out the following: Lemma 1. Fix integers k ≥ 1, n ≥ 2, r ≥ (k + 1)(n + 1), and let X ⊂ Pr be an integral nondegenerate n-dimensional variety. If X is not k-weakly defective, then τX,k is birational. Moreover, for every k ≥ 1 and n ≥ 2 we exhibit a projective n-dimensional variety being k-weakly defective but not k-defective such that τX,k is not birational (see Example 1). Next, as an application of Lemma 1, we investigate the birational geometry of a k-defective variety of arbitrary dimension: Theorem 1. Fix integers k ≥ 1, n ≥ 2, r ≥ (k+1)(n+1)−1, and let X ⊂ Pr be an integral nondegenerate n-dimensional variety which is k-defective but not (k − 1)defective. Assume that νk (X) = δk (X) and, for k ≥ 2, that X is not (k −1)-weakly defective. Then X is uniruled. Assume moreover that the general contact locus of X is irreducible. Then X is rationally connected and for νk (X) = δk (X) = 1 it is rational. We stress that the hypotheses for uniruledness cannot be removed: in Examples 2 and 3 we collect a series of non-uniruled defective varieties of any dimension which do not satisfy exactly one of the listed assumptions. An analogous remark applies to the second part of the statement, where irreducibility turns out to be essential. For instance, if X is a cone over a curve of positive genus, then two general points on X cannot be joined by a rational curve. Indeed, we suspect that a defective variety with νk = δk and reducible general contact locus should be a cone (this is certainly the case in dimension up to four by [2], proof of Proposition 4.2, and [9], § 14). This research is part of the T.A.S.C.A. project of I.N.d.A.M., supported by P.A.T. (Trento) and M.I.U.R. (Italy).

Birational geometry of defective varieties

15

2. The proofs Let k be the minimal integer such that X is k-defective. Then τX,k−1 (X) is generically ﬁnite and by deﬁnition we have νk (X) = ν1 (Xk−1 ).

(1)

Moreover, by applying twice the equality δh (Y ) = dim Y − dim τY,h (Y ), ﬁrst with Y = X and h = k, then with Y = Xk−1 and h = 1, we deduce δk (X) = δ1 (Xk−1 ).

(2)

Proposition 1. Fix integers k ≥ 1, n ≥ 2, r ≥ (k + 1)(n + 1) − 1, and let X ⊂ Pr be an integral nondegenerate n-dimensional variety. Assume that k is the minimal integer such that X is k-defective. Then we have νk (X) ≥ δk (X). Proof. Assume ﬁrst k = 1. If δ1 (X) = 1, we are just claiming that ν1 (X) > 0, which is well-known (see for instance [1], Theorem 1.1). If instead δ1 (X) ≥ 2, we take a general hyperplane section H. By [2], Lemma 3.6, we have ν1 (H) = ν1 (X) − 1 and δ1 (H) = δ1 (X) − 1, so we conclude by induction. Assume now k > 1. By (1) and (2), we are reduced to the previous case, so the proof is over.

Proof of Lemma 1. Recall that τX,k is deﬁned as the projection of X from < Tp1 (X), . . . , Tpk (X) >, where pi is a general point on X and Tpi (X) denotes the tangent space to X at pi . Pick a general point pk+1 ∈ X and let q = τX,k (pk+1 ). By Proposition 1, X is not k-defective, so τX,k is generically ﬁ−1 (q) = {pk+1 , . . . , pk+d } and we want to show that d = 1. nite. Therefore τX,k Since τX,k (TPk+h (X)) = Tq (Xk ) for every h with 1 ≤ h ≤ d, it follows that < Tp1 (X), . . . , Tpk (X), TPk+h (X) > does not depend on h. On the other hand, by [1], Theorem 1.4, the general hyperplane which is tangent to X at p1 , . . . , pk+1 is not tangent to X at any other point, so we have d = h = 1 and the proof is over.

In order to construct nontrivial examples of projective varieties whose general ktangential projection is not birational, we are going to apply the following criterion: Lemma 2. Fix integers k ≥ 1, n ≥ 2, r ≥ (k + 1)(n + 1) − 1, and let X ⊂ Pr be an integral nondegenerate n-dimensional variety. Assume that X is k-weakly defective, but not k-defective. If X is not uniruled, then τX,k (X) is generically ﬁnite but not birational. Proof. Notice that X is a fortiori not (k − 1)-defective, so we can apply Proposition 3.6 in [1] with h = k to obtain that Xk is 0-defective. Hence Xk is a developable scroll (see for instance [1], Remark 3.1 (ii)), in particular it is uniruled

and it follows that Xk is not birational to X. Example 1. Let k ≥ 1, n ≥ 2, and r ≥ (k + 1)(n + 1) − 1. Take a (n − 1)dimensional variety C in Pr which is not uniruled, a linear space V ⊂ Pr with

16

Edoardo Ballico and Claudio Fontanari

dim V = k, and a smooth hypersurface H ⊂ Pr of degree d ≥ k + 2 such that V ⊂ H. Let W be the cone over C with vertex V and deﬁne X := H ∩ W . By [1], Example 4.3, X is k-weakly defective but not k-defective. We claim that X is not uniruled. Let π : X → C be the projection and take a general point p ∈ X. If R ⊆ X is a rational curve passing through p, then R is contained in a ﬁber of π, otherwise its projection would be a rational curve through a general point of C, in contradiction with the non-uniruledness of C. On the other hand, the general ﬁber of π is set-theoretically the intersection of H with a (k +1)-dimensional linear space, hence it is a smooth hypersurface of high degree, which is not covered by rational curves. Hence the claim is established and from Lemma 2 we deduce that τX,k (X) is generically ﬁnite but not birational. Proof of Theorem 1. Assume ﬁrst k = 1. If ν1 (X) = δ1 (X) = 1, from the socalled Terracini’s Theorem ([1], Theorem 1.1) it follows that the general contact locus Σ of X imposes only three conditions on hyperplanes containing it, in particular it is a plane curve. Moreover, if Σ had degree d > 2, then the general secant line to X would be a multisecant, which is a contradiction (the above argument is borrowed from [2], proof of Proposition 4.2). Hence X is uniruled and if Σ is irreducible then X is rationally connected. More precisely, if d = 1 then two general points on X would be joined by a straight line, a contradiction since X is nondegenerate. Therefore if we ﬁx a general p ∈ X and for general q ∈ X we assume that the corresponding contact locus Σpq is irreducible, then Σpq is a smooth conic and a natural birational map from X to its tangent space Tp (X) ∼ = Pn is deﬁned by sending q to the intersection point between the tangent lines to Σpq at p and q (see [9], § 15). If instead δ1 (X) ≥ 2, let H be a general hyperplane section. By [2], Lemma 3.6, we have ν1 (H) = ν1 (X) − 1 and δ1 (H) = δ1 (X) − 1, so uniruledness and rational connectivity follow by induction. Assume now k > 1. By (1) and (2), the previous cases apply to Xk−1 . On the other hand, from Lemma 1 we get a

birational map between X and Xk−1 , so the proof is over. Example 2. Here we show that the assumption X not (k − 1)-weakly defective is essential for every n ≥ 2. Fix r ≥ 3n + 2 and let C ⊂ Pr be a (n − 1)-dimensional variety which is not uniruled. Take a line L ⊂ Pr and a smooth hypersurface H ⊂ Pr of degree d ≥ 3 such that L ⊂ H. Let W be the cone over C with vertex V and deﬁne X := H ∩ W . By [1], Example 4.3, we have δ2 (X) = ν2 (X) = 1 and X is also 1-weakly defective. Moreover, arguing as in Example 1, it is easy to check that X is not uniruled. Finally we focus on the assumption νk (X) = δk (X). By Proposition 1, it is always satisﬁed in the case of surfaces; however, in higher dimension it is no more automatic. Example 3. As in [2], Example 2.2, we consider an n-dimensional variety X contained in an (n + 1)-dimensional cone W over a curve C in Pr with r ≥ 2n + 1. We have δ1 (X) = n − 2 and ν1 (X) = n − 1; moreover, if we assume g(C) ≥ 1 and

Birational geometry of defective varieties

17

we take X := H ∩ W , where H is a general hypersurface in Pr of high degree, then the same argument as in Example 1 shows that X is not uniruled.

References [1] L. Chiantini and C. Ciliberto: Weakly defective varieties. Trans. Amer. Math. Soc. 354 (2002), 151–178. [2] L. Chiantini and C. Ciliberto: Threefolds with degenerate secant variety: on a theorem of G. Scorza. M. Dekker Lect. Notes no. 217 (2001), 111–124. [3] M. Dale: Terracini’s lemma and the secant variety of a curve. Proc. London Math. Soc. (3) 49 (1984), 329-339. [4] M. Dale: Severi’s theorem on the Veronese-surface. J. London Math. Soc. (2) 32 (1985), 419–425. [5] P. Del Pezzo: Sulle superﬁcie del nmo ordine immerse nello spazio di n dimensioni. Rend. Circ. Mat. Palermo 1 (1887), 241-271. [6] F. Palatini: Sulle superﬁcie algebriche i cui Sh (h + 1)-seganti non riempiono lo spazio ambiente, Atti. Accad. Torino 41 (1906), 634-640. [7] F. Palatini: Sulle variet` a algebriche per le quali sono di dimensione minore dell’ordinario, senza riempire lo spazio ambiente, una o alcune delle variet`a formate da spazi seganti, Atti. Accad. Torino 44 (1909), 362-374. [8] G. Scorza: Determinazione delle variet` a a tre dimensioni di Sr (r ≥ 7) i cui S3 tangenti si tagliano a due a due. Rend. Circ. Mat. Palermo 25 (1908), 193-204. [9] G. Scorza: Sulle variet` a a quattro dimensioni di Sr (r ≥ 9) i cui S4 tangenti si tagliano a due a due. Rend. Circ. Mat. Palermo 27 (1909), 148-178. [10] F. Severi: Intorno ai punti doppi impropri di una superﬁcie generale dello spazio a quattro dimensioni, e ai suoi punti tripli apparenti. Rend. Circ. Mat. Palermo 15 (1901), 33–51. a degli Sh -h + 1 seganti ha dimensione minore [11] A. Terracini: Sulle Vk per cui la variet` dell’ordinario. Rend. Circ. Mat. Palermo 31 (1911), 392-396. [12] A. Terracini: Su due problemi, concernenti la determinazione di alcune classi di superﬁcie, considerati da G. Scorza e F. Palatini. Atti Soc. Natur. e Matem. Modena, V, 6 (1921-22), 3-16.

Edoardo Ballico Universit` a degli Studi di Trento Dipartimento di Matematica Via Sommarive 14 38050 Povo, Trento, Italy Email: [email protected]

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Edoardo Ballico and Claudio Fontanari

Claudio Fontanari Universit` a degli Studi di Trento Dipartimento di Matematica Via Sommarive 14 38050 Povo, Trento, Italy Email: [email protected]

Fano threefolds as hyperplane sections Mauro Beltrametti and Maria Lucia Fania

Abstract. Let (M, L) be a smooth 4-dimensional variety polarized by a very ample line bundle L. Let A be a smooth member of |L|. Assume that A is a Fano threefold of index one, with −KA ∼ = HA for some ample line bundle HA on A. Let H be the line bundle on M which extends HA . Up to sporadic examples and four special classes of pairs (M, L), and up to taking the ﬁrst reduction (M, L) of (M, L), we show that M is a Fano variety, and the cones of eﬀective 1-cycles N E(A) and N E(M ) coincide, where A is the image of A under the ﬁrst reduction map. We also show that there exists a new polarization H on M and our main result is proving that the usual adjunction process on (M, H) terminates, this leading to a coarse classiﬁcation of (M, L). 2000 Mathematics Subject Classiﬁcation: 14N30, 14J40, 14J45, 14C20

Introduction Let (M, L) be a smooth (n + 1)-dimensional variety polarized by a very ample (or ample and spanned) line bundle L. Let A be a smooth member of |L|. Assume that n ≥ 3 and that A is a given Fano variety of co-index n − q + 1, i.e., q is the largest positive integer such that −KA ∼ = qHA for some ample line bundle HA on A. By Lefschetz theorem, HA extends to a unique line bundle H on M. It is a natural question to classify the polarized pairs (M, L), as well to study positivity properties of H. A ﬁrst motivation are nowadays classical results due to Ramanujam and Sommese in the smooth case and to B˘adescu in the general case concerned with the situation when A is a projective space (q = n + 1) or a quadric (q = n) (see e.g., [3, §5.4] for complete references) and recent results of Lanteri, Palleschi and Sommese [9], [10], [11], where the case of (A, HA ) a Del Pezzo variety, i.e., q = n − 1, is completely worked out. In [2] the next case when (A, HA ) is a Mukai variety (q = n − 2), of dimension n ≥ 4, is considered. The results of [2] have been reﬁned and strengthened in [1] under the assumption that L is merely ample, as a consequence of a general comparing cones result which holds true in the range q ≥ n/2. In the present paper we deal with the remaining case q = n − 2 = 1, i.e., A is a Fano 3-fold of index one, −KA ∼ = HA . Summing up, we consider the following

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Problem 0.1. Let (M, L) be a smooth 4-fold polarized by a very ample line bundle L. Let A be a smooth member of |L|. Assume that A is a Fano 3-fold of index one, with −KA ∼ = HA for some ample line bundle HA on A. Then classify (M, L). Note that if (M, L) is a Mukai 4-fold, i.e., KM ∼ = −2L, and A is a smooth element of L, then (A, HA = L|A ) satisﬁes the condition as in the problem. We refer to such an (M, L) as the standard example. For dimension dim A ≥ 4 the adjunction theoretic approach followed in [2] and [1] gives rather strong numerical conditions, leading to a reﬁned classiﬁcation. For dim A = 3 the number of possible cases increases so much that this makes reasonable to look for a coarse classiﬁcation (see (4.2)). Up to a few sporadic cases and the standard example mentioned above, this classiﬁcation is mainly built up by four classes of special varieties: Del Pezzo 4-folds (completely classiﬁed by Fujita [5]), quadric ﬁbrations over P1 , linear P2 -bundles over smooth Del Pezzo surfaces, and Del Pezzo ﬁbrations over P1 . From this perspective, our main result is Theorem 4.1, which shows that the same adjunction process followed in [2] terminates in case dim A = 3 as well. More precisely, we approach Problem 0.1 as follows. We ﬁrst deal in §1 with the case when Pic(A)(∼ = Pic(M)) ∼ = Z, this allowing us to assume Pic(M) ∼ = Z. Up to the special case when (M, L) is a linear P3 -bundle, described in §2, we get that the ﬁrst adjoint bundle KM + 3L is nef. Then either we are as in one of the “building block” classes of special fourfolds mentioned above, or KM + 3L is big and the linear system associated to it deﬁnes the ﬁrst reduction, (M, L), ϕ : M → M , of (M, L). A key result here is Claim 3.4, which shows that our working assumptions on (M, L), HA and A ∈ |L| being Fano, are preserved on the ﬁrst reduction. This leads to the main result of §3, Proposition 3.5. This proposition shows that, up to special classes of fourfolds as above, the unique line bundle H on M which extends the ample line bundle HA ∼ = −KA is ample. In this case it turns out that M is a Fano 4-fold and the cones of eﬀective 1-cycles N E(M ) and N E(A) coincide. Then in particular, by Kleiman ampleness criterion, the ample cones of A and M coincide. In §4 we deal with the polarization H, developing a new adjunction process on (M, H). By combining this with the previous properties of the polarized pair (M, L), we ﬁnally show that the adjunction process on (M, H) terminates and we end up with the classiﬁcation result, Theorem 4.3. We work on the complex ﬁeld C and use the standard terminology in algebraic geometry. In particular, we use the additive notation for line bundles, and by KX we denote the canonical bundle of a projective manifold X. By Qn we denote a smooth hyperquadric in Pn+1 . For a complete classiﬁcation of Del Pezzo 4-folds and Mukai 4-folds occurring through the paper we refer to Fujita’s book [5, (8.11)] and to [13], [14], [15], respectively. For all background material we need we refer to [3]. We would like to thank A.J. Sommese for many helpful conversations; in particular for suggesting us Example 2.3 and for sharing his idea on how to prove Claim 3.4.

Fano threefolds as hyperplane sections

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1. The case of Picard number one Let (M, L), A be as in Problem 0.1 stated in the introduction. In this section we consider the case when Pic(M) ∼ = Z. Del Pezzo 4-folds of Picard number one in the statement below are few well known cases listed in [5, (8.11)]. Proposition 1.1. Let (M, L) be a smooth 4-fold polarized by a very ample line bundle L. Let A be a smooth member of |L|. Assume that A is a Fano 3-fold of index one, with KA ∼ = −HA for some ample line bundle HA on A. Further assume Pic(A) ∼ = Z. Then (M, L) is as follows: 1. (M, L) is a Mukai variety (the standard example); or 2. (M, L) = (M, 2H) and (M, H) is a Del Pezzo 4-fold; or 3. (M, L) ∼ = (Q4 , OQ4 (3)); or 4. (M, L) ∼ = (P4 , OP4 (4)). Proof. By Barth-Lefschetz theorem one has Pic(M) ∼ = Pic(A) ∼ = Z, and hence HA extends to an ample line bundle H on M. Since A is of index one, HA generates Pic(A), i.e., Pic(A) ∼ = Z[H]; furthermore L|A ∼ = tHA and = Z[HA ]. Then Pic(M) ∼ ∼ L = tH for some positive integer t. From −HA ∼ = KA ∼ = (KM + L)|A we get KM|A ∼ = −HA − L|A ∼ = −(1 + t)H|A , and hence −KM ∼ = (1 + t)H on M. Since 1 + t ≤ dim M + 1 = 5, we ﬁnd t ≤ 4. If t = 1, (M, L) = (M, H) is a Mukai 4-fold; if t = 2, (M, H) is a Del Pezzo variety

and L ∼ = (P4 , OP4 (4)). = 2H; if t = 3, (M, L) ∼ = (Q4 , OQ4 (3)); if t = 4, (M, L) ∼ 1.2. (Spannedness of HA ) Let (M, L), A, −KA ∼ = HA be as in Problem 0.1. Recalling that A has index 1, we know from [7, Prop. 5] and [5, (8.11)] that HA is spanned unless either (a) A is the blowing up σ : A → V with center a smooth elliptic curve, where V is a hypersurface of degree 6 in the weighted projective space P(3, 2, 1, 1, 1); or (b) A = S × P1 with S a Del Pezzo surface of degree KS2 = 1. In case (b), we know that the P1 -bundle projection A → S extends to a P2 -bundle projection M → S (see [3, §5.5]). Thus, up to the sporadic case (a) above, the Picard number one case as in Proposition 1.1, and the case when (M, L) is a P2 -bundle over a Del Pezzo surface of degree 1, we may proceed further under the working assumption that HA is spanned on A.

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2. The case when (M, L) is a linear P3 -bundle Let (M, L), A be as in Problem 0.1. According to the usual adjunction process (see [3, Chapter 7]), we have next to consider the case of a linear bundle over a curve. Proposition 2.1. Let (M, L) be a (P3 , OP3 (1))-bundle over a smooth curve C. Then C ∼ = P1 × P2 or A is the blowing up of P3 along P1 . = P1 and either A ∼ Proof. Since A is a Fano variety, and A maps onto C under the bundle projection p : M → C, we have C ∼ = P1 . Then we have M = P(E), where E := p∗ L = 3 ⊕i=0 OP1 (bi ) is an ample vector bundle over P1 , bi positive integers. Set β := 3 i=0 bi , so that det(E) = OP1 (β). Denote with LA and pA the restrictions of L and p to A, respectively. Then A = P(Q), where Q = pA ∗ LA = ⊕2i=0 OP1 (ai ) for positive integers a0 , a1 , a2 . We have on P1 the exact sequence 0 → OP1 → ⊕3i=0 OP1 (bi ) = p∗ L → ⊕2i=0 OP1 (ai ) = pA ∗ LA → 0, 3 2 giving β := i=0 bi = i=0 ai . The canonical bundle formula gives on A, 2

ai − 2 F − 3LA , −HA ∼ = KA ∼ =

(1)

i=0

where F = P2 is a ﬁber of pA : A → P1 . We can arrange the ai ’s so that 0 < a0 ≤ a1 ≤ a2 and A ∼ = P(Q(−a0 )) = P(⊕2i=0 OP1 (ai − a0 )) with ai − a0 ≥ 0 for each index i = 0, 1, 2. Write (1) as 2

∼ −KA = − (ai − a0 ) − 2 F + 3(LA − a0 F ). i=0

2 Since −KA is ample it thus follows that − i=0 (ai − a0 ) + 2 > 0 (see e.g. [3, 2 (3.2.4)]), and hence 0 ≤ i=0 (ai − a0 ) < 2. Therefore either a0 = a1 = a2 , or a0 = a1 , a2 = a0 + 1. In the ﬁrst case A ∼ = P(⊕3 OP1 ) ∼ = P1 × P2 , and in the second 3 1

case A is the blowing up of P along P , as we want. Example 2.2. With the assumptions and notations as in Proposition 2.1, let A∼ = P(⊕3 OP1 (a)) for some positive integer a. We give here a class of examples of pairs (M, L) for each even a. Consider the exact sequence 0 → OP1 → E → ⊕3 OP1 (a) → 0,

(2)

where E is a rank 4 vector bundle on P1 . We would like the vector bundle E to be ample, or equivalently very ample. Let us show that E is ample if and only if (2) is a non-splitting extension. If E is ample, then clearly (2) is not splitting. To show the converse, note that E can fail to be ample only if E has a trivial summand, E = OP1 ⊕ V. Let

Fano threefolds as hyperplane sections

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V = ⊕3i=1 OP1 (di ). Note that di ≤ a. Indeed, otherwise, up to renaming, let d1 be the largest among the di ’s. Twisting by OP1 (−d1 ) we ﬁnd 0 → OP1 (−d1 ) → OP1 (−d1 ) ⊕ OP1 ⊕ (⊕3j=2 OP1 (dj − d1 )) → ⊕3 OP1 (a − d1 ) → 0. Note that the middle term has a section, but the ﬁrst 3 and the last do not. This contradiction shows that d1 ≤ a. Therefore from i=1 di = 3a and di ≤ a we conclude that all di ’s are equal to a. Thus twisting by OP1 (−a) we get the exact sequence 0 → OP1 (−a) → OP1 (−a) ⊕ (⊕3 OP1 ) → ⊕3 OP1 → 0. Since the sequence splits, we contradict our assumption. Note also that a > 1 if we assume that (2) is a non-splitting extension. In fact in this case E is ample, and hence deg E ≥ rank E. While, if a = 1, we would have deg E = 3 = rank E − 1. Note that non-splitting extensions as in (2) exist. Given any integer b > 0, take a general section s of OP1 (b) ⊕ OP1 (b). By Bertini, s is nowhere zero. This gives a non-splitting sequence (the usual tangent bundle non-splitting sequence for b = 1) 0 → OP1 → OP1 (b) ⊕ OP1 (b) → OP1 (2b) → 0. Thus, direct summing two copies of OP1 (2b) to the middle and end terms, we get an exact sequence 0 → OP1 → OP1 (b) ⊕ OP1 (b) ⊕ (⊕2 OP1 (2b)) → ⊕3 OP1 (2b) → 0, leading to the requested class of examples. Example 2.3. Let (M, L) = (P3 × P1 , OP3 ×P1 (1, 1)), i.e., M = P(⊕4 OP1 (1)) and let L be the tautological line bundle. Take a general section of L. Since the section gives an exact sequence 0 → OP1 → ⊕4 OP1 (1) → OP1 (a) ⊕ OP1 (b) ⊕ OP1 (c) → 0 and the quotient is ample we have (a, b, c) = (1, 1, 2). Then A = P(OP1 (1) ⊕ OP1 (1) ⊕ OP1 (2)) is a divisor on M deﬁned by a section of H 0 (M, L). Moreover A = P(OP1 (1) ⊕ OP1 (1) ⊕ OP1 (2)) is the blowing up of P3 along P1 . A more general class of examples is as follows. Take any nowhere vanishing section of OP1 (a) ⊕ OP1 (b) where a > 0, b > 0. Then we have an exact sequence 0 → OP1 → OP1 (a) ⊕ OP1 (b) → OP1 (a + b) → 0, which yields the exact sequence 0 → OP1 → OP1 (a) ⊕ OP1 (b) ⊕ (⊕2 OP1 (a + b − 1)) → → OP1 (a + b) ⊕ (⊕2 OP1 (a + b − 1)) → 0. Then M = P(OP1 (a) ⊕ OP1 (b) ⊕ (⊕2 OP1 (a + b − 1))) and A = P(OP1 (a + b) ⊕ OP1 (a + b − 1) ⊕ OP1 (a + b − 1)). Note that for a = b = 1 we get the previous example.

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3. The case when KM + 3L is nef Let (M, L), A be as in the Problem 0.1. After the discussion in the previous sections, we can assume that Pic(M) ∼ = Z and that KM +(dim M−1)L = KM +3L is nef and hence κ(KM +3L) ≥ 0. Let ϕ : M → Y be the morphism with connected ﬁbers and normal image associated to the complete linear system |m(KM + 3L)| for m 0. Recall that if KM + 3L is nef and not ample the morphism ϕ is called the nefvalue morphism of (M, L). The following gives a maximal list of all possible cases when KM + 3L is nef and not big. Proposition 3.1. Let (M, L) be a smooth 4-fold polarized by a very ample line bundle L. Let A be a smooth member of |L|. Assume that A is a Fano 3-fold of index one, with KA ∼ = −HA for some ample line bundle HA on A. Further assume Pic(M) ∼ = Z and that KM + 3L is nef and not big. Let ϕ : M → Y be the nefvalue morphism of (M, L). Then either: 3 is divisible by 8; or 1. (M, L), ϕ : M → P1 , is a quadric ﬁbration, and KA

2. (M, L), ϕ : M → Y , is a P2 -bundle over either P2 or P1 × P1 . Proof. If dim ϕ(M) = 0 then (M, L) is a Del Pezzo 4-fold. Moreover, since Pic(M) ∼ Z, we see from [5, (8.11)] that (M, L) ∼ = = (P2 × P2 , OP2 ×P2 (1, 1)). This contradicts the assumption that A has index 1. If dim ϕ(M) = 1, then (M, L) is a quadric ﬁbration over a smooth curve Y and, restricting to A, (A, LA ) is a quadric ﬁbration over Y , so that Y ∼ = P1 since A is a Fano 3-fold. In this case E := (ϕA )∗ (LA ) is a locally free sheaf of rank 4 and LA embeds A into P(E) as a divisor of relative degree 2 (see [3, §4.2] and [11, §1]). We have A ∈ |2ξ − bF |, where ξ is the tautological line bundle of P(E) and F is a ﬁber of the bundle projection. Recall that LA = ξ|A and compute L3A = ξ 3 · (2ξ − bF ) = 2ξ 4 − bξ 3 · F = 2e − b, where e := deg det(E), b ∈ Z. Now, KP(E) ∼ = −4ξ + (e − 2)F and hence KA ∼ = (KP(E) + A)|A ∼ = −2LA + (e − b − 2)F. Thus 3 KA = −8L3A + 24(e − b − 2) = 8(e − 2b − 6)

is divisible by 8. If dim ϕ(M) = 2, then (M, L) is a scroll, and in fact a P2 -bundle (see [3, (4.1.3)]) over a smooth surface Y . From [19, (1.6)] we know that Y is a Del Pezzo surface. The fact that Y is either P2 or P1 × P1 follows by looking over the lists [18] as discussed in Remark 3.2 below.

Remark 3.2. Note that in the quadric ﬁbration case ϕ : M → P1 of (3.1) the 3 is divisible by 8 cut down the assumption that A has index 1 and the fact that KA

Fano threefolds as hyperplane sections

25

87 possibilities for A from the lists in [12] to 19 possible cases. Note also that the case A = P1 × F1 in [12, Table 3, no. 28] does not occur. Indeed in this case the projection map A → F1 extends to M giving a P2 -bundle structure p : M → F1 (see [3, §5.5]). Let x ∈ F1 and let p−1 (x) = P2 . Note that ϕ(p−1 (x)) = ϕ(P2 ) is a point t ∈ P1 . Therefore P2 ⊂ ϕ−1 (t) = Q3 , contradicting the fact that there are no 2-planes in a general ﬁber Q3 of ϕ. Finally, in the P2 -bundle case 2) of Proposition 3.1, restricting to A, we see that (A, LA ) is a P1 -bundle over Y , and hence (A, LA ) is described as in the lists of [18]. We will see that Y can be either P2 or P1 × P1 . Note that all cases where the rank 2 vector bundle E (as in the lists of [18]) is a direct sum do not occur. Indeed, we know that A = P(E) ∼ = P(Q), where Q = B ⊗ E for some line bundle B on Y . We want a deﬁning exact sequence 0 → OY → V → Q → 0, for some ample vector bundle V on Y (putting M = P(V), recall that OM (A) is the tautological bundle of P(V), so that the ampleness of A in M is equivalent to the ampleness of V). If E is a direct sum, then Q is also. Since Q is ample both the direct summands of Q are ample and therefore h1 (Q∗ ) = 0. This implies that the exact sequence above splits, contradicting the ampleness of V. Thus only cases no. 4, 5, 6, 7, 12, 14 of [18] could possibly occur. Case 4) of [18] does not occur since in that case A has not index 1 (see also [12, Table 2, no. 32]). The following argument rules out case 14). In that case Y is the Hirzebruch surface F1 . Let β : F1 → P2 be the blown-down map and let π : F1 → P1 be the bundle projection. Since Pic(F1 ) is generated by f = π ∗ OP1 (1) and the section E of π, we can write Q = β ∗ TP2 (q) − bE for some integers b, q (in the present case E = TP2 (−2)). Since the restriction QE is ample we conclude that b > 0. Note that f goes to a line ⊂ P2 isomorphically under β. Since E · f = 1, we have Of (−bE) ∼ = OP1 (−b). Also, β ∗ TP2 (q)|f ∼ = TP2 (q)| . Thus the exact sequence 0 → T ∼ = OP1 (2) → TP2 | → N/P2 → 0 gives TP2 | ∼ = OP1 (2) ⊕ OP1 (1), and hence TP2 (q)| ∼ = OP1 (q + 2) ⊕ OP1 (q + 1). Therefore, restricting Q to f , we get Qf ∼ = OP1 (q − b + 1) ⊕ OP1 (q − b + 2). = (OP1 (q + 2) ⊕ OP1 (q + 1)) ⊗ OP1 (−b) ∼ Thus the ampleness of Qf yields q ≥ b. Pulling back the exact sequence 0 → OP2 (q) → ⊕3 OP2 (q + 1) → TP2 (q) → 0 and tensoring with −bE, we have the exact sequence 0 → β ∗ OP2 (q) − bE → ⊕3 β ∗ OP2 (q + 1) − bE → Q → 0.

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Since q ≥ b we conclude that the middle term is nef and big, and hence h1 (⊕3 β ∗ OP2 (−q − 1) + bE) = 0. Dualizing, we ﬁnally end up with the exact sequence 0 → Q∗ → ⊕3 β ∗ OP2 (−q − 1) + bE → β ∗ OP2 (−q) + bE → 0. Since q ≥ b > 0 and h0 (β ∗ OP2 (−q) + bE) = 0 when q > 0 it follows that h1 (Q∗ ) = 0, so we are done by the usual argument. Hence out of the 21 possibilities for A in [18] the only possible cases are no. 5, 6, 7, 12. We give in Example 3.3 below examples of pairs (M, L) with (A, LA ) as in cases no. 6, 12. Example 3.3. Let (M, L), ϕ : M → Y , be as in case 2) of Proposition 3.1. Then (A, LA ) is a P1 -bundle over Y . Let us give some explicit examples from the list in [18]. Consider the exact sequence (case 6) of that list) 0 → ⊕2 OP2 (−1) → ⊕4 OP2 → E(1) → 0,

(3)

where E is a rank 2 stable vector bundle on P2 with c1 = 0, c2 = 2. Let A = P(Q) with Q := E(q) for some integer q (note that since E(1) is spanned but not ample we must have q ≥ 2). We want a deﬁning exact sequence 0 → OP2 → V → Q → 0, for some ample vector bundle V on P2 . By ampleness of V, we want the sequence to be not splitting, and hence h1 (Q∗ ) = 0. By dualizing and tensoring (3) with OP2 (1 − q), we get the exact sequence 0 → Q∗ = E ∗ (−q) → ⊕4 OP2 (1 − q) → ⊕2 OP2 (2 − q) → 0. From the associated cohomology sequence, we see that h1 (Q∗ ) = 0 for each q = 2 (and h1 (Q∗ ) = 2 for q = 2). Thus, let Q = E(2). From 0 → ⊕2 OP2 → ⊕4 OP2 (1) → Q → 0 we get a deﬁning exact sequence 0 → OP2 → V := ⊕4 OP2 (1)/OP2 → Q → 0. Note that the quotient V is an ample rank 3 vector bundle. Therefore the sequence above leads to the example (M, L) = (P(V), O(1)). Consider the exact sequence (case 12) of the list in [18]) 0 → OP1 ×P1 (−1, −1) → ⊕3 OP1 ×P1 → E(1, 1) → 0, where E is a stable rank 2 vector bundle on P1 × P1 with c1 = (−1, −1), c2 = 2, and E(1, 1) is spanned. Tensoring with OP1 ×P1 (1, 1) we get the exact sequence 0 → OP1 ×P1 → ⊕3 OP1 ×P1 (1, 1) → E(2, 2) → 0. Note that Q := E(2, 2) is very ample and the sequence above leads to the example (M, L) = (P(⊕3 OP1 ×P1 (1, 1)), O(1)) = (P1 × P1 × P2 , O(1, 1, 1)). 2

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Thus from now on, we may assume that KM + 3L is nef and big, and therefore ϕ : M → M is a birational morphism, and in fact a blowing up at a ﬁnite set of points. Let L := ϕ∗ (L)∗∗ be the double dual. The new pair (M, L), together with the morphism ϕ, is called the ﬁrst reduction of (M, L). We refer to [3, §7.3] for more on the ﬁrst reduction. Recall that there exists a unique line bundle H on M which extends HA = −KA . Let H ∼ = ϕ∗ (H)∗∗ and let A := ϕ(A). Further denote by HA the restriction of H to A. Note that HA = (ϕA )∗ (HA )∗∗ and that H is the unique line bundle on M which extends HA . Claim 3.4. A is a Fano 3-fold and −KA ∼ = HA . Furthermore, either (M, L) is a Mukai 4-fold (the standard example) or HA is spanned. Proof. The assumption −KA ∼ = HA yields (KM + L + H)|A ∼ = OA , and hence ∗∗ ∼ O by Lefschetz. Therefore ϕ (K + L + H) KM + L + H ∼ = OM . Noting = M ∗ M ∗∗ ∼ and ϕ (K + L + H) + L + H we thus conclude that K K that ϕ∗ KM ∼ = M = M ∗ M . Since H is ample and ϕ is an isomorphism outside of a ﬁnite set −H KA ∼ = A A of points, the line bundle HA is ample on A. In particular, A is a Fano 3-fold (of index one with respect to HA ). By our present assumptions (see Section 1.2), HA is spanned on A and hence HA is spanned outside of a ﬁnite set of points, over which there are the positive Pulling back to A, we have dimensional ﬁbers of the restriction ϕA of ϕ to A. ϕ∗A (HA ) ∼ = HA + J, where J = 2 i Ei , Ei ∼ = P2 the exceptional divisors. We may assume that ϕA is the blowing up at a single point x. Let E := ϕ−1 (x). Note that, since HA is ample and spanned, and OE (E) ∼ = OP2 (−1), the restriction (HA + E)P2 is spanned as well. Note also that H 1 (HA ) = H 1 (KA − 2KA ) = 0 since −KA is ample. Thus from the exact sequence 0 → HA → HA + E → (HA + E)P2 → 0 we infer that H (HA + E) = H 1 (ϕ∗A (HA ) − E) = 0. Consider now the exact sequence 1

0 → HA + E → HA + J → (HA + J)P2 → 0. We have HA|P2 ∼ = −KA|P2 ∼ = −(KP2 −E)P2 ∼ = OP2 (2), and hence (HA +J)P2 ∼ = OP2 is spanned. Therefore, recalling that by the above H 1 (HA + E) = 0, sections lift to span HA + J ∼ = ϕ∗A (HA ) in a neighbourhood of P2 . Since H 0 (A, ϕ∗A (HA )) ∼ = 0 H (A, HA ), it follows that HA is spanned at the point x = ϕ(E). Since HA is spanned and ϕA is an isomorphism outside of E, we thus conclude that HA is spanned by its global sections. 2 By the above, from this point on, we can work on the ﬁrst reduction (M, L) of (M, L). Since dim M = 4, the adjuntion process [3, (7.3.4)] tells us that KM + 2L is nef unless • (M, L) ∼ = (P4 , OP4 (2)). Let φ : M → X be the morphism with connected ﬁbers and normal image associated to |m(KM + 2L)|, m 0.

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• If κ(KM + 2L) = 0, then (M, L) is a Mukai variety, the standard example. • If κ(KM + 2L) = 1, then (M, L), φ : M → X, is a Del Pezzo ﬁbration over a smooth curve X. Restricting to A, we get a Del Pezzo ﬁbration φA : A → X. Therefore X ∼ = P1 since A is Fano. Thus from now on, we have (working assumption) KM + 2L is nef and κ(KM + 2L) ≥ 2,

(4)

which leads us to show the following. Proposition 3.5. Let (M, L) be a smooth 4-fold polarized by a very ample line bundle L. Let A be a smooth member of |L|. Assume that the ﬁrst reduction, (M, L), ϕ : M → M , of (M, L) exists. Assume that A is a Fano 3-fold of index one, −KA ∼ = HA , with HA ample and spanned (see Section 1.2). Let A := ϕ(A). Further assume that KM + 2L is nef and κ(KM + 2L) ≥ 2. Then the following hold. 1. A is a Fano 3-fold, −KA ∼ = HA , with HA ample and spanned; 2. the unique line bundle, H, on M which extends HA is ample; 3. M is Fano and N E(M ) = N E(A). Proof. Assertion 1) is Claim 3.4. We follow now the argument as in [2, (2.1)]. By Lefschetz theorem, the fact that −KA ∼ = HA gives −H ∼ = KM + L and hence L − + 2L. Since from the basepoint free theorem we know that m(KM + 2L) H∼ K = M is spanned for m 0, it thus follows that m(L − H) is spanned for m 0 and, by the assumptions made, κ(L − H) ≥ 2. Therefore the vanishing H 1 (M, H − L) = 0

(5)

holds true (see [17, (7.65)]). Now consider the exact sequence 0 → H − L → H → HA → 0. Since HA is spanned on A, by (5) we see that sections of Γ(A, HA ) lift to span H in a neighborhood of A; but since A is ample we conclude that H is spanned oﬀ a ﬁnite set of points. Therefore H is nef. Note that since L is ample and H is nef, it follows that −KM ∼ = H + L is ample, so that M is a Fano variety. The ampleness of H follows from the comparing cone result [1, (3.2)] (see also [8, Appendix]) as in the proof of [1, (4.1)]. Since H is ample, the same argument as in the proof of [2, (2.3)] shows the equality N E(M ) = N E(A).

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4. The adjunction process with a new polarization on M Let (M, L), A, H be as in (3.5). We know that H is in fact a polarization, this allowing us to develop a new adjunction process on (M, H). Consider the morphism σ associated to the linear system |m(KM + 4H)|, m 0. Then either σ is the trivial morphism, or σ has 1-dimensional image, and furthermore σ(M ) ∼ = (Q4 , OQ4 (1)) and hence, = P1 . In the ﬁrst case (M, H) ∼ recalling that KM + L ∼ = −H, • (M, L) ∼ = (Q4 , OQ4 (3)). If κ(KM + 4H) = 1, then (M, H), σ : M → P1 , is a scroll, and in fact a P3 -bundle (see e.g., [3, (3.2.1), (11.1.1)]). Then M = P(E), where E = σ∗ H = ⊕3i=0 OP1 (ai ) is an ample locally free sheaf of rank 4. 3 We claim that this case does not occur. Indeed, let e := i=0 ai and let F = P3 be a ﬁber of σ. Writing L = aH + bF for some integers a, b, the canonical bundle formula KM ∼ = (e − 2)F − 4H yields OM ∼ = (e + b − 2)F + (a − 3)H. = KM + L + H ∼ Since H is ample, H cannot be a (rational) multiple of ﬁbers and therefore we ﬁnd a = 3, b − 2 + e = 0, that is L ∼ = 3H + (2 − e)F . By ampleness of L we ﬁnd 2 − e > 0, contradicting e ≥ 4. Applying the adjunction process on (M, H) we thus conclude that KM + 3H is nef. Then |m(KM + 3H)| for m 0 deﬁnes a morphism ψ : M → Z with connected ﬁbers onto a normal variety Z. By general adjunction theory results, the following cases are possible. • κ(KM + 3H) = 0, (M, H) is a Del Pezzo 4-fold, L ∼ = 2H. • κ(KM + 3H) = 1, (M, H), ψ : M → P1 , is a quadric ﬁbration over P1 , and LF ∼ = OF (2) for a ﬁber F of ψ. In this case E := ψ∗ H is a locally free sheaf of rank 5 and H embeds M into P(E) as a divisor of relative degree 2 (see [3, §4.2] and [11, §1]). We have M ∈ |2ξ − bF|, where ξ is the tautological line bundle of P(E) and F = P4 is a ﬁber of the bundle projection. Recall that HM = ξ|M and compute H 4 = ξ 4 · (2ξ − bF) = 2ξ 5 − bξ 4 · F = 2e − b, where e := deg det(E), b ∈ Z. Now, KP(E) ∼ = −5ξ + (e − 2)F and hence KM ∼ = (KP(E) + M )|M ∼ = −3H + (e − b − 2)F. 4 4 = 27(2e + b + 8), so that KM is divisible by 27. Thus KM

Note that M is a Fano variety of index r ≤ 3. We claim that r = 1. If r = 3 then M is a Del Pezzo 4-fold. Since Pic(M ) ∼ = Z we see from [5, (8.11)] that M = P2 × P2 . Then for any ﬁber P2 of the product projection

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M → P2 we have ψ(P2 ) = t, t a point in P1 . Therefore P2 ⊂ ψ −1 (t) = Q3 , contradicting the fact that there are no 2-planes in a general ﬁber Q3 of ψ. If r = 2, by looking at Wisniewski’s list [20] of Mukai 4-folds with b2 ≥ 2 we 4 divisible by 27. see that none of the varieties on the list have KM We thus conclude that M is a Fano fourfold of index 1. • κ(KM + 3H) = 2, (M, H), ψ : M → Z, is a scroll over a normal surface Z, and LF ∼ = OF (2) for a ﬁber F of ψ. Note that in this case ψ is an extremal ray contraction and there are no eﬀective divisors which are components of ﬁbers of ψ. Thus all ﬁbers of ψ are 2-dimensional. It follows that ψ : M → Z is a P2 -bundle and Z is smooth. It also follows that Z is a Del Pezzo surface (see [3, (14.1.1), (3.2.1)], and [19, (1.5)]). Recall that M is a Fano variety of index r ≤ 3. ∼ P2 × P2 . Since KM + L ∼ If r = 3, we see as above that M = = −H, it thus follows that ψ is the projection on the ﬁrst factor with L ∼ = OP2 ×P2 (1, 2), H∼ = OP2 ×P2 (2, 1). If r = 2 we fall in Wi´sniewski’s list [20, Table 0.3]. Since b2 (M ) = b2 (Z) + 1, we see that for the cases 1) to 15) of that table (i.e., when b2 (M ) = 2) we must have b2 (Z) = 1 and hence Z ∼ = P2 . In cases 16), 17) of that table we have b2 (M ) = 3, while b2 (M ) = 4 in the last case 18). Let us show that cases 3), 6), 9), 11), 13), 15) and 18) of [20, Table 0.3] are not possible. In cases 3), 6), 9), M = P1 × Vd where Vd is a Del Pezzo threefold of degree d = 3, 4, 5 respectively, so that in each case KVd ∼ = OVd (−2). Let p : P1 × Vd → P1 be the projection on the ﬁrst factor and let q : P1 × Vd → Vd be the projection on the second factor. Note that KM ∼ × OVd (−2) = = OP1 (−2) < × OVd (b) and let H = = (:= p∗ OP1 (−2) ⊗ q ∗ OVd (−2)). Let L = OP1 (a) < × OVd (β). Since the restrictions LP1 and LVd are ample it must be = OP1 (α) < a > 0 and b > 0. Similarly α > 0 and β > 0 by ampleness of HP1 and HVd . × OV (b + β − 2) ∼ = Since KM + L + H ∼ = OM we have OP1 (a + α − 2) < = OM , and hence a + α − 2 = 0 and b + β − 2 = 0. It thus follows that a = α = 1 as well as b = β = 1. Hence KM + 3H is ample and therefore the map associated to a power of it cannot have 2-dimensional image. In both cases 11) and 15) the 4-fold M has two projective bundle structures and therefore [16, Theorem A] applies to give the contradiction M ∼ = P2 ×P3 . In case 13), let p : M → Q3 be the bundle projection. Then for any t ∈ Z, ψ −1 (t) = P2 maps ﬁnite-to-one on Q3 and hence p(ψ −1 (t)) = P2 (see e.g., [3, (3.1.7)]). But Q3 does not contain a P2 . In case 18) then M ∼ = P1 × P1 × P1 × P1 . By using again the relation L + H ∼ = −KM we see that L ∼ =H∼ = O(1, 1, 1, 1). Therefore KM + 3H ∼ = O(1, 1, 1, 1) is very ample, contradicting the fact that ψ has a 2-dimensional image.

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• KM + 3H is nef and big. In this case ψ : M → Z is the ﬁrst reduction map of the polarized pair (M, H). Let us ﬁnally show that the adjuntion process terminates, i.e., that under the working assumption (4), the general case above when KM + 3H is nef and big does not occur, but for a single special case. Theorem 4.1. (Termination of the adjunction process) Let (M, L) be a smooth 4-fold polarized by a very ample line bundle L. Let A be a smooth member of |L|. Assume that the ﬁrst reduction, (M, L), ϕ : M → M , of (M, L) exists. Assume that A is a Fano 3-fold of index one, −KA ∼ = HA , with HA ample and spanned (see Section 1.2). Let A := ϕ(A) and let H be the unique ample line bundle on M which extends HA ∼ = −KA as in Proposition 3.5, case 2). Further assume that KM + 2L is nef, κ(KM + 2L) ≥ 2, and KM + 3H is nef and big. Let ψ : M → Z be the ﬁrst reduction map of the polarized pair (M, H), and let H := ψ∗ (H)∗∗ . Then (Z, H ) ∼ = (P4 , OP4 (2)). Proof. By the adjunction process [3, (7.3.4)], we know that KZ + 3H is ample and that KZ + 2H is nef unless (Z, H ) ∼ = (P4 , OP4 (2)). Therefore we can assume that KZ + 2H is nef. Thus, since κ(KZ + 2H ) = κ(KM + 2H), we conclude that m(KM + 2H) is eﬀective for some positive integer m (see also [3, (7.3.4), (7.6.2), (7.6.9)]). Then writing the usual relation KM +L ∼ = −H as KM + 2H ∼ = H − L, we conclude that H − L is Q-eﬀective. On the other hand, writing L − H ∼ = KM + 2L, we see that L − H is nef. It thus follows that L − H is numerically trivial, and actually that t(L − H) is trivial for a positive integer t. Therefore

κ(L − H) = κ(KM + 2L) = 0, a contradiction.

4.2. Coarse classiﬁcation of 4-folds whose hyperplane section is a Fano 3-fold of index 1. Summing up, the above arguments lead to the following maximal list. Theorem 4.3. Let (M, L) be a smooth 4-fold polarized by a very ample line bundle L. Let A be a smooth member of |L|. Assume that A is a Fano 3-fold of index one, with KA ∼ = −HA for some ample line bundle HA on A. Further assume that HA is spanned (see Section 1.2). Then the following cases are possible. 1. (M, L) is a Mukai variety (the standard example); or 2. (M, L) = (M, 2H) and (M, H) is a Del Pezzo 4-fold of Picard number one; or 3. (M, L) ∼ = (Q4 , OQ4 (3)); or

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4. (M, L) ∼ = (P4 , OP4 (4)); or 5. (M, L) is a (P3 , OP3 (1))-bundle over a smooth curve C and either A ∼ = P1 × P2 or A is the blowing up of P3 along P1 ; or 6. (M, L), ϕ : M → P1 , is a quadric ﬁbration (see also (3.2)); or 7. (M, L), ϕ : M → Y , is P2 -bundle and either Y = P2 or Y = P1 × P1 (see also (3.2)); or 8. There exists the ﬁrst reduction (M, L), ϕ : M → M , of (M, L) and either: (a) (M, L) ∼ = (P4 , OP4 (2)); or (b) (M, L) is a Mukai variety (the standard example); or (c) (M, L), φ : M → P1 , is a Del Pezzo ﬁbration; or (d) M is a Fano 4-fold, and there exists a new polarization H on M (described as in Proposition 3.5) such that either: i. (M, H) is a Del Pezzo 4-fold, L ∼ = 2H; or ii. (M, H), ψ : M → P1 , is a quadric ﬁbration over P1 and LF ∼ = OF (2) for a ﬁber F of ψ, and M is a Fano 4-fold of index 1; or iii. (M, H), ψ : M → Z, is a P2 -bundle over a smooth Del Pezzo surface Z, and LF ∼ = OF (2) for a ﬁber F of ψ. In this case either: A. M ∼ = OP2 ×P2 (1, 2), H ∼ = OP2 ×P2 (2, 1); or = P2 × P2 , L ∼ 2 ∼ B. Z = P , b2 (M ) = 2 and M is a Fano 4-fold of index 2 described as in [20, Table 0.3] (cases no. 3, 6, 9, 11, 13, 15 of that table are not possible); or C. b2 (M ) = 3, M ∼ = P1 × V , where V is the blowing up of P3 at a point; or D. b2 (M ) = 4, M ∼ = P1 × W , where W is a divisor of bidegree 2 2 (1, 1) on P × P ; or E. M is a Fano 4-fold of index 1; or iv. There exists the ﬁrst reduction (Z, H ), ψ : M → Z, of (M, H) and (Z, H ) ∼ = (P4 , OP4 (2)). Example 4.4. Let us give an example as in case 8), (d-ii) above (we refer to [6], [4, × OQ3 (1). = (3.28)] for details). Let M = P1 × Q3 , and let H := OM (1, 1) = OP1 (1) < Then (M, H), p : M → P1 , is a quadric ﬁbration of degree H 4 = 8. From the relation KM + L + H ∼ = OM we infer that × OQ3 (3) − OP1 (1) < = × OQ3 (1) = OP1 (1) < = × OQ3 (2) = L = OP1 (2) < is ample. Since M is a product of Fano varieties, then M is a Fano 4-fold of index r = g.c.d.(r1 , r2 ) = 1, where r1 = 2 and r2 = 3 are the indexes of P1 and Q3 , respectively.

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References [1] M. Andreatta, C. Novelli and G. Occhetta, Connections between the geometry of a projective variety and of an ample section, to appear in Math. Nachr. [2] M.C. Beltrametti, M.L. Fania and A.J. Sommese, Mukai varieties as hyperplane sections, Proceedings of the Fano Conference, Torino, Italy, 2002 ed. by A. Collino, A. Conte and M. Marchisio, Universit` a degli Studi di Torino, Dipartimento di Matematica, Torino, (2004), 185–208. [3] M.C. Beltrametti and A.J. Sommese, The Adjunction Theory of Complex Projective Varieties, Expositions in Mathematics, 16, W. de Gruyter, Berlin, (1995). [4] T. Fujita, Classiﬁcation of polarized manifolds of sectional genus two, in Algebraic Geometry and Commutative Algebra, in Honor of Masayoshi Nagata, (1987), 73– 98, Kinokuniya, Tokyo. [5] T. Fujita, Classiﬁcation Theories of Polarized Varieties, London Math. Soc. Lecture Note Ser. 155, Cambridge University Press, (1990). [6] P. Ionescu, Embedded projective varieties of small invariants, III, in Algebraic Geometry, Proceedings of Conference on Hyperplane Sections, L’Aquila, Italy, 1988, ed. by A.J. Sommese, A. Biancoﬁore, and E.L. Livorni, Lecture Notes in Math. 1417 (1990), 138–154, Springer-Verlag, New York. [7] V.A. Iskovskikh and V.V. Shokurov, Biregular theory of Fano 3-folds, in Proceedings of the Algebraic Geometry Conference, Copenhagen 1978, Lectures Notes in Math. 732 (1979), 171–182, Springer-Verlag. [8] J. Koll´ ar, Appendix to the paper of C. Borcea: Homogeneous vector bundles and families of Calabi-Yau threefolds, II, Several Complex Variables and Complex Geometry, 1989, Proc. Symp. Pure Math. 52, Part 2 (1991), 83–91. [9] A. Lanteri, M. Palleschi, and A.J. Sommese, On triple covers of Pn as very ample divisors, in Classiﬁcation of Algebraic Varieties, Proceedings L’Aquila, 1992, ed. by C. Ciliberto, E.L. Livorni, and A.J. Sommese, Contemp. Math. 162 (1994), 277–292. [10] A. Lanteri, M. Palleschi, and A.J. Sommese, Double covers of Pn as very ample divisors, Nagoya Math. J. 137 (1995), 1–32. [11] A. Lanteri, M. Palleschi, and A.J. Sommese, Del Pezzo surfaces as hyperplane sections, J. Math. Soc. Japan 49 (1997), 501–529. [12] S. Mori and S. Mukai, Classiﬁcation of Fano 3-folds with b2 ≥ 2, Manuscripta Math. 36 (1981), 147–162. [13] S. Mukai, Biregular classiﬁcation of Fano threefolds and Fano manifolds of coindex 3, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), 3000–3002. [14] S. Mukai, New classiﬁcation of Fano threefolds and manifolds of coindex 3, preprint, 1988. [15] S. Mukai, Contribution to: Birational geometry of algebraic varieties. Open problems, The 23rd Int. Symp. of the division of Math. of the Taniguchi Foundation, Katata, August 1988. [16] E. Sato, Varieties which have two projective space bundle structures, J. Math. Kyoto Univ. 25 (1985), 445–457.

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[17] B. Shiﬀman and A.J. Sommese, Vanishing Theorems on Complex Manifolds, Progr. Math. 56 (1985), Birkh¨ auser, Boston. [18] M. Szurek and J. Wi´sniewski, Fano bundles of rank 2 on surfaces, Compositio Math. 76 (1990), 295–305. [19] M. Szurek and J. Wi´sniewski, Fano bundles over P3 and Q3 , Paciﬁc J. Math. 141 (1990), 197–208. [20] J. Wi´sniewski, Fano 4-folds of index 2 with b2 ≥ 2. A Contribution to Mukai Classiﬁcation, Bulletin of Polish Academy of Sciences, Mathematics, 38 (1990), 173–183. Mauro C. Beltrametti Dipartimento di Matematica Via Dodecaneso 35 I-16146 Genova, Italy Email: [email protected] M.L. Fania Dipartimento di Matematica Universit` a degli Studi di L’Aquila Via Vetoio, loc. Coppito 67100 L’Aquila, Italy Email: [email protected]

Triple points imposing triple divisors and the defective hierarchy Cristiano Bocci and Luca Chiantini

Abstract. We introduce the notion of ”sub–defective” varieties as those varieties X for which a general tangent space intersects ”too many” other tangent spaces to X. We show that this notion has an unexpected link with the study of varieties with degenerate osculating behavior. Namely we show that sub-defective surfaces are precisely those surfaces whose general osculating space is bi-osculating. 2000 Mathematics Subject Classiﬁcation: 14J25

Introduction The paper is a contribution to the study of irreducible algebraic surfaces X whose projective embeddings have some particular characteristic from a diﬀerential point of view. This problem translates soon in a study of varieties whose k-th osculating spaces have some unexpected behavior. The study of these varieties should be considered as a systematic approach to the problem of interpolation in general algebraic varieties, and in multiplicity two or more. A ﬁrst instance of such degenerate behavior is given by varieties such that the k-th osculating space at a general point is indeed osculating along a positive dimensional variety. It is addressing this problem for surfaces and for k = 3 (the case of surfaces X such that the hyperplanes sections having a triple point at a general P ∈ X, in fact have a triple component) that we realized an unexpected link between the behavior of tangent and osculating spaces, that we want to present here. The case k = 2 (double points) has been classically solved in dimension 2: surfaces such that the general tangent plane is tangent along a curve (i.e. surfaces ”with degenerate Gauss image” in the classical terminology or ”0-weakly defective surfaces” in the terminology of [10]) are just cones or surfaces swept out by the tangent lines to a ﬁxed curve. Looking for tangency in more than one point, the problem is related to the study of defective varieties, i.e. varieties whose secant varieties are small. The celebrated Terracini’s lemma and the ”inﬁnitesimal Bertini

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principle” of [12] and [10] (see also [21]) show how k-defective implies that imposing tangency at k +1 general points one gets tangency along a positive dimensional subvariety. This phenomenon leads to the deﬁnition of ”weakly defective”, a condition which is eﬀectively weaker than ”defective” and is extensively studied in [10]. When the multiplicity that we impose is three or more, the situation changes. Even when the number of conditions imposed by triple points to hyperplane sections is less than the expected, nevertheless this does not imply that the sections split triple divisors. We refer to the considerably huge literature on the subject, especially to [8], [3], [1], [6], [7], [13], [5] and to the wide bibliography of [11]. There are examples of smooth surfaces such that by imposing just one triple point at a general P ∈ X, the number of conditions we obtain is 5 (while the expected value is 6) and nevertheless the general hyperplane section with a triple point has no triple divisors. Among them, let us cite ruled surfaces (in this case the sections have double divisors) or Togliatti’s example in [20], where the general section is reduced. Classically these surface are said to satisfy a Laplace equation. Conversely, there are examples of surfaces such that by imposing one triple point at a general P ∈ X to hyperplane sections, one always gets inﬁnitely many triple points. A smooth surface of this type has been studied by Lanteri and Mallavibarrena (see example 1.1), but there are many other irreducible surfaces (see Example 1.2) for which the condition holds. Let us notice here that Segre’s conjecture for linear systems in the plane (see [18] or [13]) says that when X is rational, linearly normal and general multiple points impose less conditions than expected, then the corresponding hyperplane sections are non–reduced. So, let us start with the special case of just one triple point (here Segre’s conjecture is established) on general surfaces and study what happens when this ”non-reducedness” is extreme, i.e. surfaces X such that: (*) X satisﬁes no Laplace equations and for a general point P ∈ X, the system of hyperplane sections which have a triple point at P consists of divisors having a triple component. Classically, the problem was addressed by M. Castellani in the paper [9]. In fact, in this short note of 1922, Castellani studies an even more general problem: the classiﬁcation of surfaces whose general osculating P5 is bi-osculating. The result achieved is stated in theorem 3.1 below and is correct. Castellani’s theorem was revisited and extended by D. Franco and G. Ilardi in [14]. Castellani’s classiﬁcation of surfaces satisfying condition (*) is however stronger than Franco-Ilardi’s one, as it predicts that X lie in a developable scroll Y and in this sense it provides a necessary and suﬃcient condition, i.e. a complete classiﬁcation. We should say that we feel quite uncomfortable with Castellani’s procedure, for we feel that the argument for the structure of Y (which, by the way, is missing in

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[9]) is not completely trivial and requires in fact some analysis of the geometry of fourfolds in P6 which are scrolls in 3-spaces. Thus in the ﬁrst section we are led to reconsider brieﬂy Castellani’s theorem, rephrased in modern language, providing a complete argument. Theorem 0.1. (Castellani) X ⊂ Pr satisﬁes condition (*) if and only if it lies in a fourfold Y with the following properties: Y is a scroll spanned by 1-dimensional family of P3 ’s; and Y is developable (i.e. the tangent spaces to Y are constant along ﬁbers); and there exists a smooth branch of Y passing through the general point of X. The structure of developable fourfolds is outlined in Remark 1.5. Notice that by Barth–Lefschetz theorem, these scrolls cannot be smooth in P6 (see e.g. the introduction of [15]). Notice that by taking a general projection of X to P6 , the resulting surface maintains property (*) and, after the projection, the system of hyperplane sections with a triple point at a general P is a singleton. So we will work only in the case r = 6. Section 2.3 is devoted to the particular case of surface for which the unique divisor in |H − 3P | is a triple curve. Here the family of these triple curves turns out to be a pencil, so the classiﬁcation argument becomes easier and it is outlined in Theorem 2.3. The last section is the main motivation for this paper. It is devoted to point out an unexpected link between the defective osculating behavior and some unexpected behavior of tangent spaces. Castellani’s and Franco-Ilardi’s papers concern the more general problem of varieties for which the general osculating space is bi–osculating, i.e. for P ∈ X general there is Q = P such that the osculating spaces at P, Q coincide. It turns out that there are two types of such surfaces: those such that the general osculating space osculates along a curve (i.e. the ones classiﬁed in section 1, lying in a developable fourfold scroll) and another class of surfaces which lie in some special ruled threefold. Surfaces in this second class have the property that for P general there is Q = P such that the tangent spaces at P, Q meet along a line. This is a remarkable property of the tangent spaces to X, which is highly unexpected for surfaces in P6 and shows some link with classical 1-defectivity. So we are led to consider a generalization of the deﬁnition of defective varieties. For a general point P of the surface X, write ∂i (P ) for the set of points Q = P such that the tangent spaces to X at P, Q meet in dimension ≥ i. ∂i (P ) corresponds to the set of points whose image in the Gauss map sits in some Schubert cycle deﬁned by the tangent space at P . Since the codimension of the Schubert cycles is known, it turns out then that these varieties ∂i (P ) are equipped with an expected dimension. We say that X is sub-defective when the actual dimension of ∂i (P ) does not coincide with the expected one.

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Defective varieties are obviously sub-defective, but sub-defectivity is much weaker than defectivity: there are sub-defective curves, there are many sub-defective surfaces in P4 , and so on. In P6 a surface is sub-defective if either dim(∂1 (P )) ≥ 0 or dim(∂0 (P )) ≥ 1. The former case corresponds to surfaces classiﬁed by Castellani and Franco–Ilardi (see Theorem 3.5). So in section 3 we study a classiﬁcation of surfaces for which ∂0 (P ) has dimension 1. Surprisingly enough, it turns out that these are precisely those surfaces which are contained in some developable fourfold scroll, i.e. surfaces satisfying (*). Thus ﬁnally, surfaces in P6 such that every osculating space is bi–osculating are precisely the sub-defective surface (of both classes). This concludes the circle of ideas exploited in the present paper. It also opens a series of interesting questions which we plan to explore in some forthcoming research. In particular we plan to come back to a more systematic exploration of sub-defective varieties, which seem by themselves an interesting class of special varieties satisfying some diﬀerential equations. We hope that their understanding could spread some further light in our knowledge of the osculating behavior. At the end, let us thank all people who gave us several important hints during the preparation of the paper. Among them we would like to cite mainly Ciro Ciliberto, and also Rick Miranda, Antonio Lanteri, Edoardo Ballico and Claudio Fontanari.

1. The ﬁrst classiﬁcation Let us start by providing examples of surfaces in P6 which satisfy the basic assumption (*) of the introduction. As far as we know, the unique known example of a smooth (and even rational) surface in P6 which satisﬁes our assumptions is the following surface studied by Lanteri and Mallavibarrena: Example 1.1. (See [17], case ii of §5) Let X be the blow up of P2 at 8 general 8 points Pi , i = 1, . . . , 8. The linear system H := |9 − i=1 3Ei | ( corresponds to a line), of curves of degree 9 passing three times through the points, embeds X in P6 . If we impose another triple point, then H − 3P = −3K, where K is the canonical class of X. In particular H − 3P contains only one divisor, which consists in a triple curve. Indeed the unique section of H − 3P corresponds on the plane to the cubic FP passing through all the points Pi and P , counted with multiplicity 3. Thus for P general we have |H − 3P | = {3FP } and in particular WP = 0. For our purposes, it is useful to say something more on the geometry of this example. As P varies, the curves FP move in the pencil of cubics through the eight points Pi ’s. Each FP is embedded as a plane cubic in P6 . If we project X

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from the plane of the curve FP , we obtain the map P2 → P3 given by the linear system of sextics with 8 general double points. It is known that the image is a quadric cone. Thus X ⊂ P6 lies in many cones with vertex of dimension 3 over a conic. Let us continue exploiting this example from another point of view. The eight points determine a pencil of cubic curves which meet in a ﬁxed ninth point Q. 8 The subsystem V = |9 − i=1 3Ei − 2Q| maps X to P3 , contracting the curves of the pencil. Indeed if A is any elliptic curve of the pencil, the elements of V meet A in one point, which cannot move. Hence V maps to a curve of P3 , which is necessarily a twisted cubic curve: the triple embedding of the image of the pencil. Thus X ⊂ P6 projects to a cubic curve of P3 . It is easy to ﬁnd immediately many other irreducible surfaces which satisfy our conditions, by using the properties of the previous example, in the spirit of [9]. Example 1.2. Let X ⊂ P6 be a non-degenerate irreducible surface which lies in a four-dimensional cone over a curve Γ ⊂ P3 with vertex a plane π. If we denote by H the system of hyperplane sections of X, then for a general point P ∈ X, the system H(−3P ) contains an element of type 3FP + WP . In particular, if X satisﬁes no Laplace equations, then it satisﬁes our assumptions. In fact, let Q ∈ Γ be the image of P in the projection from the vertex T of the cone and call FP the ﬁber of the projection passing through P . If TP is the osculating plane to Γ at Q, then the span < T, TP > is a linear space of dimension at most 5, which meets X along a divisor of type 3FP + WP . Here WP decomposes in a union of ﬁbers over the points of TP ∩ Γ diﬀerent from Q. Example 1.3. As a particular case of the previous example, we can take as Γ a rational cubic curve in P3 . In this case we get, for P general, H(−3P ) = {3FP }, that is WP = 0. Also Example 1.1 is a particular case of this situation. Let us prove that the suﬃcient condition of Castellani’s theorem holds. Theorem 1.4. Let X ⊂ P6 be an irreducible surface which satisﬁes no Laplace equations. Assume that X lies in a fourfold Y which is ﬁbered by a 1-dimensional family of P3 ’s. Assume that Y is developable, i.e. the tangent spaces to Y are constant along the ﬁbers. Assume also that there exists a smooth branch of Y passing through a general point of X. Then if |H| is the hyperplane linear system on X, for P ∈ X general we have that the unique divisor of |H − 3P | is triple along the curve FP of intersection between X and the ﬁber of Y through P . Proof. Fix a general plane π ⊂ P6 . We have a (rational) map ρ : X π deﬁned as follows: for P ∈ X general, P sitting in the ﬁber YP of Y , call ρ(P ) the intersection between π and the tangent space to Y at P . As the tangent space to Y is constant along YP , the map ρ has positive dimensional ﬁbers, hence its image

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is either a curve or a point. We will prove that if ρ(X) is not a point, then we will prove that calling LP the tangent line to ρ(X) at ρ(P ), the osculating space to X at P sits in the span HP =< YP , LP >. This will conclude the proof in this case, since HP is a P5 which is constant along FP = YP ∩ X. Then we ﬁnish by proving that ρ(X) cannot be a point. Assume for a moment that ρ(X) is not a point. Observe that the tangent space TY,P to Y at P is the space generated by YP and ρ(P ). Thus also the tangent plane to X at P sits in the space generated by YP and ρ(P ). Take local coordinates x, y for X around P . We may assume that x is the direction of the tangent line to F and y is the direction of some tangent line rP which points outward YP . By construction, rP ⊂< YP , ρ(P ) >. The osculating space to X at P is generated by P, Px , Py , Pxx , Pxy , Pyy . By construction HP contains the tangent space to X at P , hence it contains P, Px , Py . Pxx sits in the osculating plane to FP , which is contained in YP , hence in HP . Pxy = Pyx and Pyy sit in the ruled threefold traced by the line rP as we move P on X. When we do that inﬁnitesimally, then P moves in the tangent plane to X while ρ(P ) moves in the tangent line to ρ(X) at ρ(P ). It is clear then that both Pxy = Pyx and Pyy sits in HP and our claim is settled. It remains to prove that ρ(X) is not a point A. But the argument above can be easily rephrased to show that in this case the osculating space to X at P would lie inside the span < YP , A >, which is a P4 , hence X would satisfy a Laplace equation.

Remark 1.5. We notice that we have two ways of constructing fourfolds in P6 which are developable scrolls. Namely the intersection of Y with a general linear space of dimension 4 is a ruled surface R whose tangent spaces are constant along the ruling. This may happen only if R is either a cone or the developable surface of tangent lines to a curve E (see e.g. [10]). In the ﬁrst case, Y is necessarily a cone with vertex of dimension 2, hence a cone over a space curve. The second case contains fourfolds swept out by hyperosculating P3 ’s to a ﬁxed curve C. Remark 1.6. Of course the previous example covers surfaces contained in threefolds Y which are developable scrolls in planes. Such threefolds are either cones over space curves or they are swept out by osculating planes to a ﬁxed curve. Next, we want to show that the converse of Theorem 1.4 holds. Thus let X be a surface in P6 which satisﬁes the assumption (*) of the introduction. It is easy to see that if |H − 3P | = {3FP + WP }, then P ∈ FP and P ∈ WP , since otherwise any divisor which is triple at P is also of multiplicity 4 at P (a Laplace condition). Our task is to show that X lies in a developable fourfold ﬁbered by P3 ’s.

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Remark 1.7. When P is replaced by another general point Q ∈ FP , by our assumptions and by generality, 3FP + WP is still the unique hyperplane section which is triple at Q. Hence the divisors {FP : P ∈ X} deﬁne a 1-dimensional family of hyperplanes in P6 , i.e. a curve in the dual space (P6 )∗ . Call R this curve. By Bertini’s theorem, it is clear that R cannot be a line. We have an obvious (rational) ﬁbration ρ : X R. Proposition 1.8. For P general, consider the tangent line to R at ρ(P ). Then this line deﬁnes a pencil of hyperplane sections of X, all of them containing twice the divisor FP . In particular there exists a linear space of dimension 4 which is tangent to X at every point of FP . Proof. The curve R is contained in the (reduced) Severi variety S of hyperplanes which cut X in a divisor with a point of multiplicity 3. In fact, by our assumptions, it is clear that R is an irreducible component of S. So the tangent line to R at a general point ρ(P ) sits inside the tangent space to S. For Q ∈ FP general, it is well known by local analysis that the tangent space to S at ρ(Q) = ρ(P ) sits inside the set of hyperplanes which are tangent to X at Q (see e.g. [10], Prop. 2.3). The conclusion follows.

In a similar way we can prove the following: Proposition 1.9. For P general, the curve FP is contained in some linear space of dimension 3. Proof. By what we said in the previous proposition, it turns out that the developable surface Z of tangent lines to R (which is not a line itself) sits in the (reduced) Severi variety of hyperplanes which cut X in a divisor with a singular point. Fix a general A ∈ Z and take the (general) point P ∈ X such that A belongs to the tangent line to R at ρ(P ). Then the hyperplane section corresponding to A is of the form 2FP + WA , i.e., by assumptions, the hyperplane corresponding to A has indeed a double point along FP . This double point is locally a cusp. Now move A generically in Z. Then for Q ∈ FP general 2FP + WA moves in a family of divisors with a cusp around Q. As the family of hyperplane sections of X with a cusp at a general point has dimension 2, Z is a component of this family. It is well known that the tangent space to the family of curves with cusps, at a point corresponding to a divisor with a cusp at Q, is contained in the linear system of hyperplanes through Q. Thus the tangent plane to Z at a general point A is formed by hyperplanes, all of which pass through the general point of FP . It

follows that FP sits inside a 2-dimensional family of hyperplanes. The curves in the family {FP } may be even more degenerate. Consider indeed Example 1.1, where each FP is a plane curve. On the other hand, notice that FP is not a line, since any ruled surface satisﬁes a Laplace equation.

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Before concluding the proof of Castellani’s classiﬁcation theorem 0.1, we need a couple of properties of scrolls, which seem to us unavoidable in the sequel (see the foundational paper [19] for a classical reference). Proposition 1.10. Let Y ⊂ Pr be a scroll, ﬁbered by a 1-dimensional family of Ps ’s. Let Y0 be a general ﬁber of the family. Then for all P ∈ Y0 which are smooth points of Y , the locus ΣP = {Q ∈ Y0 : the tangent space to Y at P is also tangent to Y in Q} is a linear subspace. If Y is any variety, then it is classically known that the contact locus of a general tangent space is linear (see e.g. [16], (5.34)). Notice that, however, this would not be suﬃcient for our purposes: we need to know what happens for any P ∈ Y . Proof. Fix A, B general in ΣP . We are done if we prove that the line < A, B > also lies in ΣP . Fix a general Q = aA + bB ∈< A, B >. Let R be the curve in the Grassmannian G(s, r) which corresponds to the scroll Y , so that Y0 is the ﬁber over 0 ∈ R. Find a small arc P (z) ⊂ Y around P = P (0), such that P (z) ∈ Yz for all z in a neighborhood of 0 in R. The tangent space TP to Y at P is thus spanned by Y0 and the tangent line to the arc P (z) at P . Find arcs A(z) and B(z) in Y , around A = A(0) and B = B(0). Since the tangent line A to A(z) at A sits in the tangent space to Y at A, by assumption it sits in TP . Similarly does the tangent line B to B(z) at B. Now consider the arc Q(z) = aA(z) + bB(z). Its tangent line spans, together with Y0 , the tangent space TQ to Y at Q. But by construction sits in the space spanned by Y0 , A and B . Thus TQ ⊂ TP .

Corollary 1.11. If the general ﬁber YP of Y has a subvariety FP which does not sit in the singular locus and such that the tangent space to Y is constant at the general point of FP , and moreover < FP >= YP , then Y is developable. In order to apply the previous facts to our situation, we need to explore what happens when FP sits in the singular locus of Y . In this case, of course, we cannot use the tangent space to Y , since its dimension jumps at singular points. Remark 1.12. The previous results still work when we consider points P which are singular, but with a smooth branch of Y passing through them. Indeed the proof of the proposition still works, provided we substitute the tangent space to Y with the tangent space to the branch. Now we are ready to prove the converse of Theorem 1.4. Theorem 1.13. Let X ⊂ P6 be an irreducible surface which satisﬁes no Laplace equations. Call |H| the hyperplane linear series. Assume that for P ∈ X general, the unique divisor of |H − 3P | has a triple component FP . Then X sits in a fourfold Y which is a developable scroll, ﬁbered by P3 ’s.

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Proof. Call YP the span of a general divisor FP . By Proposition 1.9, we know that the dimension of YP is either 2 or 3. If YP has dimension 3, we take Y to be the closure of the union of the family {YP }, for P ∈ X general. FP could in principle be contained in the singular locus of Y . We show that nevertheless there exists a branch of Y which is smooth at a general point of FP . Observe that FP sits in the singular locus only if for P ∈ FP general, there exist Q such that P ∈ YQ . If Q is inﬁnitesimally close to P , then this means that FP sits in the focal locus of the scroll, which is linear. So this case is excluded for P ∈ FP general, for < FP >= YP . Then Q is a point of X far from FP . It turns out immediately that P ∈ FQ + WQ and hence, by generality, P ∈ WQ , otherwise we had another hyperplane divisor 3FQ + WQ triple at P . This means that among the ﬁbers of the scroll passing through P there is one well-determined ﬁber: the one corresponding to FP , i.e. the one which contains the triple part of the unique hyperplane divisor which is triple at P . The movement of this ﬁber determines a branch of Y which is smooth at P . The tangent space to (the branch of) Y at P is 4-dimensional and spanned by YP and the tangent space to X at P . By Propositiom 1.8, there exists a linear space ΛP of dimension 4 which contains FP and is tangent to X at any point of FP . Thus ΛP coincides with the tangent space to (the branch of) Y at a general point of FP . Therefore this tangent space is ﬁxed along FP . Since FP spans YP , in any event by Proposition 1.10 and its corollary (or Remark 1.12), it follows that the tangent space to Y along YP is constant. The case dim(YP ) = 2 is similar: here the union of the YP ’s deﬁnes a developable threefold which clearly sits in a developable fourfold.

2. The case WP = 0 In this section we consider the subcase in which the unique divisor of the hyperplane linear series |H| with a triple point at the general P is of the form 3FP . There is an alternative way to classify this case, which leads to a more precise statement. Indeed we prove that in this case X sits in a cone. Our assumption is now: for P general, |H − 3P | only contains a triple curve 3FP . Proposition 2.1. As P moves, the divisors FP are linearly equivalent. Proof. The divisors FP determine a (rational) map from X to a curve C. If A, A are general points of C, and F, F are the corresponding ﬁbers of the map X → C, then 3F ≡ 3F implies 3A ≡ 3A . Since the points are general, while the jacobian of a curve of positive genus has only ﬁnitely many torsion points, this is impossible

unless A ≡ A . Hence C is rational and F ≡ F .

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Observe that a similar statement fails when WP = 0, because we cannot conclude that the family {FP } is linear. Indeed the statement is false even for general surfaces sitting in the cone over a space curve of degree ≥ 4. Proposition 2.2. The divisors FP , P ∈ X, span a linear series W ⊂ |FP | of (projective) dimension 1. Proof. Consider the complete linear system |FP |. By the previous proposition it is not trivial. Assume that it has dimension at least 3. Then the divisors FP determine a curve Γ contained in the projective space Ps associated to (the dual of) |FP |. By assumption, the hyperplane linear system V contains all the divisors 3FP . Thus it turns out then that the triple embedding of the curve Γ spans (at most) a P6 . On the other hand, notice that no projective curve Γ ∈ Ps has a triple embedding which spans a projective space of dimension smaller or equal than 6, except for the following two cases: - s = 1; - s = 2 and Γ is a conic. Indeed consider a general projection of Γ to a plane: its triple embedding too generates a space of dimension ≤ 6. But clearly all triple embedded curves in P2 , except lines and conics, span a subspace of dimension at least 8. We want just to exclude the case s = 2. Let us argue by contradiction. Call πΓ the plane spanned by Γ, which determines by duality a sublinear system of |FP |. Let φ : X → πΓ be the induced map. Assume that φ maps X to a curve X . Then φ followed by the triple embedding of X determines a sublinear space of dimension 6 in V , which thus coincides with V . This implies that also the hyperplane series on X contracts it to a curve, a contradiction. Assume φ is dominant. As Γ corresponds to a conic, for a general Q ∈ X there are two divisors FP , FP ∈ Γ passing through Q, thus one has two diﬀerent hyperplane

divisors 3FP , 3FP which are triple in Q, a contradiction. We are now ready to prove: Theorem 2.3. Let X ⊂ P6 be a surface such that for a general point P ∈ X, the system of hyperplane sections with a triple point in P contains a single element which is a triple curve. Then X lies in a cone of dimension 4 over the twisted cubic with vertex a plane π. Proof. By Proposition 2.2, the divisors FP determine on X a 1-dimensional system parameterized by Γ = P1 . The triple embedding of Γ spans a projective space W of dimension 3 which is a subspace of |H|. The corresponding sublinear system of the hyperplane series sends thus X to the triple embedding of the line Γ. The conclusion follows.

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Example 1.3 shows that also the converse holds: for surfaces lying in cones of dimension 4 over a twisted cubic curve in P3 , triple points impose triple divisors. We notice that, as showed at the beginning, the Example 1.1 of Lanteri and Mallavibarrena ﬁts in the previous classiﬁcation.

3. A generalization: the defective hierarchy Castellani’s paper contains indeed a classiﬁcation of the more general class of surfaces X, such that the following condition holds: (**) X satisﬁes no Laplace equations and for a general point P ∈ X, a general hyperplane sections which have a triple point at P also has (at least) another triple point Q which is a regular point of X. It is clear that condition (*) is stronger than condition (**). It turns out indeed that the condition is not that stronger (see [14] Section 3 for a more general analysis). Proposition 3.1. (Castellani, Franco–Ilardi) Let X ⊂ P6 be a surface which satisﬁes condition (**) above. Then either: (a) X also satisﬁes condition (*), or (b) for P ∈ Xgeneral, there exists a P3 which is tangent to X at P and at another point Q = P . Proof. Fix P ∈ X general and take Q such that the osculating space HP to P is also osculating to Q. Assume also that there exist only ﬁnitely many points of X at which HP osculates (otherwise we are in case (a)). Consider the singular incidence variety: I = {(H, P, Q) : H osculates in P, Q} ⊂ (P6 )∗ × X 2 and look at the projection p : I → (P6 )∗ . From our assumptions it follows that I has dimension 2. As explained in the proof of Proposition 1.8, the tangent spaces to p(I) at a general point H = p(H, P, Q) sits in the subvariety of hyperplanes which are tangent to X at P, Q. Thus we have a 2-dimensional linear system of such hyperplanes, hence the span of the tangent spaces to X at P, Q is contained

in a P3 . We want to stress here that condition (b) of Proposition 3.1 can be considered as an instance of the following set of ”defective properties”: Deﬁnition 3.2. Let X ⊂ Pr be a variety of dimension n. For a general P ∈ X deﬁne the set ∂i (P ) as follows: ∂i (P ) = {Q ∈Reg(X) : P = Q and the tangent spaces to X at P, Q meet in dimension ≥ i}.

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∂i (P ) corresponds to the set of points which are sent to cuspidal points in the projection from the tangent space at P . As the expected dimension of ∂i (P ) is max{−1, min{n, n − (i + 1)(r − 2n + i)}}, then we give the following: Deﬁnition 3.3. We say that X is sub-defective if for some i and for P ∈ X general, we have: dim(∂i (P )) > max{−1, min{n, n − (i + 1)(r − 2n + i)}}. Notice that sub-defective also means that the Gauss image of X in the corresponding Grassmannian intersects improperly some Schubert cycles. Remark 3.4. X is 1-defective precisely when, for i = max{0, 2n − r + 1} and P general, ∂i (P ) contains X. For any n-dimensional variety, the expected dimension of ∂n (P ) is −1. In other words we never expect that a general tangent space is bi–tangent. Varieties for which the general tangent space is bi–tangent are thus sub–defective. Observe that an n-dimensional variety is developable (i.e. 0-weakly defective) when dim(∂n (P )) ≥ 1. So developable varieties are sub–defective. A surface X ⊂ P6 satisﬁes condition (b) of Theorem 3.1 when ∂1 (P ) has dimension at least 0, for P ∈ X general. As the expected dimension of ∂1 (P ) is −1, in this case, these surfaces are sub-defective. Many other simple facts about sub–defectivity could be easily derived directly from the deﬁnition, but they are not exploited here. We just consider the relevant case for our purposes, namely surfaces in P6 . When we consider a surface X ⊂ P6 , the expected dimension of ∂1 (P ) is −1 while the expected dimension of ∂0 (P ) is 0. Consequently sub-defective surfaces in P6 have either: dim(∂1 (P )) ≥ 0, or dim(∂0 (P )) ≥ 1. The former case corresponds to surfaces satisfying condition (b) of Theorem 3.1. A classiﬁcation of such surfaces is provided classically in Castellani’s paper, and rephrased and generalized by Franco and Ilardi in [14]. We just state the result, and refer to [14], Theorem 3.14, for the proof. Theorem 3.5. Let X be a surface X ⊂ P6 for which dim(∂1 (P )) ≥ 0. Then X sits in a ruled threefold, whose tangent spaces along a general ruling are contained in some P3 . Conversely assume that X is a surface which sits in a ruled threefold, whose tangent spaces along rulings span a P3 . Assume furthermore that the general ruling meets X in more than one point. Then dim(∂1 (P )) ≥ 0. Indeed Theorem 3.14 of [14] is much more precise on the nature of the ruled threefold Y containing X: it is proven there that Y must be a Laplace congruence of lines (refer to [14] for a deﬁnition).

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47

Thus we are led to consider more closely the case dim(∂0 (P )) ≥ 1. It corresponds to surfaces satisfying the following condition: (***) for P ∈ X general one ﬁnds a curve XP of points whose tangent spaces meet the tangent space at P . Remark 3.6. As observed in [9], it is clear by the classiﬁcation theorem that surfaces satisfying condition (a) of Proposition 3.1 also satisfy condition (***), i.e. they are sub-defective. Indeed the tangent spaces to X along the ﬁbers of the scroll Y all sit in the ﬁxed P4 which is tangent to the ﬁber. We present here a description of sub-defective surfaces for which (***) holds. Proposition 3.7. Sub-defective surfaces in P6 satisfying dim(∂0 (P )) ≥ 1 sit in a developable scroll Y of dimension 4, ﬁbered in P3 ’s. Conversely let X be a surface contained in such a scroll Y and assume that at a general P ∈ X there is a smooth branch of Y passing through P . Then X is sub-defective with dim(∂0 (P )) ≥ 1. Proof. We may assume that X is not 1-defective, for in this case it is a cone. Consider the incidence variety: J = {(H, P, Q) : H is tangent to X at P, Q} ⊂ (P6 )∗ × X 2 . Notice that we assume dim(J) ≥ 4 . Then over a general pair (P, Q) with Q ∈ ∂0 (P ), J contains the following two subvarieties: J1 obtained by ﬁxing P and moving Q generically, outside ∂0 (P ); J2 ﬁbered in linear spaces and obtained by ﬁxing P and moving Q along ∂0 (P ). Any of these subvarieties has dimension at least 4. Now take (H, P, Q) ∈ J1 (P ) ∩ J2 (P ) and consider the projection p of J to P6 . Since p(J) consist of hyperplanes tangent to X at P, Q, P is general and Q = P is a smooth point of X (by construction), even if Q is not general, nevertheless the Zariski tangent space T to p(J) at H sits in the space of hyperplanes which pass through P, Q. Thus necessarily dim(T ) ≤ 4. It follows that the dimension of p(J) around H cannot be bigger than 4. We can show now that dim(p(J2 )) < 4 around H. This is clear when dim(p(J)) < 4. If dim(p(J)) = 4, then also dim(T ) = 4, so that p(J) is smooth at H. If dim(p(J2 )) = 4, then this may only happen if p(J1 ) sits inside p(J2 ), around H. But this is impossible, for we can move Q generically outside ∂0 (P ) and this moves H outside p(J2 ). It follows that the ﬁbers of p at a general H ∈ p(J2 ) have a positive dimensional part contained in J2 . This means that H is tangent to X at P and along a curve F ⊂ XP . We want to prove that F sits in some P3 . The image p(J2 ) contains the pencil of hyperplanes through the span of the tangent spaces at P, Q. Thus it contains a ruled surface. The tangent plane to this surface at H determines a 2-dimensional family of hyperplane sections of X with a double point around any point of F .

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Cristiano Bocci and Luca Chiantini

Thus, as in the proof of Proposition 1.9, this net has F in the base locus, i.e. dim(< F >) ≤ 3. Now call YP the span of F and move P generically. We get a 1-dimensional family of linear spaces, which deﬁne a scroll Y of dimension at most 4. As in the proof of Theorem 1.13, by Proposition 1.10 and Proposition 1.12, it turns out that Y has a smooth branch at the general point of X and it is developable. Whatever the dimension of Y could be, the necessary condition is established. Conversely assume that X sits in a developable fourfold scroll Y and assume that there exists a smooth branch of Y through a general point of X. Then for P ∈ X general, call YP the ﬁber of the scroll passing through P . It is clear that for all Q ∈ YP ∩ X, the tangent spaces to X at P, Q are contained in the tangent space

of Y along YP , which is a P4 . Remark 3.8. It follows from the proof that, for P general, XP passes through P . A fact which is not obvious a priori. Corollary 3.9. The surfaces X ⊂ P6 for which triple points impose triple divisors are thus the sub-defective surfaces for which dim(∂0 (P )) ≥ 1.

References [1] J. Alexander and A. Hirschowitz polynomial interpolation in several variables, J. Alg. Geom. 4 (1995), 201-222. [2] E. Arbarello and M. Cornalba Footnotes to a paper of B. Segre, Math. Ann. 256 (1981), 341-362. [3] E. Ballico and C. Fontanari, On a Lemma of Bompiani, math.AG/0406318. [4] E. Ballico, C. Bocci and C. Fontanari, Zero–dimensional subschemes of ruled varieties, Centr. Europ. J. Math. 2 (4), (2004), 538–560. [5] A. Bernardi, M.V. Catalisano, A. Gimigliano and M. Id´a, Osculating varieties of Veronesean and their secant varieties, math.AG/0403132. [6] C. Bocci, Special linear systems and special eﬀect varieties, Tesi di Dottorato, Univ. di Torino (2004). [7] C. Bocci, R. Miranda Topics on interpolation problems in Algebraic Geometry, to appear (2004). [8] E. Bompiani, Determinazione delle superﬁcie integrali di un sistema di equazioni parziali lineari e omogenee, Rend. Ist. Lomb. 52 (1919), 610–636. [9] M. Castellani, Sulle superﬁcie i cui spazi osculatori sono biosculatori, Rend. Accad. Lincei XXXI (1922), 347–350. [10] L. Chiantini and C. Ciliberto, Weakly defective varieties, Trans. Amer. Math. Soc. 354 (2002), 151–178.

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[11] C. Ciliberto, Geometric aspects of polynomial interpolation in more variables and of Waring’s problem, Proceedings of the European Congress in Math., Barcelona 2000, Progress in Math. Birh¨ auser 1 (2001), 289–316. [12] C. Ciliberto and A. Hirschowitz Hypercubiques de P4 avec sept pointes singulieres generiques, C. R. Acad. Sci. Paris 313 (1991), 135–137. [13] C. Ciliberto and R. Miranda, The Segre and Harbourne-Hirschowitz conjectures, NATO Sci. Series II Math. Phys. Chem. 36 (2001), 37-51. [14] D. Franco and G. Ilardi, On multiosculating spaces, Comm. in Alg. 29(7) (2001), 2961-2976. [15] T. Fujita, A note on scrolls of smallest embedded codimension, alg-geom/9507003. [16] P. Griﬃths and J. Harris, Algebraic geometry and local diﬀerential geometry, Ann. Sc. Ec. Norm. Sup. 12 (1979), 355-432. [17] A. Lanteri and R. Mallavibarrena, Jets of antimulticanonical bundles on Del Pezzo surfaces of degree ≤ 2, Algebraic Geometry, A Volume in Memory of Paolo Francia, De Gruyter (2002), 258–276. [18] B. Segre, Alcune questioni su insiemi ﬁniti di punti in Geometria Algebrica, Atti Conv. Int. Geom. Alg., Torino (1961), 15–33. [19] C. Segre, Preliminari di una teoria delle variet´ a luoghi di spazi, Rend. Circ. Mat. Palermo 31 (1911), 87-121. [20] E. Togliatti, Alcuni esempi di superﬁcie algebriche degli iperspazi che rappresentano un’equazione di Laplace. Comm. Math. Helv. 1 (1929), 255–272. [21] F. L. Zak, Tangents and secants of varieties, Transl. Math. Monog. 127 (1993). Cristiano Bocci Dipartimento di Matematica Universit` a di Milano Via Saldini 50, 20133 MILANO (Italy) Email: [email protected] Luca Chiantini Dipartimento di Scienze Matematiche e Informatiche Universit` a di Siena Pian dei Mantellini 44, 53100 SIENA (Italy) Email: [email protected]

M15 is rationally connected Andrea Bruno and Alesssandro Verra

Abstract. The authors prove that M15 is rationally connected: at ﬁrst they prove that the projection of the canonical model of a general curve of genus 15 from a divisor moving in a pencil of degree 9 lies on a smooth canonical and regular surface. Then they prove the unirationality of M14,2 . Finally they use this result and Lefschetz pencils on the above surfaces in order to connect two general points of M15 via a suitable chain of rational curves. 2000 Mathematics Subject Classiﬁcation: 14H10

1. Introduction Let D be a smooth, irreducible complex projective curve, it is certainly an unexpected property that D moves in a linear system | D | on a smooth irreducible surface S which is not birational to D × P1 . For a curve D of genus g with general moduli such a property is indeed equivalent to the existence of a rational curve R such that [D] ∈ R ⊂ Mg , where Mg is the moduli space of D and [D] denotes the moduli point of D. Due to the fundamental theorem of Eisenbud, Harris and Mumford on the Kodaira dimension of Mg , there is no rational curve through a general point of Mg if g ≥ 23. Therefore we have not to expect the above property for a given curve D of genus g ≥ 23. Of course the existence of a rational curve through a general point of Mg just means that Mg is a uniruled variety. The uniruledness of Mg is somehow expected for g ≤ 23, more precisely it is implied by two famous conjectures: the conjecture that every variety of negative Kodaira dimension is uniruled and the so called slope conjecture on eﬀective divisors of Mg . The latter one implies that Mg has negative Kodaira dimension for g ≤ 22 (see [5] and [3]). In spite of being expected, the uniruledness of Mg for g ≤ 22 persists to be an open problem if 16 ≤ g ≤ 22. Let us brieﬂy recall some history and some known facts about this matter. For g ≤ 14 one knows more: Mg is not only uniruled but also unirational. The proof of the unirationality of Mg goes back to Severi

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for g ≤ 10, while the cases between 11 and 14 admit more recent proofs due to Sernesi (g = 12), Chang and Ran (g = 11, 13) and Verra (g = 14) ([8], [10], [2], [12]). In addition Chang and Ran showed that the Kodaira dimension of Mg is negative for g = 15, 16 and asked about the uniruledness of these moduli spaces (see [2]). In this note we positively answer this question in the case of genus 15. Actually we will show a stronger result: Theorem 1.1. (MAIN) M15 is rationally connected. The proof relies on a property which can be observed for a general curve D of any genus g ≤ 15, namely that there exists an embedding D ⊂ S ⊂ Pr where S is a smooth, regular, canonical surface and | D | is at least 2-dimensional. For instance it is possible to show that a general D of genus g ≤ 10 admits such an embedding in a quintic surface S ⊂ P3 . Moreover it is shown in [12] that, for g = 11, 12, 14, S can be chosen among the few other canonical surfaces which are also complete intersections (the case g = 13, though not covered, admits a completely analogous description). Let D be a general curve of genus 15 and let L be a general line bundle on D having degree 9 and satisfying h0 (L) = 2, it will be shown in section 3 that D ⊂ S ⊂ P6 where S is a complete intersection of four quadrics and OD (D) ∼ = L. Birationally speaking we can consider the moduli space W of pairs (D, L), which is open in the r with r = 1, d = 9, g = 15. Let universal Brill-Noether locus Wd,g h : | D |→ W be the natural map sending A to the isomorphism class of (A, OA (A)). It turns out that h is generically ﬁnite onto its image and that the same is true for f · h, where f : W → M15 is the forgetful map, (see Lemma 4.5). This implies a quite interesting property: through a general point of W always passes a rational surface. In particular it follows that both W and M15 are uniruled. To prove that M15 is rationally connected we consider the divisor Δ0 ⊂ M15 parametrizing isomorphism classes of nodal stable curves of arithmetic genus 15. As is well known Δ0 is dominated by the moduli space M14,2 of 2-pointed curves of genus 14. We show in section 3 that M14,2 is unirational. Hence Δ0 is unirational and the proof of the rational connectedness of M15 can be sketched as follows.

M15 is rationally connected

53

Let xi = [Di ], i = 1, 2, be general in M15 and let Li be a line bundle on Di with deg(Li ) = 9 and h0 (Li ) = 2. As above we have an embedding Di ⊂ Si ⊂ P6 ∼ Li . where Si is a smooth complete intersection of four quadrics and ODi (Di ) = In the 2-dimensional linear system | Di | we can choose a Lefschetz pencil Pi containing Di and Di , where Di is a nodal curve deﬁning a general point xi of Δ0 . In particular this implies that xi is smooth for M15 , Pi deﬁnes an irreducible rational curve Ri in M15 containing xi and xi . Since Δ0 is unirational x1 and x2 are connected by an irreducible rational curve R . Therefore x1 , x2 are connected by a chain R = R1 ∪ R2 ∪ R of irreducible rational curves. Moreover U ∩ R is connected, where U is the regular locus of M15 . Hence M15 is rationally connected. We do not know at the moment whether M15 is unirational, nor if this is plausible. In view of aﬀording the uniruledness of Mg , in the cases where this is unknown, it could be interesting to have some indications about the possible embeddings D ⊂ S ⊂ Pr , where S is a canonical surface and D is general of genus g ∈ [16, 22]. Finally it seems worth to address the following natural Problem 1.2. For which values of g is Mg rationally connected?

2. Curves of degree 19 and genus 15 in P6 In the Hilbert scheme Hilb19,15,6 of all curves of degree 19 and genus 15 of P6 we consider the open set H whose points are stable, non degenerate, smoothable curves and its open subset D = {D ∈ H / h1 (TP6 ⊗ OD ) = 0}. Assume that D = ∅ and consider any D ∈ D. As is well known the condition h1 (TP6 ⊗OD ) = 0 implies that the Kodaira-Spencer map dtD : H 0 (ND ) → H 1 (TD ) is surjective. Since dtD is the tangent map at D of the natural map t : D → M15 , it follows that t is dominant. On the other hand, for these values of the degree and of the genus of D, there exists a unique irreducible component of H which dominates M15 . Therefore D is an irreducible open subset of such a component. We will show in a moment that D is non empty. Previously we want to explain more of the geometric situation. Fix a general curve D of genus 15. Then its Brill-Noether locus W91 (D) = {L ∈ P ic9 (D) / h0 (L) = 2}

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Andrea Bruno and Alessandro Verra

is a smooth, irreducible curve. This indeed follows from the general Brill-Noether theory because the Brill-Noether number ρ(d, g, r) is one if (d, g, r) = (9, 15, 1). It is not diﬃcult to check that, since D is general, the line bundle ωD ⊗ L−1 is very ample and deﬁnes an embedding D ⊂ P6 as a curve of degree 19. D is a general point in the irreducible component of H which dominates M15 . A priori we could have h1 (TP6 ⊗ OD ) = h1 (ND ) ≥ 1 for every D in such a component, but this is not the case if D = ∅. To show that D is non empty we consider the nodal, reducible curve Do = R ∪ A where R is a rational normal sextic curve in P6 , A is a smooth, irreducible, nondegenerate curve of degree 13 and genus 8, and R ∩ A = {z1 , . . . , z8 } =: Z is a set of 8 distinct points on R. Note that these points are in general position, indeed this happens for any set of n+2 distinct points lying on a smooth, irreducible rational normal curve of Pn . Notice also that OA (1) ∼ = ωA (−p) where p is a point. We want now to show that Do ∈ D. Proposition 2.1. h1 (TP6 ⊗ ODo ) = 0. Proof. Tensoring by TP6 the standard exact sequence 0 → ODo → OR ⊕ OA → OZ → 0 and passing to the associated long exact sequence we obtain 0 → H 0 (TP6 ⊗ ODo ) → H 0 (TP6 ⊗ OA ) ⊕ H 0 (TP6 ⊗ OR ) → H 0 (TP6 ⊗ OZ ) → → H 1 (TP6 ⊗ ODo ) → H 1 (TP6 ⊗ OA ) ⊕ H 1 (TP6 ⊗ OR )) → 0. Since OR (1) is non-special, the standard Euler sequence 0 → OR → OR (1)7 → TP6 ⊗ OR → 0 implies that h1 (TP6 ⊗ OR ) = 0. It is also standard that the restriction b : H 0 (TP6 ⊗ OR ) → H 0 (TP6 ⊗ OZ ) is an isomorphism. Indeed the Euler sequence induces a diagram H 0 (OR (1))7

a −→ H 0 (TP6 ⊗ OR )

b −→ H 0 (TP6 ⊗ OZ )

such that a is surjective and b ◦ a is the natural evaluation map. Then b ◦ a is surjective and b is surjective, hence an isomorphism. From the previous arguments it then suﬃces to show that h1 (TP6 ⊗ OA )) = 0. Let A be the canonical model

M15 is rationally connected

55

of A and recall that A is obtained from A by projection from p. Consider the natural diagram 0→ 0→

H 0 (ωA (−p)) ⊗ OA ↓ ωA (−p)

→ →

H 0 (ωA ) ⊗ OA ↓ ωA

→ →

H 0 (ωA |p ) ⊗ OA ↓ ωA |p

→0 →0

where the ﬁrst two vertical arrows are given by evaluation and are both surjective. If we dualize and twist the exact sequence that one obtains from the snake lemma we obtain the sequence 0 → OA (1) → TP7 ⊗ OA (−p) → TP6 ⊗ OA → 0. We are then reduced to show that h1 (TP7 ⊗OA (−p)) = 0, but this is a consequence of the fact that, since A is canonical, h1 (TP7 ⊗ OA ) = 0 and we can factor the

natural surjection H 0 (P7 , TP7 ) → TP7 |p → 0 through H 0 (TP7 ⊗ OA ) → TP7 |p . The next theorem is a well known consequence of the vanishing of h1 (TP6 ⊗ ODo ), (see [4] 1.1). Theorem 2.2. Do is smoothable. The theorem implies that the natural morphism t : D → M15 is dominant. Let W be the moduli space of pairs (D, L) such that D is a general curve of genus 15 and L ∈ W91 (D), in addition we can consider the natural morphism u:D→W sending D ∈ D to the moduli point of (D, L), where L = ωD (−1). It is clear from the previous results and remarks that u is also dominant. So we can summarize the main achievements of this section as follows: Corollary 2.3. D is non-empty and dominates both W and M15 .

3. Embedding D in a smooth (2,2,2,2) complete intersection of P6 . In this section we show that a general D ∈ D is embedded in a smooth complete intersection of 4 quadrics and that h0 (ID (2)) = 4, where ID is the ideal sheaf of D. We will use a reducible curve Do = R ∪ A of the type considered in the previous section, proving that Do embeds in a reducible complete intersection of 4 quadrics So and that the pair (Do , So ) is smoothable. We start with a smooth, non-degenerate sextic Del Pezzo surface Y ⊂ P6 . The ideal sheaf IY of Y is generated by quadrics: this implies the next property.

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Lemma 3.1. The intersection scheme of four general quadrics containing Y is a reduced surface X ∪ Y where X is a smooth, irreducible component. Moreover the intersection scheme of X and Y is a smooth, irreducible curve B = X ∩ Y . Proof. Let σ : P → P6 be the blowing up of Y , E the exceptional divisor of σ, H the pull-back of a hyperplane by σ. Since IY is generated by quadrics the strict transform of | IY (2) | is a base point free linear system and coincides with | 2H − E | . By Bertini’s theorem the intersection of 4 general elements Q1 , . . . , Q4 ∈ | 2H − E | is a smooth surface X . Since (2H − E)6 = 4 is positive, X is connected. Furthermore we can assume that B = Q1 ∩ · · · ∩ Q4 ∩ E is a smooth, connected curve and that σ/B : B → P6 is an embedding. Then σ/X : X → P6 is an embedding too and X = σ(X ) has the following properties: (i) X ∪ Y is the complete intersection of the quadrics Qi = σ(Qi ), i = 1, . . . , 4. (ii) B = σ(B ) is the intersection scheme of X and Y . This completes the proof.

The complete intersection of four quadrics X∪Y is a canonical surface, i.e. ωX∪Y ∼ = OX∪Y (1). This implies that ωY (B) ∼ = OY (1). Since ωY ∼ = OY (−1) = ωX∪Y ⊗ OY ∼ it follows that B ∈| OY (2) | is a quadratic section of Y and a canonical curve of genus 7. The surface X is described in detail in [12], in particular X is obtained from the blowing up σ : X → P2 of 11 points e1 , . . . , e11 in general position. Let Ei be the exceptional divisor over ei , L the pull-back of a line by σ, H a hyperplane section of X then | H |=| 6L − 2(E1 + · · · + E5 ) − E6 − · · · − E11 | . It is easy to see that a general curve R ∈| 2L − E1 − E2 − E10 − E11 | is a smooth, irreducible rational sextic curve. Proposition 3.2. R is non degenerate. Proof. R is non degenerate iﬀ | H − R |=| 4L − E1 − E2 − 2E3 − 2E4 − 2E5 − E6 − E7 − E8 − E9 | is empty. Note that (H − R) · E10 = (H − R) · E11 = 0 and that (H − R)2 = −2. Then consider the contraction f : X → X of E10 and E11 . 2 Let C = f∗ C with C ∈| H − R |, then C = C 2 = −2. As is well known there is no eﬀective −2 curve on the blowing up of P2 in n ≤ 9 general points. Hence C cannot exist and | H − R | is empty.

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57

Notice also that ωX (B) ∼ = OX (H) so that H − B ∼ −3L + E1 + · · · + E11 is the canonical class. Proposition 3.3. Let IY ∪R be the ideal sheaf of Y ∪ R. Then h0 (JY ∪R (2)) = 4. Proof. Since X ∪ Y is the complete intersection of four quadrics and X ∩ Y = B, it suﬃces to show that h0 (OX (2H − B − R)) = 0. Note that 2H − B − R ∼ L−E3 −E4 −E5 +E10 +E11 . Moreover (2H −B −R)E10 = (2H −B −R)E11 = −1. This implies that an eﬀective C ∈| 2H − B − R | contains E10 and E11 . But then C = C + E10 + E11 with C ∈| L − E3 − E4 − E5 |. This is a contradiction because, since the points e3 , e4 , e5 are not collinear, | L − E3 − E4 − E5 | is empty. Hence

| 2H − B − R | is empty and h0 (OX (2H − B − R)) = 0. Proposition 3.4. | OB (R) | is a base point free pencil of degree 8. Proof. One easily computes R · B = 8. Note that KX − (R − B) ∼ H − R so that, by Serre duality, hi (OX (R − B)) = h2−i (OX (H − R)). Since R is irreducible and R ·(R −B) = −8, we have h2 (OX (H −R)) = 0. On the other hand Proposition 3.2 implies that h0 (OX (H − R)) = 0. Since χ(OX (H − R)) = 0, it follows h1 (OX (H − R)) = 0 and hence hi (OX (R − B)) = 0 for i = 0, 1, 2. Then the statement follows considering the long exact sequence of 0 → OX (R − B) → OX (R) → OB (R) → 0.

Now we consider on Y the linear system |B+N | where N is one of the 6 lines contained in Y . We have B · N = 2, pa (B + N ) = 8, deg(B + N ) = 13. Since B is a quadratic section of Y we have also h1 (OY (B)) = 0 and the exact sequence 0 → H 0 (OY (B)) → H 0 (OY (B + N )) → H 0 (ON (B)) → 0. It easily follows from the sequence that | B + N | is base point free and that its general element is smooth, irreducible. As above we consider a general R in the pencil | 2L − E1 − E2 − E10 − E11 |. Since | OB (R) | is base point free of degree 8 and R is a rational normal sextic, we can assume that the intersection scheme Z = B ∩ R is smooth and supported on 8 points in general position. Moreover we can also assume that B ∩ N and Z do not intersect. Let IZ/Y be the ideal sheaf of Z in Y . We have the following: Proposition 3.5. The base locus of | IZ/Y (B + N ) | is Z.

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Proof. Note that B+N is smooth along Z and that the linear system | OB (B+N − Z) | is base point free. Then the statement follows if the restriction ρ :| B + N |→ | OB (B + N ) | is surjective. Since h1 (OY (N )) = 0, the surjectivty of ρ follows from the long exact sequence of 0 → OY (N ) → OY (B + N ) → OB (B + N ) → 0.

The proposition implies that a general A ∈ | IZ (B + N ) | is a smooth, irreducible curve of degree 13 and genus 8. Moreover the curve Do = A ∪ R is nodal with deg(Do ) = 19, pa (Do ) = 15, Sing (Do ) = Z. We know from section 2 that Do ∈ D, where D is the open subset of the Hilbert scheme of Do parametrizing nodal, non degenerate, smoothable curves D satisfying h1 (TP6 ⊗ OD ) = 0. Proposition 3.6. h0 (IDo (2)) = 4 and hi (IDo (2)) = 0, i > 0. Proof. We have h1 (ODo (2)) = 0 and h0 (ODo (2)) = 24, this can be easily proved considering the standard exact sequence 0 → ODo (2) → OA (2) ⊕ OR (2) → OZ (2) → 0 and its associated long exact sequence. Then, by the standard exact sequence 0 → IDo (2) → OP6 (2) → ODo (2) → 0, it follows that hi (IDo (2)) = 0 for i > 1 and that h1 (IDo (2)) = 0 if and only if h0 (IDo (2)) = 4. Finally, we observe that the natural inclusion | IY ∪R (2) | ⊆ | IDo (2) | is an equality. Indeed let Q be a quadric containing Do , then Q ∩ Y contains A. But A is linearly equivalent to B + N where B is a quadratic section of Y and N is eﬀective. Therefore Q contains Y . Since R ⊂ Do , it follows that Q contains Y ∪ R and hence h0 (IDo (2)) = h0 (IY ∪R (2)). By Proposition 3.3

h0 (IY ∪R (2)) = 4 and this completes the proof. The proposition implies that a general D ∈ D is contained in a unique complete intersection S of four quadrics. More precisely one can show the following: Theorem 3.7. For a general D ∈ D one has D ⊂ S, where S is a smooth complete intersection of 4 quadrics. Moreover it holds h0 (ID (2)) = 4 for the ideal sheaf ID of D. Proof. The curve Do is contained in X ∪ Y which is a complete intersection of 4 quadrics. Moreover, by Proposition 3.6, its ideal sheaf IDo satisﬁes h0 (IDo (2)) = 4

M15 is rationally connected

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and hi (IDo (2)) = 0, i > 0. Then, by standard semicontinuity arguments, a general D ∈ D satisﬁes the same properties. In particular the base locus S of | ID (2) | is a complete intersection of 4 quadrics. It remains to show that S is smooth. Let us consider the standard exact diagram of tangent and normal bundles, 0

→

0

→

T Do ↓ TSo ⊗ ODo

→ →

TP6 ⊗ ODo ↓ TP6 ⊗ ODo

→ →

ND o ↓ NSo ⊗ ODo

→ →

TZ1 ↓ TS1o ⊗ ODo

→0 →0

where So = X ∪ Y and TZ1 , TS1o are the sheaves deﬁned by the T 1 -functor of Lichtenbaum-Schlessinger. TZ1 and TS1o are respectively supported on the singular loci of Do and of So , it is easy to check that TZ1 = OZ and that TS1o is a line bundle on B. Furthermore we have TS1o ⊗ ODo = ODo ∩B and the vertical arrow TZ1 → TS1o ⊗ ODo is the natural injection OZ → ODo ∩B . We know that Do is smoothable, let δ ∈ H 0 (NDo ) be an inﬁnitesimal deformation which is induced by an eﬀective smoothing of Do . Then, as is well known, the image of δ in TZ1 is non-zero at each z ∈ Z (see [9]). Let δ1 ∈ H 0 (NSo ⊗ ODo ) be the image of δ. Then, by the commutativity of the diagram, the image of δ1 in TS1o ⊗ ODo is non zero at each z ∈ Z ⊂ Do ∩ B. On the other hand every x ∈ Do ∩ B − Z is smooth for Do and it is obvious that we can choose δ such that δ1 (x) = 0. Finally we observe that the restriction map r : H 0 (NSo ) → H 0 (NSo ⊗ ODo ) is surjective. To see this it suﬃces to tensor by NSo the standard exact sequence 0 → IDo /So → OSo → ODo → 0. Passing to the long exact sequence the surjectivity of r follows if h1 (IDo /So ⊗ NSo ) = 0. Now NSo = OSo (2)4 because So is the complete intersection of 4 quadrics. So it suﬃces to show that h1 (IDo /So (2)) = 0. This easily follows from the long exact sequence of 0 → ISo (2) → IDo (2) → IDo /So (2) → 0 and Proposition 3.6. Since r is surjective, δ1 lifts to an inﬁnitesimal deformation σ ∈ H 0 (NSo ). By the commutativity of the diagram the image of σ in H 0 (TS1o ) is not zero at each x ∈ B ∩ Do . Finally let (St , Dt ), t ∈ T, be an eﬀective deformation of (So , Do ) induced by σ. Then such a deformation smooths Z = Sing(Do ) so that Dt is smooth. Moreover it smooths Do ∩ B as a subset of B = Sing(So ). This implies that St has at most ﬁnitely many singular

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points and that they are not in Dt for t = o. If t = o, then it is easy to show that

(Dt , St ) deforms to a pair (D, S) such that Sing(S) is empty.

4. Proof of the main theorem. In this section we prove the main theorem of this paper, i.e. that M15 is rationally connected. At ﬁrst we need to show that the moduli space M14,2 of 2-pointed curves of genus 14 is unirational. With this purpose we consider the Hilbert scheme Hilb14,8,6 of curves in P6 having degree 14 and arithmetic genus 8. It is known that a non-degenerate, linearly normal, smooth, irreducible curve C ⊂ P6 of degree 14 and genus 8 is projectively normal and generated by quadrics ([12]). Let C ⊂ Hilb14,8,6 be the open subset parametrizing all curves with the above properties. Then (see [12]): Proposition 4.1. C is irreducible and unirational. It is standard to construct a projective bundle h : P → C such that the ﬁbre of P at C is PC =| IC (2) |, IC being the ideal sheaf of C. Since C is projectively normal, dim(PC ) = 6. We have: Proposition 4.2. Let V ⊂ PC be a general 4-dimensional linear system of a general C ∈ C, then the base locus of V is C ∪ D, where D is a smooth, irreducible curve of genus 14 and degree 18. Conversely let D ⊂ P6 be a curve of genus 14 and degree 18 with general moduli, then D is in the base locus of V for some pair (C, V ).

Proof. See [12]. Using the previous results we can easily construct a rational dominant map φ : C × P6 × P6 → M14,2 .

Indeed let (C, x, y) ∈ C × P6 × P6 be a suﬃciently general element. Then we can assume that x, y are not in C and moreover that the linear system V of all quadrics containing C ∪ {x, y} is 4-dimensional. In view of the previous theorem we can

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also assume that the base locus of V is C ∪ D where D is a general curve of genus 14. Then we deﬁne φ by setting φ(C, x, y) = [D, x, y] where [D, x, y] denotes the moduli of the 2-pointed curve (D, x, y). It is clear, by Proposition 4.2, that φ is dominant. Since C is unirational, it follows that Theorem 4.3. M14,2 is unirational. Now we consider the divisor Δ0 ⊂ M15 parametrizing stable singular curves of arithmetic genus 15. As is well known, there exists a natural rational map of degree two ψ : M14,2 → Δ0 sending [D, x, y] to the moduli point of the stable curve obtained from D by glueing x to y. In particular we have: Corollary 4.4. Δ0 is unirational. Let D ∈ D be a general smooth curve of degree 19 and genus 15, we know from section 3 that D⊂S where S is a smooth complete intersection of four quadrics. Since D is not degenerated, the sheaf OD (1) is special and ωD (−1) is an element of the Brill-Noether locus W91 (D). We want to point out that | ωD (−1) | is a base-point-free pencil and that D is linearly normal. This follows because a general D ∈ D has general moduli. Then D has no g92 nor a gk1 with k ≤ 8 and hence | ωD (−1) | is a basepoint-free g91 . Moreover, by Riemann Roch, dim(| OD (1) |) = 6, therefore D is linearly normal. Since S is a canonical surface we have ωD (−1) ∼ = OD (D). Then, from the standard, exact sequence 0 → OS → OS (D) → OD (D) → 0 it follows that dim(| D |) = 2.

(4.1)

Notice also that | D | is base point free, because | OD (D) | is base point free. We expect that a general pencil in | D | is a Lefschetz pencil and that a singular element of | D | deﬁnes a general point of Δo . We will see that this is actually true. Lemma 4.5. Let D be as above, then a general singular element of | D | is an irreducible curve with exactly one ordinary node and no other singularity.

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Proof. Let f : S → P2 be the covering of degree 9 deﬁned by | D |. Since D is general, the linear series | OD (D) | has simple ramiﬁcation. Hence the branch curve B of f is reduced. This implies that a general tangent line to B intersects B transversally except for the tangency point. Hence a general singular element of | D | is integral with exactly one ordinary node.

Let us recall once more that a general smooth element D ∈ D has the following properties: (i) h0 (ID (2)) = 4 and D ⊂ S, where S is a smooth complete intersection of 4 quadrics. (ii) | ωD (−1) | is a base-point-free pencil and D is linearly normal. (iii) The Petri map μD : H 0 (ωD (−1)) ⊗ H 0 (OD (1)) → H 0 (ωD ) is injective. (iv) The family of the singular, irreducible curves Γ ∈| D | having an ordinary node as a unique singularity is non-empty, hence 1-dimensional. (i) holds by Theorem 3.6 and (iv) by the previous lemma, while (ii) has been just remarked above. The injectivity of μD follows from Gieseker-Petri theorem because D has general moduli, notice also that the Brill-Noether locus W91 (D) is a smooth, irreducible curve. The family of all curves Γ as in (iv) will be denoted as Do . Lemma 4.6. For each Γ ∈ Do (ii) holds for Γ and the Petri map μΓ is injective. ∼ OC (C) for any curve Proof. Recall that S is a canonical surface, so that ωC (−1) = C ⊂ S. A general D ∈| Γ | satisﬁes conditions (ii) and (iii), in particular | OD (D) | is a base-point-free pencil. Then it follows from the standard long exact sequence 0 → H 0 (OS ) → H 0 (OS (D)) → H 0 (OD (D)) → 0 that | D | is 2-dimensional and base-point-free. Replacing OD by OΓ in the above sequence, it follows that | ωΓ (−1) | is a base-point-free pencil. Let H be a hyperplane section of S, by Serre duality h1 (OS (H)) = h1 (OS ) = 0. Hence we have the long exact sequence 0 → H 0 (OS (H − D)) → H 0 (OS (H)) → H 0 (OD (H)) → H 1 (OS (H − D)) → 0. Since a general D ∈| Γ | is non degenerate and linearly normal, it follows that h0 (OS (H − D)) = h1 (OS (H − D)) = 0. Then, replacing as above OD by OΓ , we deduce that Γ is non degenerate and linearly normal. It remains to show that the Petri map μΓ : H 0 (OΓ (D)) ⊗ H 0 (OΓ (H)) → H 0 (OΓ (H + D))

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is injective. By the base-point-free pencil trick we have Ker μΓ ∼ = H 0 (OΓ (H −D)), 0 see [1], p.126. For the same trick we have dim Ker μD = h (OD (H − D)) for each D ∈| Γ |. Since μD is injective for a general D, the map D → h0 (OD (H − D)) is generically zero on | Γ |. Since h1 (OS (H − D) = 0, the standard exact sequence 0 → OS (H − 2D) → OS (H − D) → OD (H − D) → 0 yields the long exact sequence 0 → H 0 (OD (H − D)) → H 1 (OS (H − 2D)) → 0 → . . . . Then we have h0 (OΓ (H − D)) = h1 (OS (H − 2D)) = 0 and μΓ is injective.

Let D be the closure of D in the Hilbert scheme, then D contains the dense open set U parametrizing those integral curves D ∈ D which satisfy the previous conditions (i), (ii), (iii), (iv) and have at most one ordinary node as their only singularity. Note that Do ⊂ U as a divisor. Indeed (i) implies that U is ruled by the family of surfaces UD := U ∩ | D |, where D ∈ U . Moreover UD ∩ D0 is pure of dimension 1. We consider the natural map f : U → M15 . Since U is open in D the map f is dominant, we want to show something more: Proposition 4.7. The map f /D0 : D0 → Δ0 is dominant. Proof. It suﬃces to show that f : U → M15 has ﬁbres of constant dimension. Then, since D0 is a divisor in U and f (D0 ) is contained in the irreducible divisor Δ0 , the statement follows from a count of dimensions. For any D ∈ U let us consider the ﬁbre FD = f −1 f (D) and the natural morphism h : FD → W91 (D) ⊂ P ic9 (D) sending D ∈ FD to the point ωD (−1) of the Brill-Noether locus W91 (D). We know that D is linearly normal, therefore h−1 h(D ) is just the family of all curves projectively equivalent to D . In particular dim(h−1 h(D )) = dim(P GL(7)). On the other hand the Petri map μD : H 0 (ωD (−1)) ⊗ H 0 (OD (1) → H 0 (ωD ) is injective, i.e. its corank is one. This implies that W91 (D) is smooth of dimension one at its point ωD (−1). Let L be in a small neighborhood N of ωD (−1) in W91 (D). Then, by standard semicontinuity arguments, we can assume that | L | is a base-point-free pencil and that ωD ⊗ L−1 deﬁnes an embedding of D in P6 as a linearly normal curve D”. By the same semicontinuity arguments we can also assume that D” ∈ U so that h(D”) = L. But then N ⊂ h(FD ) and it

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follows that each irreducible component of h(FD ) is a curve. This implies that

dim(FD ) = dim(P GL(7)) + 1 = 49, for each ﬁbre FD . PROOF OF THE MAIN THEOREM. We can ﬁnally conclude this note by proving that M15 is rationally connected. Fix two general points [D1 ] and [D2 ] in M15 , then ﬁx a general Li in the Brill-Noether locus W91 (Di ), i = 1, 2. Consider the embedding Di ⊂ P6 deﬁned by the line bundle ωDi ⊗ L−1 i . Di is a general element of D, in particular Di ⊂ Si where Si is a smooth complete intersection of 4 quadrics. Take a general pencil Pi ⊂| Di | containing Di and consider one element Di0 ∈ Pi which is a singular curve. Then [Di0 ] is general in Δ0 and hence smooth for M15 . Since Δ0 is unirational, there exists an irreducible rational curve R containing [D1 ]0 and [D20 ]. Let Ri be the image of Pi in M15 and let U ⊂ M15 be the open set of regular points. Then U ∩ (R1 ∪ R2 ∪ R) is a connnected chain of rational curves joining [D1 ] to [D2 ]. This implies the

rational connectedness of M15 . Remark 4.8. The unirationality of W and M15 would follow if there exists a unirational variety V ⊂ D which intersect a general | D | in ﬁnitely many points. It is not clear to us whether such a V exists.

References [1] E. Arbarello, M. Cornalba, P. Griﬃths, J. Harris, Geometry of Algebraic Curves I, Springer-Verlag, Berlin (1984). [2] M.C. Chang, Z. Ran, Unirationality of the moduli space of curves of genus 11, 13 (and 12), Invent. Math. 76 (1984), 41-54. [3] G. Farkas, M. Popa, Eﬀective divisors on Mg and a counterexample to the Slope Conjecture, preprint (2002). [4] R. Hartshorne, A. Hirschowitz, Smoothing Algebraic Space Curves, in Algebraic Geometry, Sitges 1983 (E. Casas-Alvero, G.E. Welters, S. Xambo-Descamps eds.), Springer Lect. Notes 1124 (1985), 98-131. [5] J. Harris, I. Morrison, Moduli of Curves, Springer-Verlag Berlin (1991). [6] J. Harris, I. Morrison, Slopes of eﬀective divisors on the moduli space of curves, Invent. Math. 99 (1990), 321-335.

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[7] S. Mori, S. Mukai, The uniruledness of the moduli space of curves of genus 11, in Algebraic Geometry, Proceedings Tokio/Kyoto 1982 (M.Raynaud, T. Shioda eds.), Springer Lect. Notes 1016 (1983), 334-353. [8] E. Sernesi, L’unirazionalit´ a della variet´ a dei moduli delle curve di genere 12, Ann. Sc. Norm. Sup. Pisa 8 (1981), 405-439. [9] E. Sernesi, On the existence of certain families of curves, Invent. Math. 75 (1984), 25-57 [10] F. Severi, Vorlesungen u ¨ber Algebraische Geometrie, (translated by E. Loeﬄer), Teubner, Leipzig (1921). [11] F.O. Schreyer, F. Tonoli, Needles in a haystack: special varieties via small ﬁelds, in Mathematical computations with Macaulay 2, (D. Eisenbud, D. Grayson, M. Stillman, B. Sturmfels eds.), Springer-Verlag, Berlin (2002). [12] A. Verra, The unirationality of the moduli space of curves of genus g ≤ 14, to appear in Compositio Math. (2005). Andrea Bruno Dipartimento di Matematica Universit` a di Roma III Largo Murialdo, 00146 Roma, Italy Email: [email protected] Alessandro Verra Dipartimento di Matematica Universit` a di Roma III Largo Murialdo, 00146 Roma, Italy Email: [email protected]

Codimension one decompositions and Chow varieties Enrico Carlini∗

Abstract. A presentation of a degree d form in n+1 variables as the sum of homogenous elements “essentially” involving n variables is called a codimension one decomposition. Codimension one decompositions are introduced and the related Waring Problem is stated and solved. Natural schemes describing the codimension one decompositions of a generic form are deﬁned. Dimension and degree formulae for these schemes are derived when the number of summands is the minimal one; in the zero dimensional case the scheme is showed to be reduced. These results are obtained by studying the Chow variety Δn,s of zero dimensional degree s cycles in Pn . In particular, an explicit formula for deg Δn,s is determined. 2000 Mathematics Subject Classiﬁcation: 14N05, 14N10

1. Introduction Usually, a homogeneous element in the polynomial ring S = C[X0 , . . . , Xn ] is presented as a sum of monomials. In other words, we use the homogeneous structure to choose a vector space basis in each homogenous piece Sd of S. Actually, we may want to write down f ∈ Sd in diﬀerent ways and this can be done also without selecting a vector space basis. Sum of powers decompositions (see [16]) are just an example: f = l1d + . . . + lsd , l1 , . . . , ls ∈ S1 . Sum of powers presentations have been widely studied classically in an attempt to produce a classiﬁcation of homogeneous polynomials. The idea was to mimic what happens for quadratic forms and diagonalization (this has not been very successful, e.g. no eﬀective algorithm is known to perform a sum of powers decomposition if d > 3 and n > 1). Nevertheless, information can be obtained on f and on its zero locus by studying properties of its sum of powers decompositions (see [27], e.g., page 252). In particular, it is useful to know how many summands are needed. ∗ The author is a member of the PRIN01 project “Spazi di moduli e teoria di Lie” of the University of Pavia.

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n+d 1 For a generic f ∈ Sd , a parameters count shows that at least n+1 n summands are needed for a sum of powers presentation of f and before Clebsch’s paper this number was believed to be always enough. In [8], it is shown that a generic ternary quartic (i.e. n = 2, d = 4) is the sum of 6 and not of 5 powers of linear forms. The existence of defective cases makes Waring Problem for forms so interesting: For each pair (n, d) determine the minimal number of summands appearing in the sum of powers decompositions of the generic form of degree d in n + 1 variables. Defective pairs, such as Clebsch’s, were readily discovered, but the problem remained unsolved for almost one century. The complete answer was only recently found by Alexander and Hirschowitz ([1]): Theorem n+d 1.1. A generic form of degree d in n + 1 variables is the sum of s = 1 n+1 d powers of linear forms, unless • d = 2, where s = n + 1 instead of n+2 2 , or • d = 4 and n = 2, 3, 4, where s = 6, 10, 15 instead of 5, 9, 14 respectively, or • d = 3 and n = 4, where s = 8 instead of 7. As ld is a homogeneous element in the univariate ring C[l], a sum of powers decomposition can be viewed as a presentation of a form as sum of forms “essentially” involving one variable. With this in mind is natural to consider other presentations of this kind, e.g. binary decompositions, where the summands essentially involve two variables (see [2] and [3]). We can also move to the opposite end of the spectrum and consider codimension one decompositions, where the summands essentially involve one variable less than the original form. The study of codimension one decompositions is the object of this paper. In section 2, we will address and solve the analogous Waring type problem obtaining the following results: Theorem 1.2. The generic form of degree d in n + 1 variables is the sum of ≥ 0} codimension one forms and no fewer. min{s : ns − d−s+n n Corollary 1.3. Let n ≥ 2. The minimal number of summands appearing in the codimension one decompositions of the generic form of degree d in n + 1 variables is the expected one (see Deﬁnitions 2.5 and 2.7) if and only if d = 2, 3 for any n ≥ 2 or d = 4, 5, 6 and 8 for n = 2.

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In particular, the corollary shows how codimension one and sum of powers decompositions are deeply diﬀerent. In the sum of powers case the expected number of summands almost always works with some exceptions, as shown in Theorem 1.1. But, for codimension one decompositions, exactly the opposite happens: only in some cases the expected number of summands works and almost all the pairs are defective. In section 3, we introduce and study a natural scheme, VSH (see Deﬁnition 3.1), describing the codimension one decompositions of a generic form. This is mostly done in the spirit of [23] and we obtain dimension and degree formulae: Theorem 1.4. Let F ⊂ Pn be a generic degree d hypersurface and let s = smin (n, d) (see Deﬁnition 2.5), then ; • dim VSH(F, s) = ns − d−s+n n ns−1 n(s−1)−1 • deg VSH(F, s) = n−1 · · . . . · 1. n−1 In the zero dimensional cases we also get reducedness: Proposition 1.5. Let F ⊂ Pn be a generic degree d hypersurface and let s = smin (n, d) (see Deﬁnition 2.5). If VSH(F, s) is zero dimensional, then it is reduced. These results are obtained through a careful study of the Chow variety Δn,s and of some special linear sections of this. In particular, we obtain a description of the tangent space in a generic point and an explicit degree formula (see Proposition 3.4). En passant, we recall the connection with higher secant varieties. It is well known that sums of powers decompositions are deeply related with the study of higher secant varieties of Veronese varieties, e.g. see [13]. The same is true for codimension one decompositions (see Remark 2.6 and what follows) and, in general, for additive decompositions, but we have to replace the Veronese variety with other suitable objects (e.g., in this paper, we use the varieties Vˆn,d , see Remark 2.4). Interest in the study of higher secant varieties of classical varieties (Segre varieties, Grassmannians) has been recently renewed in [4],[5] and [17]. But also new families of non-classical varieties are of interest as shown here and in [10] in connection with Algebraic Statistic. I wish to thank Aldo Conca and Anthony Geramita for their help with the algebraic claim in the proof of Proposition 3.4: the latter for giving me an idea of a proof and the former for showing me a much better proof than the one I had. The referee’s comments were extremely useful. In particular, I wish to thank her/him for showing to me the connection with Chow varieties. The hospitality of the Mathematics Department of the University of Genoa and the ﬁnancial support of the Mathematics Department of the University of Pavia were appreciated. Notation: we work with the polynomial ring S = C[X0 , . . . , Xn ] and its ring of diﬀerential operators T = C[∂0 , . . . , ∂n ], i.e. ∂i acts as the partial derivation n ∂ ∂xi . In particular, S1 and T1 are dual to each other and we let P = PT1 and

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ˇ n = PS1 . A form f ∈ Sd deﬁnes a hypersurface F = V (f ) ⊂ Pn and linear spaces P (f ⊥ )s ⊂ PTs which we will denote as F ⊥ ⊂ PTs with abuse of notation. We work over the complex number ﬁeld C, but any algebraically closed ﬁeld of characteristic 0 could be used instead (in positive characteristic problems arise because of the coeﬃcients produced by diﬀerentiating).

2. Waring Problem for codimension one decompositions In what follows we need some basic facts about apolarity theory (see [13]). In particular, we consider the polynomial rings S = C[X0 , . . . , Xn ] and T = C[∂0 , . . . , ∂n ] where S has a T -module structure given by the diﬀerentiation action, which we denote with “◦”. Given a form f ∈ Sd , f ⊥ = {D ∈ T : D◦f = 0} ⊂ T denotes the homogeneous ideal of derivations annihilating f and T /f ⊥ is an artinian Gorenstein ring with socle in degree d. Given a derivation D ∈ Td , D−1 = {f ∈ S : D ◦ f = 0} is a graduated sub-T -module. The apolarity pairing Sd × Td → C is perfect. The link between apolarity and the study of polynomial decompositions is given by the classical Apolarity Lemma (for a proof see [23]): Lemma 2.1 (Apolarity Lemma). Let f ∈ Sd , then the following are equivalent: 1. f = l1d + . . . + lsd , where the li ’s are pairwise non-proportional linear forms; ˇ n. 2. f ⊥ ⊃ IX , where IX is the ideal of the set of s points X = {l1 , . . . , ls } ⊂ P Deﬁnition 2.2. Let g ∈ Sd , then g is called a codimension one form if (g ⊥ )1 = 0. Given a form f ∈ Sd , a codimension one decomposition of f is a presentation f = fˆ1 + . . . + fˆs where fˆi , i = 1, . . . , s, are codimension one forms of degree d. Remark 2.3. If g ∈ Sd is a codimension one form, then there exists a linear change of variables Xi → Yi , i = 0, . . . , n, such that g(Y ) only involves n variables, i.e. g(Y ) ∈ C[Y1 , . . . , Yn ]. Remark 2.4. Codimension one forms can be nicely described in geometric terms. ˇ n → PSd be the d-uple embedding and denote by νd (H) ⊂ PSd Let νd : PS1 = P ˇ n . Then the variety the linear span of the image of a hyperplane H ⊂ P νd (H) Vˆn,d = H∈Pn

parameterizes the codimension one forms in PSd . Notice that Vˆn,d is a determi matrix of nantal variety deﬁned by the maximal minors of a (n + 1) × n+d−1 n

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linear forms (which is also a catalecticant matrix, But, in general, it is see [14]). not standard determinantal and dim Vˆn,d = n + d+n−1 . n−1 No algorithm is known to determine a codimension one decomposition of a given form, thus it is interesting to study quantitative aspects of such a presentation, e.g. the number of summands s. As in the case of sum of powers decompositions, we begin by studying the behavior of generic forms (see Remark 2.6): Deﬁnition 2.5. smin (n, d) is the minimal number of summands appearing in the codimension one decompositions of the generic form of degree d in n + 1 variables. Remark 2.6. To make clearer what we mean by generic, it is useful to recall some geometry. Let Sect (Vˆn,d ) be the variety of t + 1 secants, i.e. the closure of the union of the Pt ’s spanned by points of Vˆn,d . Then, the generic f ∈ Sect (Vˆn,d ) is a sum of t + 1 codimension one forms. In these terms, t = smin (n, d) − 1 is the smallest integer such that Sect (Vˆn,d ) = PSd . Studying the decomposition of any form one completely loses this nice geometric interpretation and the problem gets considerably harder (even in the sums of power case no complete solution is known!). In this paper we will only deal with the generic case. An estimate for smin can be easily determined. As dim Secs−1 (Vˆn,d ) ≤ s dim Vˆn,d + s − 1 the condition Secs−1 (Vˆn,d ) = PSd gives an inequality and solving it we get 1 d+n smin (n, d) ≥ . n n + d+n−1 n−1 Deﬁnition 2.7. The expected value of smin (n, d) is sexp (n, d) =

d+n . n n + d+n−1 n−1 1

If smin (n, d) = sexp (n, d) the pair (n, d) is said to be defective. The Waring Problem for codimension one decompositions can be stated as follows: For each pair (n, d) determine the minimal number of codimension one forms needed for the decomposition of the generic form of degree d in n + 1 variables, i.e. compute smin (n, d). Apolarity provides us with a strong tool to study codimension one decompositions: Lemma 2.8 (Codimension One Lemma). Let f ∈ Sd . Then the following are equivalent:

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1. f = fˆ1 + . . . + fˆs , where the fˆi ’s are codimension one forms of degree d such ˆ ⊥ that (fˆi )⊥ 1 = (fj )1 for i = j; 2. there exists L1 · . . . · Ls ∈ f ⊥ , where the Li ’s are pairwise non-proportional linear forms. Proof. If f admits a codimension one decomposition with the property above, choose non-proportional linear forms Li ∈ (fˆi )⊥ 1 , i = 1, . . . , s. Then (L1 · . . . · Ls ) ◦ f = 0 and the claim follows. Conversely, assume that there exists L1 · . . . · Ls ∈ f ⊥ such that the hyperplanes {Li = 0}, i = 1, . . . , s, are distinct and choose Nn,d = d+n−1 generic points on each of them. If we denote by X the resulting set of n−1 sNn,d points, then its deﬁning ideal IX satisﬁes (IX )t ⊆ (f ⊥ )t , for t > d. But the inclusion also holds for t ≤ d by Bezout’s theorem. Thus, the Apolarity Lemma yields f=

n,d s N

(lij )d

i=1 j=1

Nn,d where Li ◦ ( j=1 (lij )d ) = 0, i = 1, . . . , s, by construction. Hence the claim holds.

Remark 2.9. This lemma produces a useful bound for smin almost without eﬀort. Given a form f ∈ Sd the ring T /f ⊥ is known to be artinian Gorenstein and to have socle in degree d. In particular, for a generic form, f ⊥ is generated in degree no smaller than d+1 2 and hence the Codimension One Lemma implies d+1 that smin (n, d) ≥ 2 . We will use Lemma 2.8 and basic Algebraic Geometry techniques to give an answer to the Waring Problem for codimension one decompositions. Deﬁnition 2.10. Let Δn,s ⊂ PTs be the variety of totally decomposable forms of degree s in n + 1 variables, i.e. a point of Δn,s represents a form which can be written as the product of s linear forms. Remark 2.11. Δn,s is the Chow variety of zero-dimensional degree s cycles in Pn and in these terms it has been widely studied in [12]. In particular, it is shown there how to ﬁnd equations for Δn,s set-theoretically, but it is known that these equations do not generate the deﬁning ideal. Remark 2.12. If we consider the symmetrization of the Segre product s

PT1 × . . . × PT1 and we embed it in PTs , then we get Δn,s . In particular, this shows that dim Δn,s = ns.

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With this setting we can develop a strategy to study codimension one decompositions in general. Given f ∈ Sd , we consider the smooth points of Δn,s lying in the linear space (f ⊥ )s : if there are any, then f is the sum of s codimension one forms, otherwise it is not. Using this we get a formula for smin , thus solving the Waring Problem for codimension one decompositions. Theorem 2.13. The generic form of degree d in n + 1 variables is the sum of ≥ 0} codimension one forms and no fewer. min{s : ns − d−s+n n Proof. The key part of the proof is Lemma 2.8: a form f ∈ Sd is the sum of s codimension one forms if and only if f ⊥ contains a totally decomposable form of degree s without repeated factors. Or, more geometrically, if and only if the linear space F ⊥ = (f ⊥ )s intersects Δn,s in at least a smooth point. Consider the incidence correspondence Σ = {(f, D) : D ∈ F ⊥ , D = L1 · . . . · Ls } ⊂ PSd × Δn,s and the incidence diagram ΣD DD DDβ DD D! Δn,s . PSd d−s+n (use β to show that Σ is a projective Clearly dim Σ = dim PSd + ns − n bundle over Δn,s having ﬁber over D the projectivized of (D−1 )d ). Moreover, for ≥ 0}, the map α is surjective (a dimension count s ≥ s¯ = min{t : nt − d−t+n n shows that Δn,s ∩ F ⊥ = ∅ for any f ). Let Σ0 = {(f, D = L1 · . . . · Ls ) : D ∈ F ⊥ , Li ∼ Lj for some i = j} and notice that Σ0 has codimension 2 in Σ. For s ≥ s¯ and f generic, a dimension argument yields α−1 (f ) \ Σ0 = ∅, hence the claim holds.

|| || | | ~| | α

Remark 2.14. Notice that, given f ∈ Sd , the variety (f ⊥ )s ∩ Δn,s contains all the information about all the possible codimension one decompositions of the form f involving s summands. We will investigate this in the next section. Remark 2.15. In the case of binary forms (n = 1), the theorem gives smin (1, d) = min{s : 2s − d − 1) ≥ 0} = d+1 2 as was well known to Sylvester. As a folklore result, it is interesting to compare smin and sexp , thus measuring how codimension one decompositions are defective. Compared to the sum of powers case the result is quite surprising. Corollary 2.16. Let n ≥ 2. The minimal number of summands appearing in the codimension one decompositions of the generic form of degree d in n + 1 variables is the expected one (i.e. smin (n, d) = sexp (n, d)) if and only if d = 2, 3 for any n ≥ 2

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or d = 4, 5, 6 and 8 for n = 2. Proof. The proof is mainly an exercise in arithmetic. First we notice that the ∗ quantity s∗ (d) = d+1 2 (see Remark 2.9) also satisﬁes the inequality s (d) ≥ sexp (n, d) (†). Then the result follows by studying the equalities smin (n, d) = s∗ (d) () and sexp (n, d) = s∗ (d) (). The n = 2 case is contained in [2], thus we restrict our analysis the1 n > 2 cases. d+nto 1 ≤ n (d + n), to show (†) it is enough to show that s∗ (d) ≥ As n+ d+n−1 n ( n−1 ) 1 n (d + n). For d = 2k this is equivalent to (n − 2)k ≥ 0. While, for d = 2k + 1, we get the inequality (n − 2)k ≥ 1. Thus (†) holds for n > 2 and d > 1. ∗ To study (), notice that smin (n, d) > s∗ (d) if and only if s∗ (d)n− d−s n(d)+n < 0. For d = 2k + 1 the last inequality is equivalent to n(k + 1) < k+n k , which holds for k = 2andcan be proved to hold for n > 2 and k ≥ 2 by induction k+n−1(use the which fact that n+k k+1 ≥ n). For d = 2k we have the inequality n(k + 1) < k holds for n > 4 and k ≥ 3 and for n = 3, 4 and k > 3. In conclusion, () could only possibly holds for (n, d) = (3, 6), (4, 6), (n, 2), (n, 3), (n, 4) for n > 2 and it is easy to check that this is actually the case. Finally, by direct substitution, we verify for which pairs () and () have common solutions.

3. How many decompositions? In the previous section we solved the Waring Problem for codimension one decompositions, i.e. we determined the minimal number of summands appearing in the decompositions of a generic form. Once we know that a generic f can be written as the sum of smin codimension one forms, it is natural to study in how many ways such a decomposition can be obtained. Actually, we are interested in an even more general question: How can we describe the codimension one decompositions of f involving s summands? The answer is suggested by Remark 2.14: Deﬁnition 3.1. Let F ⊂ Pn be a generic degree d hypersurface, then the variety of sums of codimension one forms of F with respect to s is the scheme-theoretic intersection VSH(F, s) = F ⊥ ∩ Δn,s . Remark 3.2. Recall that F = V (f ) for a generic f ∈ Sd and that F ⊥ denotes the projectivization of the appropriate homogeneous piece of f ⊥ . We deﬁne VSH

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75

in terms of F rather than of f because the variety is a controvariant of the form under the action of P GL(n + 1). Remark 3.3. Strictly speaking, VSH(F = V (f ), s) does not describe all the possible codimension one decompositions of f . Indeed, given a reduced point L1 · . . . · Ls ∈ VSH(F, s) we know that there exist codimension one forms fˆ1 , . . . , fˆs such that s

Li ◦ fi = 0, i = 1, . . . , s and f = fˆi , i=1

but the forms fˆi ’s are not uniquely determined by the Li ’s. It is not diﬃcult to see that all the codimension one decompositions of f are described by a projective bundle over VSH(F, s). To carry on our analysis we need to study the variety Δn,s in some detail. Besides the general results in [12] (see Remark 2.11), very few is known about Δn,s : for n = 2 and d = 3 an invariant theory description of the deﬁning ideal is contained in [6]; in [3] some geometric properties are described for n = 2, any d, and we generalize the ideas contained there. Proposition 3.4. Let F = L1 · . . . · Ls ∈ Δn,s be a generic point. If we let ˇ n , then the tangent space to Δn,s in F is XF = ∪i=j {Li = 0} ∩ {Lj = 0} ⊂ P ˇ n passing through XF }. TF (Δn,s ) = |sH − XF | = {hypersurfaces of degree s in P Moreover deg Δn,s

= =

ˇn} #{s-uples of hyperplanes through ns generic points in P ns−1 n(s−1)−1 · . . . · 1. n−1 · n−1

Proof. Using the diﬀerential of a parametric description of Δn,s it is immediate to see that TF (Δn,s ) is the projectivization of the vector space L1 · . . . · Li · . . . · Ls : Li ∈ T1 , i = 1, . . . , s . We claim that: the deﬁning ideal of XF , IXF = ∩i=j (Li , Lj ), is generated by the degree s − 1 elements {L1 · . . . · Li · . . . · Ls : i = 1, . . . , s}. This is enough to get the desired description of the tangent space. ˇ n and consider To determine deg Δn,s choose ns generic points P1 , . . . , Pns ∈ P H(P1 , . . . , Pns ) ⊂ PTd which denotes the linear system of degree s hypersurfaces through them. If Y = H(P1 , . . . , Pns ) ∩ Δn,s is zero dimensional and smooth, then its cardinality is the degree of the variety of totally decomposable forms. By the genericity of the Pi ’s, Y is clearly a set of points of the desired cardinality. Notice that G = R1 · . . . · Rs ∈ Y is a singular point if and only if TG (Δn,s ) ∩ H(P1 , . . . , Pns ) = G.

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Or, in other terms, if and only if there are degree s hypersurfaces passing through XG = ∪i=j {Ri = 0} ∩ {Rj = 0} and the Pi ’s beside {G = 0}. Notice that such an ˇ n in s − 1 codimension 2 linear spaces element meets the hyperplane {Ri = 0} ⊂ P and in n points in generic position, thus {Ri = 0} is a component. Hence G is the unique element with the required property and the degree formula is proved. Proof of the claim. The proof is purely algebraic. In the polynomial ring C[Y1 , . . . , Ys ], clearly we have (Y1 · . . . · Yi · . . . · Ys : i = 1, . . . , s) = ∩i=j (Yi , Yj ) and the ideal can be shown to be Cohen-Macaulay. By specializing we get Li · . . . · Ls : i = 1 . . . , s) ⊆ ∩i=j (Li , Lj ) (L1 · . . . · where we know that the leftmost ideal is Cohen-Macaulay. Comparing degrees and dimensions the equality follows.

Remark 3.5. This Proposition allows us to easily compute the degree of the varieties of secant lines to the quadratic Veronese varieties. In fact, it is well known that Sec1 (Vn,2 ) = Δn,2 and hence deg Sec1 (Vn,2 ) = 2n−1 n−1 . Using this result we can get some information on the varieties parameterizing codimension one decompositions: Theorem 3.6. Let F ⊂ Pn be a generic degree d hypersurface and let s = smin (n, d), then ; • dim VSH(F, s) = ns − d−s+n n ns−1 n(s−1)−1 · . . . · 1. • deg VSH(F, s) = n−1 · n−1 Proof. With the notation of the Proof of Theorem 2.13, we have VSH(F = V (f ), s) = α−1 (f ). The dimension claim readily follows. Once dim VSH(F, s) is known, we realize that the intersection F ⊥ ∩ Δn,s is proper and hence we obtain the degree formula.

The variety of totally decomposable forms contains singularities in codimension 2. Thus, as soon as dim VSH ≥ 2, the scheme parameterizing codimension one decomposition is singular and possibly not reduced. Nevertheless, we can get smoothness in one remarkable case: Proposition 3.7. Let F ⊂ Pn be a generic degree d hypersurface and let s = smin (n, d). If VSH(F, s) is zero dimensional, then it is smooth (and hence reduced). Proof. The non-smoothness condition is a closed condition on the form f ∈ Sd deﬁning F . Hence we prove the Proposition by exhibiting forms with the required property. d−a+n Notice that the zero dimension assumption yields the relation ns = (). n

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77

d li . Choose ns generic linear forms l1 , . . . , lns and consider the form g = Notice that VSH(G = V (g), s) contains at least deg Δns points. Hence, in the case dim VSH(G, s) = 0, the variety is also smooth (the form g is in general very degenerate as it is the sum of few powers, in particular Theorem 3.6 does not apply and the dimension has to be determined by other means). Let X = ˇ n and denote by I its deﬁning ideal. Clearly g ⊥ ⊃ I by the {l1 , . . . , lns } ⊂ P X X Apolarity Lemma and, if equality holds in degree s, then dim VSH(G, s) = 0. By standard computation we get s+n s+n d−s+n − ns and dimC (g ⊥ )s ≥ − dimC (IX )s = n n n and we want to show that the last inequality is an equality (this suﬃces by ()). If the inequality is strict, then there exists D ∈ (g ⊥ )d−s . Thus, D◦f =

ns

ci D(li )lis = 0,

i=1

where ci D(li ) ∈ C are not all zero as (IX )d−s = 0 by genericity. Hence a cond s tradiction, as the li ’s can be chosen s+nin such a way that l1 , . . . , lns are linearly independent in PSs (notice that n − ns = dimC (IX )s > 0). In conclusion,

(g ⊥ )s = (IX )s and dim VSH(G, s) = 0, as required. Remark 3.8. The previous Proposition has a “negative” consequence in the spirit of 19th century invariant theory: there is no reasonable way (not even taking multiplicities in account) to produce canonical forms via codimension one decompositions. In other words, the decomposition is never unique (for similar results see, for sum of powers, [19] and, for partially symmetric tensors, [11]). Remark 3.9. For n = 2 it is easy to show that VSH is zero dimensional in inﬁnitely many cases (dim VSH = 0 is a degree two equation having integer solutions). If n > 2, experiments suggests that zero dimensionality only occurs for n = 3, 6. Notice that the number of summands, s, rapidly increases and that the number of decompositions, #, is soon intractable. This table shows what the numbers look like for n ≤ 100, d ≤ 34: d

n

s

#

5 8 17 20 25 34

2 2 6 2 2 3

3 5 14 14 18 28

15 945 b 213458046676875 221643095476699771875 a

where a and b have 86 digits!

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References [1] Alexander, J. and Hirschowitz, A., Polynomial interpolation in several variables, J. Algebraic Geom. 4 n. 2 (1995), 201–222 [2] Carlini, E., Geometric aspects of some polynomial decompositions, Ph.D. Thesis, University of Pavia (2004) [3] Carlini, E., Binary Decompositions and Varieties of Sums of Binaries, preprint (2004) to appear in Journal of Pure and Applied Algebra [4] Catalisano, M. V., Geramita, A. V. and Gimigliano, A., Ranks of tensors, secant varieties of Segre varieties and fat points, Linear Algebra Appl. 355 (2002), 263–285 [5] Catalisano, M. V., Geramita, A. V. and Gimigliano, A., Secant Varieties of Grassmann Varieties, math.AG/0208166 (2003) [6] Chipalkatti, J. V., Decomposable ternary cubics, Experiment. Math. 11 n.1 (2002), 69–80 [7] Ciliberto, C., Geometric aspects of polynomial interpolation in more variables and of Waring’s problem, European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math. 201 (2001), 289–316 ¨ [8] Clebsch, A., Uber Curven vierter Ordnung, J. f¨ ur Math. 59 (1861), 125–145 [9] Dixon, A. and Stuart, T., On the reduction of the ternary quintic and septimic to their canonical forms, Proc. London Math. Soc. 4 n. 2 (1906), 160–168 [10] Eriksson, N., Ranestad, K., Sturmfels, B. and Sullivant, S., Phylogenetic algebraic geometry math.AG/0407033 (2004) [11] Fontanari, C., On Waring’s problem for partially symmetric tensors, math.AG/0407384 (2004) [12] Gel’fand, I. M., Kapranov, M. M. and Zelevinsky, A.V., Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkh¨ auser Boston, MA (1994) [13] Geramita, A. V., Inverse systems of fat points: Waring’s problem, secant varieties of Veronese varieties and parameter spaces for Gorenstein ideals, The Curves Seminar at Queen’s, Vol. X (Kingston, ON, 1995), Queen’s Papers in Pure and Appl. Math. 102 (1996), 2–114 [14] Geramita, A. V., Catalecticant varieties, Commutative algebra and algebraic geometry (Ferrara), M. Dekker Lecture Notes in Pure and Appl. Math. 206 (1999), 143–156 [15] Hilbert, D., Letter adresse` e´ a M. Hermite, Gesam. Abh. II (1888), 148–153 [16] Iarrobino, A. and Kanev, V., Power sums, Gorenstein algebras, and determinantal loci, Springer Lecture Notes in Mathematics 1721 (1999) [17] Landsberg, J. M. and Manivel, L., On the ideals of secant varieties of Segre varieties, Found. Comput. Math. 4 (4) (2004), 397–422 [18] McLachlan, R. I., Quispel, G. and Reinout W., Splitting methods, Acta Numer. 11 (2002), 341–434

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[19] Mella, M., Singularities of linear systems and the Waring Problem, math.AG/0406288 (2004) [20] Mukai, S., Fano 3-folds, Complex projective geometry (Trieste, 1989/Bergen, 1989), London Math. Soc. Lecture Note Ser. 179 (1992), 255–263 [21] Mukai, S., Polarized K3 surfaces of genus 18 and 20, Complex projective geometry (Trieste, 1989/Bergen, 1989), London Math. Soc. Lecture Note Ser. 179 (1992), 264–276 [22] Palatini, F., Sulla rappresentazione delle forme ternarie medinate la somma di potenze di forme lineari, Rom. Acc. L. Rend. 12 n. 5 (1903), 378–384 [23] Ranestad, K. and Schreyer, F.-O., Varieties of sums of powers, J. Reine Angew. Math. 525 (2000), 147–181 [24] Reye, T., Geometrischer Beweis des Sylvesterschen Satzes:“Jede quatern¨ are cubische Form ist darstellbar als Summe von f¨ unf Cuben linearer Formen”, J. Reine Angew. Math 78 (1874) 114–122 [25] Richmond, H. W. , On canonical forms, Quart. J. Pure Appl. Math. 33 (1904), 331–340 ¨ [26] Rosanes, J., Uber ein Prinzip der Zuordnung algebraischer Formen, J. reine angew. Math 76 (1873), 312–330 [27] Salmon, G., A Treatise on Higher Plane Curves, Reprinted by Chelsea Publishing Co. (1879) [28] Salmon, G., A Treatise on the Analytic Geometry of Three Dimensions I,II, Longman, Green and Co., London (1912) [29] Scorza, G., Sopra la teoria delle ﬁgure polari delle curve piane del quarto ordine, Ann. di Mat. 2 n. 3 (1899), 155-202 Enrico Carlini Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino, Italy Email: [email protected]

Higher secant varieties of Segre-Veronese varieties M.V. Catalisano, A.V. Geramita and A. Gimigliano

Abstract. In this paper we consider the Segre-Veronese varieties, i.e., the embeddings of Pn1 × · · · × Pnt in the projective space PN via divisors of multi-degree (d1 , . . . , dt ), i ), and we study the dimension of their higher secant varieties. We give (N = Π din+n i the dimensions of all the higher secant varieties of P1 × P1 embedded by divisors of any bi-degree (d1 , d2 ). We ﬁnd that Pr × Pk , embedded by divisors of bi-degree (k + 1, 1), has no deﬁcient higher secant varieties, and we give several examples of defective and Grassmann defective Segre-Veronese varieties. 2000 Mathematics Subject Classiﬁcation: 14M99, 15A69, 13D40

Introduction The higher secant varieties of a projective variety X are naturally occurring varieties which often reveal interesting aspects of the projective geometry of X. It is, thus, no surprise that they have been the object of study by many of the most outstanding geometers of the 19th and 20th centuries. A ﬁrst natural question to ask about these higher secant varieties of X is: what are their dimensions? Since there is, by counting parameters, a natural guess as to what one should expect this dimension to be, the question becomes: when is the expected dimension the actual dimension, and if it is not, why not? In the case when X is one of the Segre varieties there is much recent interest in this question, and not only among geometers. In fact, this particular problem is strongly connected to questions in representation theory, coding theory, algebraic complexity theory (see our paper [6] for some recent results as well as a summary of known results, and also [3]) and, surprisingly enough, also in algebraic statistics (e.g. see [16] and [15]). We address a generalization of this problem here; more precisely we will study the higher secant varieties of X = Pn1 × · · · × Pnt = Pn , n = (n1 , . . . , nt ) i − 1) by the morphism νn,d embedded in the projective space PN (N = Π din+n i given by OPn (d), where d = (d1 , . . . , dt ) (di positive integers). We denote the

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M. Virginia Catalisano, Antony V. Geramita and Alessandro Gimigliano

embedded variety νn,d (X) by Xn,d , and call it a Segre-Veronese variety and the embedding a Segre-Veronese embedding (see [4]). In Section 1 we recall some classical results of Terracini regarding secant varieties and their defects. We also introduce the fundamental observation (Theorem 1.5) which allows us to convert certain questions about ideals of varieties in multiprojective space to questions about ideals in standard polynomial rings. In Section 2 we concentrate on t = 2, 3 and let d be arbitrary. In Corollary 2.3 we give the dimensions of all the higher secant varieties for Xn,d (where n = (1, 1), and d is arbitrary). This is a consequence of our determination of the Hilbert function for all fat point schemes in P2 of the form d1 P1 + d2 P2 + 2R1 + · · · + 2Rs whose support consists of s + 2 generic points of P2 . For n = (r, k), d = (k + 1, 1) we ﬁnd that Xn,d has no deﬁcient higher secant varieties (Theorem 2.9) and this gives some interesting and surprising conclusions about Grassmann defectivity for certain Veronese varieties. We also state our theorem (the proof will appear elsewhere) which gives the dimensions of all the higher secant varieties to X(1,1,1),(a,b,c) for any positive integers a, b and c (Theorem 2.12). Section 3 is dedicated to results on secant varieties of Segre-Veronese varieties which can be deduced from results for the Segre varieties and by studying the multigraded Hilbert function of a scheme of 2-fat points in Pn . We give several examples of defective and Grassmann defective Segre-Veronese varieties. Finally, in Section 4 we describe a way of thinking about the points of SegreVeronese varieties as (partially symmetric) tensors. Our approach is essentially this (see §1): we use Terracini’s Lemma (as in [6], [8]) to translate the problem of determining the dimensions of the higher secant varieties of Xn,d into that of calculating the value, at d = (d1 , . . . , dt ), of the Hilbert function of generic sets of 2-fat points in Pn . Then we show, by passing to an aﬃne chart in Pn and then homogenizing in order to pass to Pn , n = n1 + · · · + nt , that this last calculation amounts to computing the Hilbert function of a very particular subscheme of Pn in degree d = d1 + · · · + dn . Finally, we study the postulation of these special subschemes of Pn . We wish to warmly thank Monica Id` a, Luca Chiantini and Ciro Ciliberto for many interesting conversations about the questions considered in this paper.

1. Preliminaries; the multiprojective-aﬃne-projective method Let us recall the notion of higher secant varieties. Deﬁnition 1.1. Let X ⊆ PN be a closed irreducible and non-degenerate projective variety of dimension n. The sth higher secant variety of X, denoted X s (or sometimes Secs−1 (X)), is the closure of the union of all linear spaces spanned by s independent points of X.

Higher secant varieties of Segre-Veronese varieties

83

Recall that, for X as above, there is an inequality concerning the dimension of X s . Namely, dim X s ≤ min{N, sn + s − 1} and one “expects” that the inequality should, in general, be an equality. When X s does not have the expected dimension, X is said to be (s−1)-defective, and the positive integer δs−1 (X) := min{N, sn + s − 1} − dim X s is called the (s − 1)-defect of X. Probably the most well known defective variety is the Veronese surface, X, (in P5 ) for which δ1 (X) = 1. As a generalization of the higher secant varieties of a variety, one can also consider the following varieties. Deﬁnition 1.2. Let X ⊆ PN be a closed, irreducible and non-degenerate projective variety of dimension n. Suppose that we are given integers 0 ≤ k ≤ s − 1 < N . The (k, s − 1)-Grassmann secant variety, denoted Seck,s−1 (X), is the Zariski closure (in the Grassmannian of k-dimensional linear spaces of PN ) of the set { l ∈ G(k, N ) | l lies in the linear span of s independent points of X }. In case k = 0 we get X s = Sec0,s−1 (X)(= Secs−1 (X)). As a generalization of the analogous result for the higher secant varieties, one always has dim Seck,s−1 (X) ≤ min{sn + (k + 1)(s − k − 1), (k + 1)(N − k)}, with equality being what is generally “expected”. When Seck,s−1 (X) does not have the expected dimension then we say that X is (k, s − 1)-defective and in this case we deﬁne the (k, s − 1)-defect of X as the number: δk,s−1 (X) = min{sn + (k + 1)(s − k − 1), (k + 1)(N − k)} − dim Seck,s−1 (X). (For general information about these defectivities see [12] and [14].) In this note we study the defectivities of Xn,d . In his paper [27], Terracini gives a link between these two kinds of defectivity for a variety X as above (see [14] for a modern proof): Proposition 1.3. (Terracini) Let X ⊂ PN be a reduced and irreducible nondegenerate projective variety of dimension n. Let σ : X × Pk → P(k+1)(N +1)−1 be the (usual) Segre embedding. Then X is (k, s− 1)-defective with defect δk,s−1 (X) = δ if and only if σ(X × Pk ) is (s − 1)-defective with (s − 1)-defect δs−1 (X × Pk ) = δ. The most important classical result about higher secant varieties is Terracini’s Lemma (see [26], [6]):

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M. Virginia Catalisano, Antony V. Geramita and Alessandro Gimigliano

Terracini’s Lemma: Let (X, L) be a polarized, integral scheme; and suppose that L embeds X into PN , then: TP (X s ) = TP1 (X), . . . , TPs (X) , where P1 , . . . , Ps are s generic points on X, and P is a generic point of P1 , . . . , Ps (the linear span of P1 , . . . , Ps ); here TPi (X) is the projectivized tangent space of X in PN . Let Z ⊂ X be a scheme of s generic 2-fat points, i.e., a scheme deﬁned by the ideal sheaf IZ = IP2 1 ∩ · · · ∩ IP2 s ⊂ OX , where P1 , . . . , Ps are s generic points of X. Since there is a bijection between hyperplanes of the space PN containing the linear space TP1 (X), . . . , TPs (X) and the elements of H 0 (X, IZ (L)), we have: Corollary 1.4. Let X, L, Z, be as in Terracini’s Lemma; then dim X s = dim TP1 (X), . . . , TPs (X) = N − dimk H 0 (X, IZ (L)). Let X = Pn1 × · · · × Pnt = Pn and let Xn,d ⊂ PN be the embedding of X given by L = OX (d), d = (d1 , . . . , dt ). By applying Corollary 1.4 to our case, we get: s = H(Z, d) − 1, dim Xn,d

where Z ⊂ Pn is a set of s generic 2-fat points, and where for all j ∈ Nt , H(Z, j) is the multigraded Hilbert function of Z, i.e. H(Z, j) = dimk Rj − dimk H 0 (Pn , IZ (j)), where R = k[x0,1 , . . . , xn1 ,1 , . . . , x0,t , . . . , xnt ,t ] = ⊕R(d1 ,...,dt ) is the multi-graded homogeneous coordinate ring of Pn . Now let n = n1 + · · · + nt and consider the birational map g : Pn An , where: ((x0,1 , . . . , xn1 ,1 ), . . . , (x0,t , . . . , xnt ,t )) −→ −→ (

xn ,1 x1,2 xn ,2 x1,t xn ,t x1,1 x2,1 , ,..., 1 ; ,..., 2 ;...; ,..., t ) . x0,1 x0,1 x0,1 x0,2 x0,2 x0,t x0,t

This map is deﬁned in the open subset of Pn given by {x0,1 x0,2 · · · x0,t = 0}. Let S = k[z0 , z1,1 , . . . , zn1 ,1 , z1,2 , . . . , zn2 ,2 , . . . , z1,t , . . . , znt ,t ] be the coordinate ring of Pn and consider the embedding An → Pn whose image is the chart An0 = {z0 = 0}. By composing the two maps above we get: f : Pn − −− → Pn , with ((x0,1 , . . . , xn1 ,1 ), . . . , (x0,t , . . . , xnt ,t )) −→

Higher secant varieties of Segre-Veronese varieties

−→ (1,

85

xn ,1 x1,2 xn ,2 x1,t xn ,t x1,1 ,..., 1 ; ,..., 2 ;...; ,..., t ) = x0,1 x0,1 x0,2 x0,2 x0,t x0,t

= (x0,1 x0,2 · · · x0,t , x1,1 x0,2 · · · x0,t , x0,1 x1,2 · · · x0,t , . . . , x0,1 · · · x0,t−1 xnt ,t ). Let Z ⊂ Pn be a zero-dimensional scheme which is contained in the aﬃne chart {x0,1 x0,2 · · · x0,t = 0} and let Z = f (Z). We want to construct a scheme W ⊂ Pn such that dim(IW )d = dim(IZ )(d1 ,...,dt ) , where d = d1 + · · · + dt . Let us recall that the coordinate ring of Pn is S = k[z0 , z1,1 , . . . , zn1 ,1 , z1,2 , . . . , zn2 ,2 , . . . , z1,t , . . . , znt ,t ], and let Q0 , Q1,1 , . . . , Qn1 ,1 , Q1,2 , . . . , Qn2 ,2 , . . . , Qnt ,t be the coordinate points of Pn . Consider the linear space Πi ∼ = Pni −1 ⊂ Pn , where Πi =< Q1,i , . . . , Qni ,i >. The deﬁning ideal of Πi is: IΠi = (z0 , z1,1 , . . . , zn1 ,1 ; . . . ; zˆ1,i , . . . , zˆni ,i ; . . . ; z1,t , . . . , znt ,t ) . Let Wi be the subscheme of Pn denoted by (d − di )Πi , i.e. the scheme deﬁned by d−di the ideal IΠ . Since IΠi is a prime ideal generated by a regular sequence, the i d−di ideal IΠi is saturated (and even primary for IΠi ). Notice that Wi ∩ Wj = ∅ for i = j. Theorem 1.5. Let Z, Z , W1 , . . . , Wt be as above, and let W = Z + W1 + · · · + Wt ⊂ Pn . Let IW ⊂ S, and IZ ⊂ R, be the ideals of W and Z, respectively. Then we have: dimk (IW )d = dimk (IZ )(d1 ,...,dt ) where d = d1 + · · · + dt . Proof. First note that the part of multi-degree (d1 , . . . , dt ) of the ring R is 1 −s1 2 −s2 t −st N1 )(xd0,2 N2 ) · · · (xd0,t Nt ) , R(d1 ,...,dt ) = (xd0,1

where the Ni vary among all the monomials in (x1,i , . . . , xni ,i )si , for all si ≤ di . By dehomogenizing (via f above) and then substituting zi,j for (xi,j /x0,j ), and ﬁnally homogenizing with respect to z0 , we see that R(d1 ,...,dt ) z0d−s1 −···−st M1 M2 · · · Mt where the Mi vary among all the monomials in (z1,i , . . . , zni ,i )si . Claim: (IW1 +···+Wt )d = (IW1 ∩ · · · ∩ IWt )d = z0d−s1 −···−st M1 M2 · · · Mt where the Mi vary among all the monomials in (z1,i , . . . , zni ,i )si , for all si ≤ di . Proof of Claim: ⊆: Since both vector spaces are generated by monomials, it is enough to show that the monomials of the left hand side of the equality are contained in the right hand side of the equality. Consider M = z0d−s1 −···−st M1 M2 · · · Mt (as above). We now show that this si (for j = i) and that monomial is in IWi (for each i). Notice that Mj ∈ IΠ i

86

M. Virginia Catalisano, Antony V. Geramita and Alessandro Gimigliano (d−s −···−s )+(s +···+sˆ +···+s )

d−s1 −···−st d−si t 1 i t z0d−s1 −···−st ∈ IΠ . Thus, M ∈ IΠi 1 = IΠ . i i d−di Since si ≤ di we have d − di ≤ d − si and so M ∈ IΠi as well, and that is what we wanted to show. ⊇ : To prove this inclusion, consider an arbitrary monomial M ∈ Sd . Such an M can be written M = z0α0 M1 · · · Mt where Mi ∈ (z1,i , . . . , zni ,i ) is a monomial of degree αi . Now, M ∈ (IW1 +···+Wt )d means M ∈ (IWi )d for each i, hence

α0 + α1 + · · · + αˆi + · · · + αt ≥ d − di for i = 1, . . . , t. Since α0 + α1 + · · · + αt = d, then d − αi ≥ d − di for each i, and so αi ≤ di for each i. That ﬁnishes the proof of the claim. Now, since Z and Z are isomorphic (f is an isomorphism between the two aﬃne charts {z0 = 0} and {x0,1 x0,2 · · · x0,t = 0}), it immediately follows (via the

two diﬀerent dehomogenizations) that (IZ )(d1 ,...,dt ) ∼ = (IW )d . Note that the scheme W that we have constructed in Pn has two parts: the part W1 + · · · + Wt (which we shall call the part at inﬁnity and denote W∞ ) and that part Z , which is isomorphic to our original zero-dimensional scheme Z ⊂ Pn . Thus, if Z = ∅ (and hence Z = ∅) we obtain from the theorem that dimk (IW∞ )d = dimk R(d1 ,...,dt ) , It follows that

H(W∞ , d) =

d = d1 + · · · + dt .

d1 + · · · + dt + n d1 + n1 dt + nt ··· . − n1 nt n

With this observation made, the following corollary is immediate:

Corollary 1.6. Let Z and Z be as above and write (as above) W = Z + W∞ . Then H(W, d) = H(Z, d) + H(W∞ , d). When Z is given by s generic 2-fat points, we obtain:

Corollary 1.7. Let Z ⊂ Pn be a generic set of s 2-fat points and let W ⊂ Pn be as in Theorem 1.5. Then we have: s = H(Z, (d1 , . . . , dt )) − 1 = N − dim(IW )d , dim Xn,d

where N = Πti=1

di +ni di

− 1.

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87

2. On Segre-Veronese varieties with two or three factors First we consider the case P1 × P1 , i.e. t = 2, n = (n1 , n2 ) = (1, 1), for all d = (d1 , d2 ) ∈ N2 . In this case we get that the Πi ’s are points. So, we can assume that Π1 = A1 = (0, 1, 0) and that Π2 = A2 = (0, 0, 1) in P2 and let Z = 2R1 + · · · + 2Rs ⊂ P1 × P1 be a set of s generic 2-fat points. We may also assume that Ri = ((1, αi ), (1, βi )), so that f : P1 × P1 → P2 is such that: f (Ri ) = Pi = (1, αi , βi ) ∈ P2 . Consequently, we have that A1 , A2 , P1 , . . . , Ps are a set of generic points of P2 . Let W = d2 A1 + d1 A2 + 2P1 + · · · + 2Ps ⊂ P2 d2 d1 be the scheme deﬁned by the ideal sheaf IW = IA ∩ IA ∩ IP2 1 ∩ · · · ∩ IP2 s . (In our 1 2 earlier notation, the subscheme W∞ of W is given by: W∞ = d2 A1 + d1 A2 ). From Corollary 1.6 we obtain

H(W, d1 + d2 ) = H(W∞ , d1 + d2 ) + H(Z, (d1 , d2 )). But observe that the scheme W∞ is regular in degree d1 + d2 and so d1 + 1 d2 + 1 H(W∞ , d1 + d2 ) = + 2 2 and hence H(W, d1 + d2 ) = H(Z, (d1 , d2 )) +

d1 + 1 d2 + 1 + . 2 2

If we now consider the Segre-Veronese variety Y = X(1,1),(d1 ,d2 ) = ν(1,1),(d1 ,d2 ) (P1 × P1 ) ⊂ Pd1 d2 +d1 +d2 and apply Corollary 1.7 we obtain (with Z as above): d1 + 1 d2 + 1 s dim Y = H(Z, (d1 , d2 )) − 1 = H(W, d1 + d2 ) − − −1 2 2 = deg W − h1 (IW (d1 + d2 )) −

d1 + 1 d2 + 1 − −1 2 2

= 3s − 1 − h1 (IW (d1 + d2 )) = (d1 + 1)(d2 + 1) − h0 (IW (d1 + d2 )) − 1. Thus to understand the dimensions of the secant varieties to these SegreVeronese varieties we are naturally brought to consider the Hilbert function of the subschemes W of P2 . The theorem we have about these schemes is the following:

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Proposition 2.1. Let W be the subscheme of P2 deﬁned as W = d1 P1 + d2 P2 + 2R1 + · · · + 2Rs where d1 ≥ d2 ≥ 2, and P1 , P2 , R1 , . . . , Rs are s + 2 generic points of P2 . Let t ≥ d1 + d2 . Then h0 (IW (t))·h1 (IW (t)) = 0 (i.e. W has maximal Hilbert function in degree t) except when t = d1 + d2 , d1 even, d2 = 2, s = d1 + 1,

(*)

in which case h0 (IW (t)) = h1 (IW (t)) = 1. Proof. Let’s ﬁrst consider the case when t = d1 + d2 , d1 is even, and d2 = 2. Let d1 = 2a. If s = d1 + 1, let C be the unique irreducible (rational) curve of degree a + 1 through aP1 + P2 + R1 + · · · + Rs . By Bezout’s Theorem, 2C is the unique curve · · · + 2Rs , hence of degree t = 2a + 2 passing through W = 2aP 2P2 + 2R1 + 1+ 1 − deg W + h (I h0 (IW (t)) = 1. Moreover, since h0 (IW (t)) = t+2 W (t)), we easily 2 get h1 (IW (t)) = 1. Since h0 (IW (t)) = 1 for s = d1 + 1, then for s > d1 + 1 we obviously have h0 (IW (t)) = 0. For s = d1 , the expected dimension of h0 (IW (t)) is 3. Let Q ∈ P2 be a generic point, and let W = W + Q. Since there is a unique irreducible (rational) curve of degree a + 1 through aP1 + P2 + R1 + · · · + Rs + Q (call it C), we have (again by Bezout’s Theorem) that the sections of IW (t) correspond to curves which have C as a ﬁxed component. Hence one easily gets that h0 (IW (t)) = 2. Since Q is generic, it follows that h0 (IW (t)) = 3 and hence that h1 (IW (t)) = 0. Since h1 (IW (t)) = 0 for s = d1 , h1 (IW (t)) = 0 for s < d1 . From now on we may assume that we are not in case t = d1 + d2 , d1 even, and d2 = 2. We work by induction on d1 + d2 . For 2 ≤ d1 ≤ 3 and 2 ≤ d2 ≤ 3 the result is well known (see, for instance, [1] or [19]). Let d1 + d2 ≥ 6, and we may assume that d1 ≥ 4. Set s1 =

1 3

t+2 2

−

d1 +1 2

−

d2 +1 2

,

W1 = d1 P1 + d2 P2 + 2R1 + · · · + 2Rs1 ,

s2 =

1 3

t+2 2

−

d1 +1 2

−

d2 +1 2

,

W2 = d1 P1 + d2 P2 + 2R1 + · · · + 2Rs2 .

Since s1 and s2 (if not equal) are consecutive, if we prove that both h1 (IW1 (t)) and h0 (IW2 (t)) are 0, we will be done, since if s ≥ s2 we will continue to have h0 (IW2 (t)) = 0, while if s ≤ s1 we will continue to have h1 (IW1 (t)) = 0. Let q, r ∈ N be such that t + 1 − d1 = 2q + r,

0 ≤ r ≤ 1.

We ﬁrst show that q + r < s1 . In fact, since t ≥ d1 + d2 , we get:

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89

1 t+2 d1 + 1 d2 + 1 t + 1 − d1 − r −r s1 − q − r = − − − 3 2 2 2 2 ≥

1 2 1 (t − d21 − d22 + 2d1 − d2 − 1 − 3r) ≥ (2d1 d2 + 2d1 − d2 − 1 − 3r) 6 6 =

1 ((2d1 − 1)(d2 + 1) − 3r) > 0. 6

Let

t+2 d1 + 1 d2 + 1 v := − − − 3s1 = h0 (IW1 (t)) − h1 (IW1 (t)) 2 2 2

and observe that, by the deﬁnition of s1 , we have 0 ≤ v ≤ 2. Since v ≥ 0 and we want to prove that h1 (IW1 (t)) = 0, notice that this is equivalent to proving h0 (IW1 (t)) = v. When v = 0 we have W1 = W2 . Since proving h1 (IW1 (t)) = 0 is equivalent to proving h0 (IW1 (t)) = 0 we will automatically get h0 (IW2 (t)) = 0. When v = 1 then s2 > s1 and so if we prove h0 (IW1 (t)) = 1 then we easily get 0 h (IW2 (t)) = 0. When v = 2, if we can show that h0 (IW1 (t)) = 2 this will only give (as we mentioned) h1 (IW1 (t)) = 0. Thus, we will have to prove directly, in this case, that h0 (IW2 (t)) = 0. Thus, the procedure is clear: prove h0 (IW1 (t)) = v for v = 0, 1, 2 and h0 (IW2 (t)) = 0 when v = 2. It turns out that we have to consider not only v but also r and the method of proof breaks down into two rather separate cases: (v, r) = (0, 1) and (v, r) = (0, 1). The ﬁrst case will be taken care of by the usual kind of specialization and residuation arguments, while the second case will require the more sophisticated analysis pioneered by J. Alexander and A. Hirschowitz in [1] (and baptized the methode d’Horace diﬀerentiel). Case 1: (v, r) = (0, 1) Let L be a generic line passing through P1 . Let Q be a generic simple point, and let Q if r = 1 Qr = ∅ if r = 0. Add Qr to the scheme W1 , and let W1 = W1 + Qr (for r = 0, W1 = W1 ). Let ˜1, . . . , R ˜q , R ˜ q+1 , Q ˜ 1 be generic points on L (of course, Q ˜ r = ∅ if r = 0). R Now specialize the schemes W1 and W2 to the schemes ˜ 1 + · · · + 2R ˜q + Q ˜ r + 2Rq+1 + · · · + 2Rs1 1 = d1 P1 + d2 P2 + 2R W 2 = d1 P1 + d2 P2 + 2R ˜ 1 + · · · + 2R ˜ q+r + 2Rq+r+1 + · · · + 2Rs2 W respectively. Observe that this is possible since q + r ≤ s1 .

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M. Virginia Catalisano, Antony V. Geramita and Alessandro Gimigliano

In order to prove that h0 (IW1 (t)) = v and also, when v = 2, that h0 (IW2 (t)) = 0, it suﬃces to prove that h0 (IW1 (t)) = v − r

and

h0 (IW2 (t)) = 0.

i ∩ L) ≥ d1 + 2q + r = t + 1, the line L is a ﬁxed component for Since deg(W i . Hence the curves of degree t containing W h0 (IWi (t)) = h0 (IResL Wi (t − 1))

(i = 1, 2)

i is the residual scheme of W i with respect to L, that is: where ResL W 1 = (d1 − 1)P1 + d2 P2 + R ˜1 + · · · + R ˜ q + 2Rq+1 + · · · + 2Rs1 ResL W 2 = (d1 − 1)P1 + d2 P2 + R ˜1 + · · · + R ˜ q+r + 2Rq+r+1 + · · · + 2Rs2 . ResL W Let 1 − (R ˜1 + · · · + R ˜ q ) = (d1 − 1)P1 + d2 P2 + 2Rq+1 + · · · + 2Rs1 Z1 = ResL W 2 − (R ˜1 + · · · + R ˜ q+r ) = (d1 − 1)P1 + d2 P2 + 2Rq+r+1 + · · · + 2Rs2 . Z2 = ResL W Now we need a criterion for adding to the zero-dimensional schemes Z1 , Z2 , a set of generic points lying on the line L and estimating h0 of the resulting schemes. We recall the following lemma (see [9] or [7]): Lemma 2.2. Let Z ⊂ P2 be a zero-dimensional scheme, and let P1 , . . . , Px be generic points on a line L. If dim(IZ+P1 +···+Px−1 )t > dim(IResL Z )t−1 then dim(IZ+P1 +···+Px )t = dim(IZ )t − x. In order to use this lemma to compute h0 (IResL Wi (t − 1)), we have to know dim(IZi )t−1 , and dim(IResL Zi )t−2 , where ResL Z1 = (d1 − 2)P1 + d2 P2 + 2Rq+1 + · · · + 2Rs1 and ResL Z2 = (d1 − 2)P1 + d2 P2 + 2Rq+r+1 + · · · + 2Rs2 . By an easy computation we get t+1 q+v−r −deg Zi = h0 (IZi (t−1))−h1(IZi (t−1)) = q + 2r − 1 2 and

for i = 1, for i = 2, v = 2,

t − deg ResL Zi h (IResL Zi (t − 2)) − h (IResL Zi (t − 2)) = 2 0

1

91

Higher secant varieties of Segre-Veronese varieties

−q + v − 2r = −q + r − 1

for i = 1, for i = 2, v = 2

(the only cases that interest us). Now we apply the induction hypothesis to the schemes Zi and ResL Zi (i = 1, 2). 1 −r + v − r ≥ 3−3r Since t ≥ d1 + d2 ≥ d1 + 2, we have q + v − r = t+1−d 2 2 + v. But, since we are in Case 1, v > 0, or v = r = 0 and hence q + v − r > 0. It follows that h0 (IZ1 (t − 1)) − h1 (IZ1 (t − 1)) > 0 and so we are not in the exceptional case (*). Thus, we have h0 (IZ1 (t − 1)) · h1 (IZ1 (t − 1)) = 0 and hence h1 (IZ1 (t − 1)) = 0 and h0 (IZ1 (t − 1)) = q + v − r.

(1)

1 −r Moreover q + 2r − 1 = t+1−d + 2r − 1 ≥ 1+3r > 0. Hence, for v = 2, 2 2 1 h (IZ2 (t − 1)) − h (IZ2 (t − 1)) > 0 and once again we are not in the exceptional case (*). Thus, we get h0 (IZ2 (t − 1)) · h1 (IZ2 (t − 1)) = 0 and so h1 (IZ2 (t − 1)) = 0, and

0

h0 (IZ2 (t − 1)) = q + 2r − 1.

(2)

If h0 (IResL Z1 (t − 2)) = h1 (IResL Z1 (t − 2)) = 1, then, by the induction hypothesis, we must have t − 2 = (d1 − 2) + d2 , i.e. t = d1 + d2 , with d1 even, d2 = 2, and h0 (IResL Z1 (t − 2)) − h1 (IResL Z1 (t − 2)) = 0. Hence r = t + 1 − d1 − 2q = d2 + 1 − 2q and, since d2 = 2 is even, we get r = 1. So h0 (IResL Z1 (t − 2)) − h1 (IResL Z1 (t − 2)) = −q + v − 2r −d2 −t − 1 + d1 + r + v − 2r = + v − 2 ≤ −1, = 2 2 a contradiction. It follows that h0 (IResL Z1 (t − 2)) · h1(IResL Z1 (t − 2)) = 0. Thus, since −q + v − ≤ 12 , we have −q + v − 2r ≤ 0, hence h1 (IResL Z1 (t − 2)) = q − v + 2r, 2r ≤ 1−3r 2 and h0 (IResL Z1 (t − 2)) = 0.

(3)

Finally, if v = 2, h0 (IResL Z2 (t − 2)) − h1 (IResL Z2 (t − 2)) = −q + r − 1 ≤

3r − 5 ≤ −1, 2

and once again we are not in case (*). Hence, by the induction hypothesis, h0 (IResL Z2 (t − 2)) · h1 (IResL Z2 (t − 2)) = 0. So h1 (IResL Z2 (t − 2)) = q − r + 1, and h0 (IResL Z2 (t − 2)) = 0.

(4)

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M. Virginia Catalisano, Antony V. Geramita and Alessandro Gimigliano

Now we are ready to use Lemma 2.2. We have dim(IZ1 +R˜ 1 +···+R˜ q−1 )t−1 ≥ dim(IZ1 )t−1 −(q−1) = (q+v−r)−(q−1) = v−r+1 > 0 (where the ﬁrst inequality is always true, the ﬁrst equality comes from (1) and the last inequality comes from (v, r) = (0, 1)). Thus, by Lemma 2.2, since dim(IZ1 +R˜ 1 +···+R˜ q−1 )t−1 > dim(IResL Z1 )t−2 = 0 (this last equality from (3)), we have dim(IResL W1 )t−1 = = dim(IZ1 +R˜ 1 +···+R˜ q )t−1 = dim(IZ1 )t−1 − q = (q + v − r) − q = v − r. Hence h0 (IW1 (t)) = h0 (IResL W1 (t − 1)) = v − r, as we wanted to show. Now let v = 2. Again (by Lemma 2.2) we have for r = 0, dim(IZ2 +R˜ 1 +···+R˜ q−2 )t−1 ≥ dim(IZ2 )t−1 − (q − 2) = (q − 1) − (q − 2) = 1 > dim(IResL Z2 )t−2 = 0, (where the ﬁrst inequality is always true, the ﬁrst equality is from (2) and the last equality is from (4)) while for r = 1 we have, dim(IZ2 +R˜ 1 +···+R˜ q )t−1 ≥ dim(IZ2 )t−1 −q = (q+1)−q = 1 > dim(IResL Z2 )t−2 = 0. Putting these together gives dim(IResL W2 )t−1 = ⎧ ⎪ dim(IZ2 +R˜ 1 +···+R˜ q )t−1 ≤ dim(IZ2 +R˜ 1 +···+R˜ q−1 )t−1 = dim(IZ2 )t−1 − (q − 1) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ for r=0 ⎨ ⎪ ⎪ ⎪ dim(IZ2 +R˜ 1 +···+R˜ q +R˜ q+1 )t−1 = dim(IZ2 )t−1 − (q + 1) = 0, ⎪ ⎪ ⎪ ⎩ for r = 1 It follows that h0 (IW2 (t)) = h0 (IResL W2 (t − 1)) = 0, as we wanted to show. Case 2: v = 0, r = 1. We need only prove that h0 (IW1 (t)) = v = 0. 1 , obtained from W1 by moving Specialize the scheme W1 to the scheme W R1 , . . . , Rq onto a line L passing through P1 . That means, 1 = d1 P1 + d2 P2 + 2R ˜ 1 + · · · + 2R ˜ q + 2Rq+1 + · · · + 2Rs1 , W ˜ s are generic points on L. As in the previous case, in order to prove where the R i that dim(IW1 )t = 0, we will prove that dim(IW1 )t = 0. We will make use of the Horace diﬀerential method.

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93

˜ q+1 be a generic point on L, and let D2,L (R ˜ q+1 ) = 2R ˜ q+1 ∩ L. Now let: Let R 1 − 2Rq+1 ) + D2,L (R ˜ q+1 ) Z = ResL (W ˜1 + · · · + R ˜ q + D2,L (R ˜ q+1 ) + 2Rq+2 + · · · + 2Rs1 , = (d1 − 1)P1 + d2 P2 + R 1 − 2Rq+1 ) ∩ L + R ˜ q+1 = (d1 P1 + 2R ˜ 1 + · · · + 2R ˜q + R ˜ q+1 ) ∩ L. Z = (W By the Horace Diﬀerential Lemma (see [1]) if dim(IZ )t−1 = dim(IZ )t = 0, then dim(IW1 )t = 0. Since deg Z = d1 + 2q + 1 = t + 1, then obviously dim(IZ )t = 0. So we are only left to prove that dim(IZ )t−1 = 0. For q = 1, we have t = d1 + 2, d2 = 2 and, since we are assuming that we are not in case t = d1 + d2 , d1 even, d2 = 2, it must be that d1 is odd. Since deg(Z ∩ L) = (d1 − 1) + q + 2 = d1 + 2, the forms of (IZ )t−1 correspond to curves which have the line L as a ﬁxed component. It follows that dim(IZ )t−1 = dim(IResL Z )t−2 , where ResL Z = (d1 − 2)P1 + d2 P2 + 2Rq+2 + · · · + 2Rs1 . We have

t − deg ResL Z = 1 − q = 0. h (IResL Z (t − 2)) − h (IResL Z (t − 2)) = 2 0

1

By the induction hypothesis, since d1 − 2 is odd, we have h0 (IResL Z (t − 2)) · h1 (IResL Z (t− 2)) = 0. It follows that h0 (IResL Z (t− 2)) = h1 (IResL Z (t− 2)) = 0, hence dim(IZ )t−1 = 0. Now assume q > 1. Let ˜1 + · · · + R ˜ q ) = (d1 − 1)P1 + d2 P2 + D2,L (R ˜ q+1 ) + 2Rq+2 + · · · + 2Rs1 X = Z − (R ˜ q+1 + 2Rq+2 + · · · + 2Rs1 . X = (d1 − 1)P1 + d2 P2 + 2R We have h0 (IX (t − 1)) − h1 (IX (t − 1)) =

t+1 − deg X = q − 1 > 0. 2

By the induction hypothesis, since h0 (IX (t − 1)) > h1 (IX (t − 1)), we get that h0 (IX (t − 1)) · h1 (IX (t − 1)) = 0. It follows that h1 (IX (t − 1)) = 0, and h0 (IX (t − 1))= q − 1. Since X ⊂ X, and deg X = deg X − 1, we immediately get h1 (IX (t − 1)) = 0, so h0 (IX (t − 1)) = q. Now consider ResL X = (d1 − 2)P1 + d2 P2 + 2Rq+2 + · · · + 2Rs1 . We have

t − deg ResL X = −q + 1 < 0. h (IResL X (t − 2)) − h (IResL X (t − 2)) = 2 0

1

94

M. Virginia Catalisano, Antony V. Geramita and Alessandro Gimigliano

By the induction hypothesis, since h0 (IResL X (t − 2)) < h1 (IResL X (t − 2)), we have h0 (IResL X (t − 2)) · h1 (IResL X (t − 2)) = 0, so h1 (IResL X (t − 2)) = q − 1, and h0 (IResL X (t − 2)) = 0. Finally, (again by Lemma 2.2) since dim(IX +R˜ 1 +···+R˜ q−1 )t−1 ≥ ≥ dim(IX )t−1 − (q − 1) = q − (q − 1) > dim(IResL X )t−2 = 0, we have dim(IX +R˜ 1 +···+R˜ q )t−1 = dim(IX )t−1 − q = q − q = 0. ˜1 + · · · + R ˜ q = Z , and thus we are done. But X + R

Corollary 2.3. Let Y = X(1,1),(d1 ,d2 ) = ν(1,1),(d1 ,d2 ) (P1 × P1 ), d1 ≥ 1, d2 ≥ 1, then Y s has the expected dimension, except for d1 = 2a, d2 = 2,

and s = d1 + 1 (a any positive integer ).

In this case Y s is defective, and dim Y s = 3s − 2 (its defectiveness is 1). Proof. Let W = d1 P1 + d2 P2 + 2R1 + · · · + 2Rs . As we saw above, dim Y s = 3s − 1 − h1 (IW (d1 + d2 )) = (d1 + 1)(d2 + 1) − 1 − h0 (IW (d1 + d2 )). Hence, for d1 ≥ 2, d2 ≥ 2, the corollary immediately follows from Proposition 2.1. So, to prove the corollary, we need only consider (IW )d1 +d2 when at least one of d1 and d2 , is 1. We may assume d2 = 1. Since the lines P1 Ri (1 ≤ i ≤ s) are ﬁxed components for the curves of degree d1 + 1 containing W , we have 0 for d1 + 1 < s h0 (IW (d1 + 1)) = h0 (IW (d1 + 1 − s)) for d1 + 1 ≥ s where W = (d1 − s)P1 + P2 + R1 + · · · + Rs . Now

d1 + 1 − s + 2 h (IW (d1 +1−s))−h (IW (d1 +1−s)) = −deg W = 2d1 −3s+2 2 0

1

and also

d1 + 1 + 2 h (IW (d1 + 1)) − h (IW (d1 + 1)) = − deg W = 2d1 − 3s + 2. 2 0

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Hence, since h0 (IW (d1 + 1 − s)) · h1 (IW (d1 + 1 − s)) = 0, we have also h0 (IW (d1 + 1)) · h1 (IW (d1 + 1)) = 0,

and the conclusion follows.

Remark 2.4. 1) We want to thank Monica Id` a for showing us a direct proof of Corollary 2.3 which does not use Proposition 2.1 ([23]). 2) The ﬁrst classiﬁcation of surfaces with some defective secant variety was achieved by Terracini (see [28], §2), hence one could, undoubtedly, deduce Corollary 2.3 from that work as well. One should note, however, that Proposition 2.1 is signiﬁcantly stronger than Corollary 2.3. Notice also that the recent paper [10] extends Terracini’s result, and gives a complete classiﬁcation of all the weaklydefective surfaces. Although we did not comment, in Proposition 2.1, on the values of the Hilbert function when t < d1 + d2 , the next proposition will give, as a consequence (see Theorem 2.7) the values of the Hilbert function in those cases as well. Proposition 2.5. Let d1 ≤ t < d1 + d2 , and d1 ≥ d2 ≥ 2, and let W be the subscheme of P2 deﬁned as W = d1 P1 + d2 P2 + 2R1 + · · · + 2Rs . Let r = max{d1 + d2 − t; 0}, Then

r˜ = max{d1 + 2 − t; 0}.

$ t+2 r r˜ h (IW (t)) = max − deg W + +s ; 0 , 2 2 2 0

and, if h0 (IW (t)) > 0,

r r˜ +s , h (IW (t)) = 2 2 1

except for t = d1 + 2, d1 = 2a + r, d2 = 2 + r,

and s = 2a + 1 (a any positive integer ), in which case h0 (IW (t)) = 1, h1 (IW (t)) = 1 + r2 .

Proof. To simplify the notation, let t+2 r r˜ x= − deg W + +s . 2 2 2

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If we are not in the exceptional case, we have to prove that h0 (IW (t)) = max{x; 0}. Since, for t = d1 , we have r = d2 and r˜ = 2, while for t = d1 + 1, we have r = d2 − 1 and r˜ = 1, and for t > d1 + 1, we have r˜ = 0, we get ⎧ for t = d1 ⎨d1 − d2 − 2s + 1 2(d − d − s + 2) − s for t = d1 + 1 x = 1 2 (5) r ⎩ t+2 − deg W + 2 for t > d1 + 1. 2 Now the line P1 P2 , and the lines P1 Ri (1 ≤ i ≤ s) are ﬁxed components, with multiplicity r and r˜, respectively, for the curves of degree t containing W . So these curves must have degree at least r + s˜ r and hence for t < r + s˜ r , h0 (IW (t)) = 0. But t < r + s˜ r ⇔ t − r − s˜ r ≤ −1, and ⎧ for t = d1 ⎨d1 − d2 − 2s (6) t − r − s˜ r = d1 − d2 − s + 2 for t = d1 + 1 ⎩ 2t − d1 − d2 > 0 for t > d1 + 1 so, from (5), in case t < r + s˜ r, we easily have that h0 (IW (t)) = max {x; 0}. Now assume t − r − s˜ r ≥ 0. Taking away the ﬁxed lines, we get r )) h0 (IW (t)) = h0 (IW (t − r − s˜ where r ; 0})P1 + (d2 − r)P2 + (2 − r˜)R1 + · · · + (2 − r˜)Rs W = (max{d1 − r − s˜ ⎧ ⎨(d1 − d2 − 2s)P1 = (max{d1 − d2 − s + 1; 0})P1 + P2 + R1 + · · · + Rs ⎩ (d1 − r)P1 + (d2 − r)P2 + 2R1 + · · · + 2Rs

for for for

t = d1 t = d1 + 1 t > d1 + 1.

r = d1 −d2 −2s (≥ 0), and by (5) d1 −d2 −2s+1 = For t = d1 , since by (6) t−r−s˜ x, we immediately get h0 (IW (t − r − s˜ r)) =

d1 − d2 − 2s + 2 d1 − d2 − 2s + 1 − 2 2

= d1 − d2 − 2s + 1 = x.

Since we are in case t−r−s˜ r ≥ 0, it follows that h0 (IW (t−r−s˜ r)) = max {x; 0} . r = d1 −d2 −s+2. If d1 −d2 −s+2 = 0, then For t = d1 +1, by (6) we have t−r−s˜ r = −1, W = P2 + R1 + · · ·+ Rs and x = −s, hence h0 (IW (t − r − s˜ r)) = d1 − r − s˜ 0 h (IW (0)) = 0 = max {x; 0} . If d1 − d2 − s + 2 > 0, then W = (d1 − d2 − s + 1)P1 + P2 + R1 + · · · + Rs and, since W has maximal Hilbert function, we easily get h0 (IW (d1 − d2 − s + 2)) $ d1 − d2 − s + 4 d1 − d2 − s + 2 = max − − s − 1; 0 2 2 = max{2(d1 − d2 − s + 2) − s; 0} = max {x; 0} .

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Finally, let t > d1 + 1, so W = (d1 − r)P1 + (d2 − r)P2 + 2R1 + · · · + 2Rs . r)) has the expected dimension except when By Proposition 2.1, h0 (IW (t − r − s˜ t − r − s˜ r = (d1 − r) + (d2 − r),

(d1 − r) = 2a,

(d2 − r) = 2,

s = (d1 − r) + 1

i.e. since r˜ = 0 and t − r = 2t − d1 − d2 , for t = d1 + 2,

d1 = 2a + r,

d2 = 2 + r,

s = 2a + 1.

In this case, h (IW (t − r)) = h (IW (t − r)) = 1. Hence h0 (IW (t)) = 1, moreover, − deg W , we get since h0 (IW (t)) − h1 (IW (t)) = t+2 2 t+2 d1 + 4 d1 + 1 d2 + 1 + deg W = 1 − h1 (IW (t)) = 1 − + + + 3s 2 2 2 2 0

1

= 1 − 3(2a + r) − 6 +

r+3 r + 3(2a + 1) = 1 + . 2 2

If we are not in the exceptional case, then h0 (IW (t − r)) has the expected t−r+2 0 − deg W ; 0}. So by an easy dimension, that is h (IW (t − r)) = max{ 2 computation we get: $ t−r+2 d2 − r + 1 d1 − r + 1 h0 (IW (t − r)) = max − − − 3s; 0 2 2 2 $ t+2 r = max − deg W + ; 0 = max{x; 0}, 2 2

and we are done.

Remark 2.6. Since for t < d1 we obviously have h0 (IW (t)) = 0, we can put together Proposition 2.1 and Proposition 2.5 to give a complete description of the Hilbert function of all subschemes of P2 having the form W = d1 P1 + d2 P2 + 2R1 + · · · + 2Rs whose support is a set of s + 2 generic points. Theorem 2.7. Let {P1 , P2 , R1 , . . . , Rs } be a generic set of s + 2 points in P2 and let W be the subscheme W = d1 P1 + d2 P2 + 2R1 + · · · + 2Rs . Suppose that d1 ≥ d2 ≥ 2, and deﬁne r = max{d1 + d2 − t; 0}, and r˜ = max{d1 + 2 − t; 0}. t+2 Then: 2 % r r˜& for t < d1 i) H(W, t) = ; deg W − − s for t ≥ d1 min t+2 2 2 2 except for t = d1 + 2, d1 − d2 even , s = d1 − d2 + 3, − 1; in which case H(W, t) = t+2 2

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ii) The Hilbert function of W is NOT maximal if and only if, either ⎧ d1 −d2 +1 $ ⎪ for d2 = 2; ⎨ 2 d1 − d2 + 1 (d1 − 1)(d2 − 1) s < max , = ⎪ 2 3 ⎩ (d1 −1)(d2 −1) for d2 > 2, 3 or s = d1 − d2 + 3

and d1 − d2

is even .

Proof. All one needs for the proof of this theorem has been gathered in Proposition 2.1, Corollary 2.3 and Proposition 2.5, and so we leave the putting together of these results to the reader.

Remark 2.8. This theorem gives many new examples of non-homogeneous regular linear systems in P2 and can be considered as complementary to the papers [5], [13], [24], [20], [29] and to a result of [2]. Note also that this theorem gives further conﬁrmation of the conjecture of Harbourne and Hirschowitz (see [19] for a statement of the conjecture). Now let us consider another case: namely the products Pr × Pk . We have the following: Theorem 2.9. Let r, k ≥ 1, and let Y be the Segre-Veronese variety (k + 1) + r (k + 1) − 1. Y = X(r,k),(k+1,1) = ν(r,k),(k+1,1) (Pr × Pk ) ⊂ PN , N = r Then for any s ≥ 1, Y s has the expected dimension. the expected dimension of Y s (i.e. s dim Y + Proof. First note that for s = r+k k (s − 1))is exactly N , hence the statement will follow if we prove that Y s = PN , r+k for s = k . Consider the scheme W = Π1 + (k + 1)Π2 + 2P1 + · ·· + 2Ps ⊂ Pr+k , where Π1 ∼ = Pr−1 and Π2 ∼ = Pk−1 are linear spaces and s = r+k k . By Corollary 1.7, we get that dim Y s = N − h0 (Pr+k , IW (k + 2)). From what we have seen before, we will be done if dim(IW )k+2 = 0. We proceed by double induction on k and r. Consider the case k = 1 (any r) ﬁrst. When k = 1, W is the scheme theoretic union: W = Π1 + 2Π2 + 2P1 + · · · + 2Ps ⊂ Pr+1 ; where Π1 Pr−1 , Π2 is a point and s = r + 1. For this case (i.e. k = 1) it is enough to show that (IW )3 = {0}. We do this by induction on r. For r = 1 we trivially have h0 (P2 , IW (3)) = 0 since W consists of three generic 2-fat points and one generic simple point.

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When r > 1, let H ⊂ Pr+1 be the hyperplane H = P1 , . . . , Pr+1 (H its equation) and consider the residue exact sequence: ×H

0 → IW (2) −→ IW (3) −→ IW ∩H,H (3) → 0 where W = Π1 + 2Π2 + P1 + · · · + Pr+1 . We get h0 (H, IW ∩H,H (3)) = 0 by induction. It is enough to show that h0 (Pr+1 , IW (2)) = 0. To see why this is so, ﬁrst observe that a quadric containing W must be singular at Π2 and hence must be a quadric cone. This cone must also contain the (generic) Pr−1 Π1 and hence must contain the hyperplane Π1 , Π2 . Thus, the quadric has to be the product of two linear forms. The other linear form must vanish at Π2 , P1 , . . . , Pr+1 , which is a set of r + 2 generic points of Pr+1 . This last statement gives us the desired contradiction. Hence the case k = 1 (r arbitrary) is done. Now consider the case r = 1 (k arbitrary). In this case P1 × Pk → PN (where N = (k + 2)(k + 1) − 1) and W = Π1 + (k + 1)Π2 + 2P1 + · · · + 2Pk+1 ⊂ Pk+1 where Π1 is a point, Π2 Pk−1 and s = k + 1. We must show that (IW )k+2 = {0}. The hyperplanes Hi = Pi , Π2 , i = 1, . . . , k, are ﬁxed components for the hypersurfaces given by the forms in (IW )k+2 , hence (by removing such ﬁxed components), we get dim(IW )k+2 = dim(IΠ1 +P1 +···+Pk+1 )1 = 0. So the case r = 1 (k arbitrary) is done. Now we can consider r, k ≥ 2 and work by double induction on these integers. Let H ⊂ Pr+k be a hyperplane such that Π2 ⊂ H and Π1 is not contained to generic in H. Let Π1 = H ∩ Π1 ∼ = Pr−2 . Specialize P1 , . . . , Ps , s = r+k−1 k points on H and consider the residue exact sequence: 0 → IZ (k + 1) → IW (k + 2) → IW ∩H,H (k + 2) → 0, where Z = Π1 + kΠ2 + P1 + · · · + Ps + 2Ps +1 + · · · + 2Ps ⊂ Pr+k and W ∩ H = Π1 + (k + 1)Π2 + 2P1 + · · · + 2Ps |H . We have h0 (IW ∩H,H (k + 2)) = 0 by induction on r; so we will be done if h0 (IZ (k + 1)) = 0, since the above sequence would yield h0 (IW (k + 2)) = 0. Now take a hyperplane H ⊂ Pr+k with Π1 ⊂ H and Π2 not contained in H ; let Π2 = H ∩ Π2 ∼ = Pk−2 , then specialize Ps +1 , . . . , Ps to generic points on H and consider the residue exact sequence: 0 → IZ (k) → IZ (k + 1) → IZ∩H ,H (k + 1) → 0, · ·+P Pr+k and Z∩H = Π1 +kΠ2 +2Ps +1 +· · ·+2Ps |H . where Z = kΠ2 +P1 +· s ⊂ r+k r−1+k 0 = r+k−1 Notice that s − s = k − k k−1 , so that h (IZ∩H ,H (k + 1)) = 0 by induction on r and k.

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All that is left to prove is that h0 (IZ (k)) = 0. The sections of IZ (k) correspond to degree k hypersurfaces in Pr+k which, in order to contain kΠ2 , have to be cones with Π2 as vertex. Let H ∼ = Pr be a generic r-dimensional linear subspace of Pr+k ; then we have 0 h (IZ (k)) = h0 (IZ ,H (k)), where Z ⊂ H ∼ = Pr is the projection of Z into H from Π2 . We have Z = Q1 + · · · + Qs + Qs +1 + · · · + Qs , where Qs +1 , . . . , Qs are generic in H , while Q1 , . . . , Qs are contained in the linear space H ∩ H ∼ = Pr−1 (where they are generic). A hypersurface F of degree k in H cannot contain H (F Q1 , . . . , Qs without containing all H ∩ r+k−1 intersects H ∩ H insomething of degree k, but the Qi , i = 1, . . . , s = are generic in H ∩ H , so they are k not contained in a hypersurface of degree k of Pr−1 = H ∩ H ). So, if F vanishes on Q1 , . . . , Qs , we have that F is the union of H ∩ H and a hypersurface F of degree k − 1 vanishing on Qs +1 , . . . , Qs , but again this cannot happen because generic points in Pr , so no form of degree k − 1 vanishes at them. they are r+k−1 k−1 0 Thus h (IZ ,H (k)) = 0 and we are done.

From this result we immediately get the following: Corollary 2.10. Let r, k ≥ 1, and let X be the (k+1)-ple (Veronese) embedding of Pr . Then X is not (Grassmann) (k, s − 1)-defective, for any s. Proof. By Proposition 1.3, this statement is equivalent to Theorem 2.9.

Remark 2.11. 1) Applying Corollary 2.10 to the case k = 1, r arbitrary, we obtain that the quadratic Veronese varieties, ν2 (Pr ), are not (1, s − 1)-defective for any s. This r+2 means that in the Grassmannian of P1 ’s of P( 2 )−1 , those lines in the secant Ps−1 ’s to ν2 (Pr ), have the expected dimension. This is true despite the fact that all the secant varieties to ν2 (Pr ) are defective! This is somewhat counter-intuitive, in that one might expect that if the secant Ps−1 ’s to a variety X were “not as many as one expects” then the linear subspaces of them would not be as many as one expects either. Our result shows that this kind of reasoning is faulty. On the other hand, there are varieties that have the expected behaviour with respect to all their secant varieties, but their Grassmannian of lines in the secant Ps−1 ’s have dimension less than expected. For instance, consider the surface X(1,1),(2,1) ⊂ P5 : all the secant varieties of X(1,1),(2,1) have the expected dimension (see Corollary 2.3), but it is classically known that the Grassmannian of lines in the secant P2 ’s to X(1,1),(2,1) is defective (as it follows from Proposition 1.3, and the 4th example in the list of defective varieties of §3). The paper [12] contains a complete classiﬁcation of the irreducible surfaces X ⊂ P5 for which the Grassmannians of lines contained in the secant P2 ’s to X is defective, and the curious behaviour of the smooth surfaces of minimal degree in P5 , i.e. scrolls and Veronese surface, is pointed out.

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2) Let’s now apply Corollary 2.10 to the case k = 2, r = 2, in which case we ﬁnd that the Veronese variety ν3 (P2 ) ⊂ P9 is not (2, s − 1)-defective, for any s. That is, the P2 ’s in the secant Ps−1 ’s to ν3 (P2 ) all have the expected dimension in the Grassmannian of 2-dimensional linear spaces in P9 . In particular, the P2 ’s in the secant P4 ’s to ν3 (P2 ) have the correct dimension in G(2, 9). As we will see below (and, in fact, as was known to Terracini) if Y = X(2,1),(3,1) ⊂ P19 then Y 5 = P19 as expected, i.e. δ4 (Y ) ≥ 1. Applying Proposition 1.3, we obtain that ν3 (P2 ) is (1, 4)-defective with δ1,4 (ν3 (P2 )) ≥ 1. I.e. the P1 ’s in the secant P4 ’s to ν3 (P2 ) do not have the expected dimension. We look at the secant P4 ’s to ν3 (P2 ) and we ﬁnd that the P2 ’s in them are exactly as many as we expect – but the P1 ’s in those P4 ’s are not as many as we expect. In the case t = 3, i.e. P1 ×P1 ×P1 = P(1,1,1) , the situation for all Segre-Veronese embeddings can be analyzed with the same methods used above. More speciﬁcally, if we want to consider the secant varieties of Xn,d , n = (1, 1, 1), d = (d1 , d2 , d3 ), we have to know (see Theorem 1.5) the Hilbert function (in degree d = d1 + d2 + d3 ) of the scheme of fat points Ws = (d2 + d3 )A1 + (d1 + d3 )A2 + (d1 + d2 )A3 + 2P1 + · · · + 2Ps ⊆ P3 , where A1 , A2 , A3 are coordinate points. A complete description of what happens is given by the following result: Theorem 2.12. Let d1 ≥ d2 ≥ d3 ≥ 1, and Y = X(1,1,1),(d1 ,d2 ,d3 ) be a SegreVeronese embedding of P1 × P1 × P1 . Then Y s has the expected dimension, except for: (d1 , d2 , d3 ) = (2, 2, 2), (d1 , d2 , d3 ) = (2α, 1, 1),

and

and

s = 7;

s = 2α + 1 (α any positive integer ).

In these cases Y s is defective, and its defect is 2 in the ﬁrst case and 1 in the second. The proof of the theorem uses the same kind of procedures as Theorem 2.9. A complete proof can be found in [7]. We also note that [11] gives a geometric description of all defective 3-folds. Corollary 2.13. Let d1 , d2 ≥ 1, and Y = X(1,1),(d1 ,d2 ) be a Segre-Veronese embedding of P1 ×P1 . Then Y is not Grassmann defective, except when (d1 , d2 ) = (2α, 1), and in this case Y is (1, 2α)-defective. Proof. By Proposition 1.3, this statement is equivalent to Theorem 2.12.

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3. Other results s The correspondence between the dimension of Xn,d and the Hilbert function of n a scheme Z made of s generic 2-fat points in P (see Corollary 1.7) allows us s from previous results on Segre varieties and from to deduce results on dim Xn,d properties of Hilbert functions. Namely we have (notation as in §1):

Proposition 3.1. Let Xn,d be a Segre-Veronese variety and s ≥ 2 be such that s s = ns+s−1 (the expected dimension). Then Xn,b has the (same) expected dim Xn,d dimension for any b such that bi ≥ di ∀i ∈ {1, . . . , t}. Proof. This is an immediate consequence of Corollary 1.7, since our hypothesis on s amounts to saying that for a generic scheme Z ⊂ Pn made of s 2-fat points, Xn,d H(Z, d) = s(n + 1) = length Z and this trivially implies that H(Z, b) = s(n + 1) =

length Z for all b such that bi ≥ di ∀i ∈ {1, . . . , t}. As an immediate consequence we have: Proposition 3.2. Let 1 ≤ n1 ≤ n2 ≤ · · · ≤ nt , and let Y = Xn,d be a SegreVeronese embedding of Pn = Pn1 × · · · × Pnt . If we are not in the case t = 2, d = (1, 1), then dim Y s = s(n1 + · · · + nt + 1) − 1 for all s ≤ n1 + 1. Proof. If t ≥ 3, from [6], Proposition 2.3, we have that H(Z, 1) = s(n1 + · · · + nt + 1) = length Z, so the conclusion of this Proposition follows from Proposition 3.1. So, all that remains is to consider the situation when t = 2. For this case we know that for d = (1, 1), Y s is defective for all 2 ≤ s ≤ n1 (e.g. see [6], Proposition 2.3 again). Thus, we will be done (again by Proposition 3.1) if we can show that n1 +1 n1 +1 and X(n are not defective. X(n 1 ,n2 ),(1,2) 1 ,n2 ),(2,1) Consider the set of n1 + 1 points M = {P0 , . . . , Pn1 } ⊂ Pn1 × Pn2 , where Pi is the coordinate point associated to the bihomogeneous ideal ˆi , . . . , xn1 ; y0 , . . . , yˆi , . . . , yn2 ) ℘i = (x0 , . . . , x in the ring k[x0 , . . . , xn1 ; y0 , . . . , yn2 ]. Let Z be the scheme of 2-fat points with support M and let I = ℘20 ∩ · · · ∩ ℘2n1 be the ideal associated to Z. Since I is a monomial ideal (the Pi ’s are coordinate points), we only need to show that the monomials not in I(1,2) and those not in I(2,1) are (n1 + 1)(n1 + n2 + 1) in number. The monomials not in I(1,2) are of two types: xi yi yj , with i = 0, . . . , n1 , j = 0, . . . , n2 ((n1 + 1)(n2 + 1) of them), xi yj2 , with i = j, both in {0, . . . , n1 } ((n1 + 1)n1 of them), in total a number of (n1 + 1)(n1 + n2 + 1) monomials not in I(1,2) , as required. The monomials not in I(2,1) are of two types: xi xj yk , with i = j, both in {0, . . . , n1 } and k = i or k = j ((n1 + 1)n1 of them), x2i yj , with i = 0, . . . , n1 , j = 0, . . . , n2 ((n1 + 1)(n2 + 1) of them),

in total a number of (n1 + 1)(n1 + n2 + 1) monomials not in I(2,1) as requested.

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Up to this point in this section we have only given results about Segre-Veronese varieties which are NOT defective. What follows is a (far from exhaustive) list of examples of defective Segre-Veronese varieties. The way one checks the defectivity in all these examples is the same: the s = PN , but instead numbers have been chosen in such a way that one expects Xn,d it is easy to ﬁnd a way to split d = b + c = (b1 , . . . , bt ) + (c1 , . . . , ct ) in such a way that there is a form f1 of multidegree (b1 , . . . , bt ) and a form f2 of multidegree (c1 , . . . , ct ) passing through s generic (simple) points, hence there is at least one form of multidegree d (namely, f1 f2 ) through s generic 2-fat points which was not supposed to exist. In the following list we always have m ≥ 1, and we give values s, n, d for which s is defective: Xn,d P1 × P2 , d = (1, 3), b = (0, 2), c = (1, 1), s = 5; ; P1 × Pm , d = (2k, 2), b = c = (k, 1), k ≥ 1, s = (2k+1)(m+1) 2 P2 × P2 , d = (2, 2), b = c = (1, 1), s = 8; P1 × P1 × Pm , d = (1, 1, 2), b = (1, 0, 1), c = (0, 1, 1), k ≥ 1, s = 2m + 1; P1 × Pm × Pm , d = (2k, 1, 1), b = (k, 1, 0), c = (k, 0, 1), k ≥ 1, s = km + k + m; P1 × Pr × Pm , d = (r + m, 1, 1), b = (r, 1, 0), c = (m, 0, 1), s = rm + r + m; P1 × P1 × Pm , d = (2, 2, 2), b = c = (1, 1, 1), m ≤ 3, s = 4m + 3; P2 × Pm × Pm , d = (2, 1, 1), b = (1, 1, 0), c = (1, 0, 1), s = 3m + 2; P1 × P1 × P2 × P5 , d = (2, 1, 1, 1), b = (1, 1, 1, 0), c = (1, 0, 0, 1), s = 11; P1 × P1 × P1 × P2m−1 , d = (m, 1, 1, 1), b = (m − 1, 1, 1, 0), c = (1, 0, 0, 1), m > 1, s = 4m − 1; P1 × P1 × P1 × P2m , d = (m, 1, 1, 1), b = (m − 1, 1, 1, 0), c = (1, 0, 0, 1), m ≥ 4, s = 4m − 1. Notice that the ﬁrst example above is the one we referred to in Remark 2.11. Also, the penultimate case for m = 1 is defective also, but it is not of the same s has dimension 13 and not 14); for this example see [8], kind as the others (Xn,d Example 2.2. Of course from these examples we can derive examples of Grassmann defectivity by using Proposition 1.3; we will just mention what we get from the last two cases: Corollary 3.3. Let Y = X(1,1,1),(m,1,1) be the Segre-Veronese embedding of P1 × P1 × P1 . Then Y is (Grassmann) (2m − 1, 4m − 2)-defective, for m ≥ 1 and (2m, 4m − 2)-defective, for m ≥ 4. The corollary shows that the Segre Veronese varieties given by the (2α, 1, 1)embedding of P1 × P1 × P1 is both defective (see Theorem 2.12) and Grassmann defective. This responds to a question of Ciro Ciliberto who asked if there could exist varieties with both kinds of defectivity simultaneously.

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4. Partially symmetric tensors Now we want to describe how to interpret our Segre-Veronese embeddings from the point of view of tensors (e.g. see [17] or [22] for the Veronese case and [6] for the Segre case). In order to look at the Veronese variety (or t-uple emone can consider the Segre variety bedding of Pn ) νt (Pn ) ⊂ PN , N = t+r r ν(Pr × · · · × Pr ), with t factors, in PM , M = (r + 1)t − 1, and then consider the action of the symmetric group St on PM where, if the variables in PM are {z(1,0,...,0),...,(1,0,...,0) , . . . , z(0,...,0,1),...(0,...,0,1) }, the action of an element σ ∈ St is deﬁned by σ(z(1,0,...,0),...,(1,0,...,0) , . . . , z(0,...,0,1),...,(0,...,0,1) ) = (zσ((1,0,...,0),...,(1,0,...,0)) , . . . , zσ((0,...,0,1),...,(0,...,0,1)) ). The invariant subspace of PM with respect to this action is actually a linear space ∼ = PN , and the linear equations which deﬁne it give the required symmetries for the tensors parameterized by the points of PM . We can view PM as the parameterizing space of all the (r+1)t tensors, and PN inside it as the subspace of symmetric ones: then the Segre variety and the Veronese parameterize the rank one (decomposable) tensors. Notice that the symmetric tensors of rank one correspond to forms that can be written as powers of linear forms. Notice also that when we say that a symmetric tensor has rank one, i.e that it is decomposable, we mean that it is decomposable as an element of the Tensor Algebra V ⊗ · · · ⊗ V = V ⊗t (where Pr = P(V )), not of the symmetric algebra Symt (V ). Consider for example a rational normal curve Ct ⊂ Pt ; we are used to viewing its ideal as generated by the 2 × 2 minors of a 2 × t catalecticant matrix of indeterminates (or also by the 2 × 2 minors of a diﬀerent catalecticant matrix, see e.g. [25]). From the point of view above we should look at the ideal of the t Segre embedding Vt : (P1 )t → P2 −1 , which is generated by the 2 × 2 minors of a 2 × 2 × · · · × 2 (t times) tensor (e.g. see [18] and [21]); the ideal of Ct comes from the ideal of Vt modulo the symmetry relations (given by the action of the t t symmetric group St on P2 −1 ) which deﬁne a linear space Pt in P2 −1 . This can be thought of as a more “complete” way to view those ideals, with respect to the usual way (as given by minors of catalecticant matrices) since the tensor represents their symmetries “more faithfully”. Now consider e.g. the case t = 3; we can think of “stopping halfway” between the Segre variety V3 (parameterizing 2 × 2 × 2 decomposable tensors in P7 ) and the rational normal curve Ct (which parameterizes decomposable 2 × 2 × 2 symmetric tensors) by considering the Segre-Veronese embedding V(2,1) of P1 × P1 of degree (2, 1) into P5 . We can consider the action of the symmetric group S2 on P7 which symmetrizes its variables xijk , i, j, k ∈ {0, 1} only with respect to i and j. The invariant space for this action is a linear space P5 ⊂ P7 , and it cuts V3 exactly in V(2,1) . Hence the

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variety V(2,1) parameterizes 2 × 2 × 2 “partially symmetric tensors”, i.e. tensors whose entries are symmetric only with respect to the ﬁrst two indeces. In general, consider P(r,r,...,r) = Pr × · · · × Pr , t times, and its Segre-Veronese embeddings Xn,d , n = (r, . . . , r) and d = (d1 , . . . , dt ), into the space PN , N = i Πti=1 r+d − 1. Let d = d1 + · · · + dt , and consider the Segre embedding of (Pr )d di into PM , where M = (r+1)d −1. We can view PN inside PM as the space of tensors which are invariant with respect to the action of the symmetric groups Sa1 ,. . . ,Sat on the variables with appropriate indices. So those are “partially symmetric” tensors. For example, for t = 1 we get symmetric tensors and Vn,d is the Veronese variety, while for d1 = · · · = dt = 1 they are generic tensors and Xn,d is simply the usual Segre embedding of (Pr )t . So the Segre-Veronese variety Xn,d , will parameterize the partially symmetric tensors (with respect to the action of Sd1 ,. . . ,Sdt ) in PM which are decomposable. Since those are the tensors of tensor rank 1 (e.g. see [6]), the secant varieties of Xn,d give the stratiﬁcation, by tensor rank, of those partially symmetric tensors.

References [1] J. Alexander, A. Hirschowitz. An asymptotic vanishing theorem for generic unions of multiple points. Inv. Math. 140 (2000), 303-325. [2] E.Arbarello, M.Cornalba. Footnotes to a paper of Beniamino Segre. Math. Ann. 256 (1981), no. 3, 341-362. [3] P. B¨ urgisser, M. Clausen, M.A.Shokrollahi. Algebraic Complexity Theory. Vol. 315, Grund. der Math. Wiss., Springer, 1997. [4] S ¸ .B˘ arc˘ anescu, N. Manolache. Betti numbers of Segre-Veronese singularities. Rev. Roumaine Math. Pures Appl. 26 (1981), 549–565. [5] M.V.Catalisano, A.V.Geramita, A.Gimigliano. On the Secant Varieties to the Tangential Varieties of a Veronesean. Proc. A.M.S. 130 (2001), 975-985. [6] M.V.Catalisano, A.V.Geramita, A.Gimigliano. Rank of Tensors, Secant Varieties of Segre Varieties and Fat Points. Linear Alg. Appl. 355, (2002), 261-285. See also the Errata of the Publisher: 367 (2003), 347-348. [7] M.V.Catalisano, A.V.Geramita, A.Gimigliano. Higher Secant varieties of Segre embeddings of P1 × P1 × P1 . Preprint 2003. [8] M.V.Catalisano, A.V.Geramita, A.Gimigliano. Higher Secant varieties of the Segre varieties P1 × · · · × P1 . Accepted for Publication, Journal of Pure and Applied Algebra, (2004). [9] K.Chandler. A brief proof of a maximal rank theorem for generic double points in projective space. Trans. Amer. Math. Soc. 353 (2000), 1907-1920. [10] L.Chiantini, C.Ciliberto. Weakly defective varieties. Trans. Am. Math. Soc. 354 (2001), 151-178.

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[11] L. Chiantini, C. Ciliberto. On the Classiﬁcation of Defective Threefolds, arXiv.math.AG 0312518v3, 18 Jun2004. [12] L.Chiantini, M.Coppens. Grassmannians for secant varieties. Forum Math. 13 (2001), 615-628. [13] C. Ciliberto, R. Miranda. Degenerations of linear planar systems. J. Reine Angew. Math. 501 (1998), 191-220. [14] C. Dionisi, C.Fontanari. Grassmann defectivity ` a la Terracini. Preprint (AG0112149). [15] L.D. Garcia, M. Stillman, B. Sturmfels. Algebraic Geometry of Bayesian Networks. Preprint 2003. [16] D.Geiger, D.Hackerman, H.King, C.Meek. Stratiﬁed Exponential Families: Graphical Models and Model Selection. Annals of Statistics 29 (2001), 505-527. [17] A.V.Geramita. Inverse Systems of Fat Points: Waring’s Problem, Secant Varieties of Veronese Varieties and Parameter Spaces for Gorenstein Ideals. Queen’s Papers in Pure and Applied Math. 102, The Curves Seminar at Queens’, vol. X (1996) 3-104. [18] A.V.Geramita. Catalecticant varieties. Commutative algebra and algebraic geometry (Ferrara), 143–156, Lecture Notes in Pure and Appl. Math., 206, Dekker, New York, 1999. [19] A. Gimigliano. Our Thin Knowledge of Fat Points. The Curves Seminar at Queen’s, Vol. VI (Kingston, Ont. 1989) Queen’s Papers in Pure and Applied Math., vol. 83, Queen’s University, Kingston, Ont. Canada, 1989, Exp. No. B 50 pp. [20] B. Harbourne. Problems and progress: a survey on fat points in P2 . Zero-dimensional Schemes and Applications (Naples 2000), Queen’s Papers in Pure and Appl. Math., vol. 123, Queen’s Univ., Kingston, Ont. Canada, 2002, pp. 85-132. [21] J.Harris. Algebraic Geometry, a First Course. Springer-Verlag, New York (1993). [22] A.Iarrobino, V.Kanev. Power Sums, Gorenstein algebras, and determinantal loci. Lecture Notes in Math. 1721, Springer, Berlin, (1999). [23] M.Id` a. Private communication. [24] A. Laface, L. Ugaglia. Quasi-homogeneous linear systems on P2 with base points of multiplicity 5. Canad. J. of Math. 55 (2003), 561-575. [25] M. Pucci. The Veronese variety and catalecticant matrices. J. Algebra 202 (1998), no. 1, 72–95. [26] A.Terracini. Sulle Vk per cui la variet` a degli Sh (h+1)-seganti ha dimensione minore dell’ordinario. Rend. Circ. Mat. Palermo 31 (1911), 392-396. [27] A.Terracini. Sulla rappresentazione delle coppie di forme ternarie mediante somme di potenze di forme lineari. Ann. Mat. Pur ed appl. XXIV, III (1915), 91-100. [28] A.Terracini. Su due problemi, concernenti la determinazione di alcune classi di superﬁcie, considerati da G.Scorza e da F. Palatini. Atti Soc. Natur. e Matem. Modena 6 (1921), 14-27. [29] S. Yang. Linear systems in P2 with base points of bounded multiplicity. arXiv:math.AG/0406591 v1 29 Jun 2004.

Higher secant varieties of Segre-Veronese varieties Maria Virginia Catalisano D.I.P.E.M. Universit` a di Genova, Italy Email: [email protected] Anthony V. Geramita Dept. Math. and Stats. Queens’ University, Kingston, Canada and Dip. di Matematica Universit` a di Genova, Italy Email: [email protected]; [email protected] Alessandro Gimigliano Dip. di Matematica and C.I.R.A.M. Universit` a di Bologna, Italy Email: [email protected]

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An appendix to a paper of Catalisano, Geramita, Gimigliano: The Hilbert function of generic sets of 2-fat points in P1 × P1 Adam Van Tuyl

Abstract. Implicit in the paper of Catalisano, Geramita, and Gimigliano [1] is a formula for the Hilbert function of generic sets of 2-fat points in P1 × P1 . We make this result explicit in this note. 2000 Mathematics Subject Classiﬁcation: 13D40, 14M05

We shall use the deﬁnitions and notation as found in [1]. Let d1 , d2 be two positive integers and consider the fat point scheme W = d1 Q1 + d2 Q2 + 2R1 + · · · + 2Rs ⊆ P2 where Q1 , Q2 , R1 , . . . , Rs are s + 2 points in generic position. Proposition 2.1 of [1] describes some properties of the Hilbert function of W . Furthermore, this proposition is used (see [1, Corollary 2.2]) to compute the dimensions of the secant varieties of P1 ×P1 embedded into P(d1 +1)(d2 +1)−1 by the morphism given by OP1 ×P1 (d1 , d2 ). By Terracini’s Lemma (see §1 of [1]) there is a relationship between the dimensions of these embedded varieties and the Hilbert functions of generic sets of 2-fat points in P1 × P1 . Thus, implicit in [1, Corollary 2.2] is a formula for H(Z, (d1 , d2 )) when Z = 2P1 + · · · + 2Ps ⊆ P1 × P1 is a generic set of 2-fat points and (d1 , d2 ) ≥ (1, 1). Because of the recent work on the Hilbert functions of points (for the reduced case, see [2, 6] and for the nonreduced case, see [3, 4]) in multiprojective spaces, it is of interest to have an explicit description of this formula. In fact, by coupling [1, Proposition 2.1] with a result from [5], we can describe H(Z, (i, j)) for all (i, j) ∈ N2 .

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Theorem 1. Suppose Z = 2P1 + · · · + 2Ps ⊆ P1 × P1 is a generic set of 2-fat points. Then ⎧ if j = 0, ⎪ ⎪ min{(i + 1), 2s} ⎪ ⎪ min{(j + 1), 2s} if i = 0, ⎪ ⎪ ⎪ ⎪ min{(i + 1)(j + 1), 3s} if (i, j) ≥ (1, 1) and ⎪ ⎪ ⎨ (i, j) ∈ {(2, s − 1), (s − 1, 2)}, H(Z, (i, j)) = 3s if s ≡ 0 (mod 2) and ⎪ ⎪ ⎪ ⎪ (i, j) ∈ {(2, s − 1), (s − 1, 2)}, ⎪ ⎪ ⎪ ⎪ 3s − 1 if s ≡ 1 (mod 2) and ⎪ ⎪ ⎩ (i, j) ∈ {(2, s − 1), (s − 1, 2)}. Proof. Suppose that (i, j) ∈ N2 with j = 0. Let π1 : P1 × P1 → P1 be the projection morphism given by π1 (P × Q) = P . If we set Z1 := π1 (Z) ⊆ P1 , then by [5, Lemma 4.1] we have H(Z, (i, 0)) = H(Z1 , i) for all i ∈ N. Because the support of Z is in generic position, the ﬁrst coordinates of the points P1 , . . . , Ps must all be distinct. Thus Z1 is a set of s double points in P1 , and hence H(Z, (i, 0)) = H(Z1 , i) = min{(i + 1), 2s} for all i ∈ N. Notice that if (i, j) ∈ N2 with i = 0, then the proof is the same except that the projection morphism π2 : P1 × P1 → P1 deﬁned by π2 (P × Q) = Q is used. So, we need to compute H(Z, (i, j)) when (i, j) ≥ (1, 1). By [1, Theorem 1.1] (and the discussion at the start of §2 in [1]) we have H(Z, (i, j))

= dimk R(i,j) − dimk (IZ )(i,j) = (i + 1)(j + 1) − dimk (IW )i+j

(1)

where IW is the deﬁning ideal in S = k[x, y, z] of the fat point scheme W = iQ1 + jQ2 + 2R1 + · · · + 2Rs ⊆ P2 and Q1 , Q2 , R1 , . . . , Rs are s + 2 points in generic position. The value of H(W, i + j) can be computed using [1, Proposition 2.1]. In particular, H(W, i + j) = min{dimk Si+j , deg W }

(2)

except when s = 2a + 1 and (i, j) = (2, 2a) or (2a, 2). (Note that although [1, Proposition 2.1] assumes that i ≥ j, if (i, j) ∈ N2 with i < j, then we can swap the roles of i and j to compute H(W, i + j).) In the two exceptional cases, [1, Proposition 2.1] demonstrated that h0 (IW (i + j)) = dimk (IW )i+j = 1.

(3)

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j+1 Using the fact that deg W = 3s + i+1 2 + 2 , we can use (2) and (3) to derive the following formula for dimk (IW )i+j : ⎧ ⎪ ⎪ max{0, (i + 1)(j + 1) − 3s} if (i, j) ≥ (1, 1) and ⎪ ⎪ (i, j) ∈ {(2, s − 1), (s − 1, 2)}, ⎪ ⎪ ⎨ 0 if s ≡ 0 (mod 2) and dimk (IW )i+j = (i, j) ∈ {(2, s − 1), (s − 1, 2)}, ⎪ ⎪ ⎪ ⎪ 1 if s ≡ 1 (mod 2) and ⎪ ⎪ ⎩ (i, j) ∈ {(2, s − 1), (s − 1, 2)}. We now substitute the above values into (1). If (i, j) ≥ (1, 1) and (i, j) ∈ {(2, s − 1), (s − 1, 2)}, then H(Z, (i, j))

=

(i + 1)(j + 1) − max{0, (i + 1)(j + 1) − 3s}

=

min{(i + 1)(j + 1), 3s}.

When (i, j) ∈ {(2, s − 1), (s − 1, 2)}, then (i + 1)(j + 1) = 3s. So, if s ≡ 0 (mod 2), then H(Z, (i, j)) = 3s − dimk (IW )i+j = 3s. However, if s ≡ 1 (mod 2), then H(Z, (i, j)) = 3s − dimk (IW )i+j = 3s − 1.

This completes the proof.

Acknowledgments. I would like to thank A.V. Geramita for his suggestion to write this appendix. As well, I would like to acknowledge the ﬁnancial support provided by NSERC.

References [1] M.V. Catalisano, A.V. Geramita, A. Gimigliano, Higher Secant Varieties of SegreVeronese varieties (2004). This volume, pp. 81–108. [2] S. Giuﬀrida, R. Maggioni, A. Ragusa, On the postulation of 0-dimensional subschemes on a smooth quadric. Paciﬁc J. Math. 155 (1992) 251–282. [3] E. Guardo, Fat point schemes on a smooth quadric. J. Pure Appl. Algebra 162 (2001) 183-208. [4] E. Guardo, A. Van Tuyl, Fat Points in P1 × P1 and their Hilbert functions. Canad. J. Math 56 (2004) 716–741. [5] J. Sidman, A. Van Tuyl, Multigraded regularity: syzygies and fat points. (2005) To appear in Beitr¨ age zur Algebra und Geometrie [6] A. Van Tuyl, The border of the Hilbert function of a set of points in Pn1 × · · · × Pnk . J. Pure Appl. Algebra 176 (2002) 223–247.

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Adam Van Tuyl Department of Mathematical Sciences Lakehead University Thunder Bay, ON, P7B 5E1, Canada Email: [email protected]

The 3-cuspidal quartic and braid monodromy of degree 4 coverings Fabrizio Catanese and Bronislaw Wajnryb∗

Abstract. Motivated by the study of the diﬀerential and symplectic topology of (Z/2)2 Galois covers of P1 ×P1 , we determine the local braid monodromy of natural deformations of smooth (Z/2)2 - Galois covers of surfaces at the points where the branch curve has a nodal singularity. The study of the local deformed branch curves is solved via some interesting geometry of projectively unique objects: plane quartics with 3 cusps, which are the plane sections of the quartic surface having the twisted cubic as a cuspidal curve. 2000 Mathematics Subject Classiﬁcation: 14J80, 14N25, 57R17, 57R50, 57R52, 57R17, 57M12, 58K15, 32S50, 13B99.

1. Introduction This article is a continuation of a preceding one ([C-W]), which was devoted to the proof that the so called (a, b, c)-surfaces (where we take a, b, c ∈ N with b and a + c ﬁxed) provide examples of simply connected algebraic surfaces which are diﬀeomorphic but not deformation equivalent. The (a, b, c)-surfaces are coverings of P1 × P1 of degree 4 and are deﬁned by 2 equations z2

=

f (x, y)

2

=

g(x, y),

w

(1)

where f and g are bihomogeneous polynomials , belonging to respective vector spaces of sections of line bundles: f ∈ H 0 (P1 × P1 , OP1 ×P1 (2a, 2b)) and g ∈ H 0 (P1 × P1 , OP1 ×P1 (2c, 2b)). A question which was left open in [C-W] was the symplectic equivalence of the above (a, b, c)-surfaces. To this purpose, and for more general purposes, it is important to determine the braid monodromy factorization of the branch curve cor∗ The research was performed in the realm of the SCHWERPUNKT ”Globale Methode in der komplexen Geometrie”, and of the EAGER EEC Project.

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responding to a symplectic deformation of the 4-1 covering S → P1 × P1 possessed by an (a, b, c)-surface S (note that in [C-W] one key result was the determination of the mapping class group monodromy factorization, which is a homomorphic image of the braid monodromy factorization). In this paper we approach the ﬁrst step, namely, we determine the local braid monodromy factorization of 4-1 coverings which are deformations of bidouble covers ((Z/2)2 - Galois covers). The discriminant picture that we get is somehow unexpected, in that instead of the usual swallowtail surface we obtain a rational quartic surface with a twisted cubic as cuspidal curve. We show in the last section, namely in Theorem 6.1, that such surface is projectively unique, being the tangential developable of the twisted cubic, or equivalently, the dual surface of the twisted cubic curve in P3 , or the discriminant of the general equation of degree 3. As hinted at in the ﬁrst section, the picture is not completely unexpected, especially the fact that the quartic is a discriminant surface for the general equation of degree 3, since actually Galois theory teaches us that deformed Galois covers of degree 4 are exactly the trick to relate the solvability of the general equation of degree 4 to the solvability of the general equation of degree 3. It follows that our perturbed local discriminant curve of S → P1 × P1 is a plane quartic curve Δ with three cusps, and bitangent to the line at ∞ in two real points. Viewing the curve as a small perturbation of a pair of real lines counted with multiplicity two made the determination of the braid monodromy of this aﬃne curve almost impossible. The trick which solved the problem is the following well known observation: a three cuspidal quartic over an algebraically closed ﬁeld is projectively unique, since it is the dual curve of a nodal plane cubic curve. We can then change the real picture and take a nodal cubic with an isolated double point, but with three real ﬂexes: its dual curve, once we take as line at inﬁnity the dual line of the nodal point, will be a quartic C with three real cusps, and bitangent at the line at inﬁnity in two imaginary points. For C the points with a real abscissa x which are interesting for the determination of the braid monodromy have now an ordinate y which is either real or imaginary, and it is quite easy to calculate then the braid monodromy factorization. From this one, since Δ is complex aﬃne equivalent to C, we deduce the braid monodromy factorization for Δ. Lack of time prevents us to analyse the question whether one can similarly determine the local braid monodromy factorization for a deformed abelian cover. This would also be a very useful result in the study of the diﬀerential and symplectic topology of huge classes of algebraic surfaces.

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115

2. The discriminant of a deformed bidouble cover. Consider a ring A of characteristic p = 2 and a so called simple bidouble cover of A, i.e., a (Z/2)2 - Galois ring extension A ⊂ B where B is the quotient ring of A[z, w] given, for some choice of u, v ∈ A, by z2

=

v

w2

=

u.

(2)

A deformed bidouble cover is a ﬁnite ring extension A ⊂ B given, for u, v, a, b ∈ A, by z2

=

v + aw

w2

=

u + bz.

(3)

Observe that, if a is invertible, then w = a−1 (z 2 − v), and we get the quartic equation a−2 (z 2 − v)2 − bz − u = 0, equivalent to z 4 − 2z 2 v − a2 bz + (v 2 − a2 u) = 0 and that, since p = 2, every quartic equation can be reduced, by a translation (Tschirnhausen transformation), to the above equation, for a suitable choice of v, b, u. This standard trick, which allows to deform a bidouble Galois extension to the general quartic equation is very important in Galois theory. Because, in the z, w plane we have the pencil of conics generated by the above two parabolae, and the determinant function on the parameter of the pencil provides an equation of degree three, whose Galois group corresponds to the image of the Galois group of the quartic equation under the surjection S4 → S3 with kernel (Z/2)2 . We are however interested in a ﬁner geometric question, we do not only look at algebraic extensions of function ﬁelds, indeed we look more closely at ﬁnite coverings of smooth algebraic varieties. The concepts of bidouble cover, and deformed bidouble cover have been introduced, in the global case of coverings of smooth algebraic varieties, in [Cat1], the latter under the name of natural deformations of bidouble covers. We refer the reader for details to [Cat1], and as well to [C-W] for the applications we have in mind. We observe that B is a rank 4 free A-module, with basis 1, z, w, zw, and that to each nontrivial basis element corresponds the respective multiplication matrix ⎛

0 v ⎜1 0 Mz = ⎜ ⎝0 a 0 0

0 0 0 1

⎛ ⎞ au 0 ⎜0 ab ⎟ ⎟ , Mw = ⎜ ⎝1 v⎠ 0 0

0 u 0 b 0 0 1 0

⎛ ⎞ bv 0 au bv ⎜0 ab u u⎟ ⎟ , Mzw = ⎜ ⎝0 v ab ab⎠ 0 1 0 0

⎞ uv bv ⎟ ⎟. au⎠ ab

Consider now the diﬀerent R (ramiﬁcation Cartier divisor) of the ring extension, 2z −a R = det = 4zw − ab. −b 2w We can then ﬁnd the discriminant Δ as the norm of R, thus

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⎛ ⎞ −ab/4 au bv uv ⎜ 0 ab 3ab/4 u bv ⎟ ⎟, Δ = 44 det Mzw − Id = 44 det ⎜ ⎝ ⎠ 0 v 3ab/4 au 4 1 0 0 3ab/4 and −

9 27 1 Δ = −u2 v 2 − uv(ab)2 + b2 v 3 + a2 u3 + 2 a4 b4 =: P (u, v, a2 , b2 ). 162 8 16

We see immediately that, setting α := a2 , β := b2 , P (u, v, α, β) is homogeneous of degree 4, and symmetric for the involution (u, α) ↔ (v, β), whose ﬁxed point locus is not contained in {P = 0} (this symmetry is forced by the symmetry exchanging a with b, w with z, u with v). Theorem 2.1. The quartic hypersurface P ⊂ P3 deﬁned by P (u, v, α, β) = 0 is irreducible and has as singular locus a twisted cubic curve Γ, which is a cuspidal curve for P . In particular P is projectively unique, being the dual surface of the twisted cubic. P is also the tangential developable of the twisted cubic, and the discriminant surface of the space P3 of polynomials of degree 3 on P1 . Remark 2.2. Chapter V of [SupRaz] is devoted to more general surfaces of degree 4 in P3 which have a twisted cubic as double curve. Proof. We calculate for later use 9 9 ∂P/∂u = −2uv 2 − (ab)2 v + 3a2 u2 , ∂P/∂v = −2vu2 − (ab)2 u + 3b2 v 2 8 8 and moreover 9 54 9 54 ∂P/∂α = − uv(b)2 + u3 + 2 a2 b4 , ∂P/∂β = − uv(a)2 u + v 3 + 2 a4 b2 , 8 16 8 16 whence in particular u(∂P/∂u) − v(∂P/∂v) = 3a2 u3 − 3b2 v 3 , α(∂P/∂α) − β(∂P/∂β) = a2 u3 − b2 v 3 . We conclude that the hypersurface P , i.e., {P = 0}, is irreducible, being reduced and being the image of the quadric Q := {4zw − ab = 0}. We claim now that P is singular along a twisted cubic Γ, which is of cuspidal type. In view of the irreducibility of P , we will then conclude that there is no other singular curve on P . In order to do this, let us work with aﬃne coordinates on Q setting a = 1, whence b = 4zw and z, w are aﬃne coordinates. In terms of these coordinates, α = 1, β = 16z 2w2 , u = w2 − 4z 2 w, v = z 2 − w. We conclude that F : Q → P factors through (z, w) → (s := z 2 , w) and G(s, w) := (16sw2 , w2 − 4sw, s − w).

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We calculate the derivative matrix of G, ⎛

16w2 DG = ⎝ −4w 1

⎞ 32sw 2w − 4s⎠ −1

which has rank equal to 1 exactly for w + 2s = 0. Observe moreover that in these points the kernel of DG is given by the tangent vector ∂/∂s + ∂/∂w. An immediate calculation shows that the image curve Γ is the twisted cubic (64s3 , 12s2 , 3s) and one may verify that on Γ all the four partial derivatives of P do vanish. It follows that Γ is the only singular curve of P (an irreducible quartic curve has at most three singular points), and that it is a cuspidal curve, since if we intersect P with a general plane, then we get a curve in the (s, w) plane which, at an intersection point with w + 2s = 0, is tangent to the kernel of DG, but maps with local degree one (since , for instance, the line w = s + c maps to the plane v = −c by s → (16(s + c)2 s, (s + c)(c − 3s), −c)). We conclude also easily that Sing(P ) = Γ. Since, if p were another singular point of P , any plane through p would intersect P in a reducible curve; but projection with centre p yields a double cover of the plane P2 , and a general line cannot be tangent to the branch locus, thus we get a contradiction. Let now X ⊂ P3 be a quartic surface which has a twisted cubic curve Γ as cuspidal curve: then it follows by Theorem 6.1 proved in the last section that X is unique, whence it coincides with the tangential developable of Γ. An alternative argument is as follows: if two general polar surfaces

Σi yi ∂X/∂xi = 0, Σi zi ∂X/∂xi = 0

are shown to intersect along Γ with multiplicity three, then the dual variety of X is a curve D. Unfortunately, as pointed out by the referee, this statement is not obvious for a general X as above, but indeed for our explicit surface P a direct calculation with Macaulay shows that the dual variety of P is a curve D (and also that D has degree 3, but we do not need this fact). Once we know that the dual variety of X is a curve D, by biduality, X is a developable surface which is not a cone, so X is a tangential developable, and its singular curve Γ must be the edge of regression. We conclude thus that X is the tangential developable of Γ, and that the dual variety D of X is the curve of osculating planes of Γ. We conclude also that D is then a twisted cubic curve, so X is the dual surface of a twisted cubic curve. X is also projectively unique since D is projectively unique.

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3. The 2-dimensional picture In this section we shall assume that u, v are local coordinates in the plane (or local parameters for a two-dimensional local ring A), and that a, b are local holomorphic functions at the origin, which are invertible and take small values in a neighbourhood of the origin (respectively, a, b are units of A). Then we write b = ca, and consider new coordinates U, V such that u = a2 U , v = a2 V : in these new coordinates our discriminant Δ is divisible by a8 , and after dividing by −44 a8 we obtain the function 9 27 δc := −U 2 V 2 − U V (c)2 + c2 V 3 + U 3 + 2 c4 . 8 16 In other words, we could have assumed without loss of generality that a = 1. We can further simplify the above equation by taking a cubic root λ of c, and considering new coordinates u , v with U = λ4 u0 , V = λ2 v0 : then our equation, after dividing by λ12 becomes 9 27 δ := −u20 v02 − u0 v0 + v03 + u30 + 2 . 8 16 Let us determine exactly the singular points of this curve, where for simplicity of notation we replace u0 , v0 by u, v respectively. In other words, we could have assumed from the onset a = b = 1, and we consider 9 27 δ(u, v) := P (u, v, 1, 1) = −u2 v 2 − uv + v 3 + u3 + 2 , 8 16 and our previous calculation of u(∂P/∂u) − v(∂P/∂v) shows that the singular points satisfy u3 = v 3 ; since however the origin does not lie on our curve, we may set, for such a singular point, u = ζv, where ζ is a cubic root of 1, and v = 0. We look now at (∂P/∂u) = −2ζv 3 − 98 v + 3ζ 2 v 2 = 0, but disregarding the root 9 2 ζ − 32 ζv = (v − 34 ζ)2 = 0. v = 0; whence, we get the equation v 2 + 16 Thus we conclude that the three cuspidal points are the three points u=

3 2 3 ζ , v = ζ (where ζ 3 = 1). 4 4

Our goal is to understand the local braid monodromy of the curve δ = 0, for a good projection given by a linear form x. In order to understand the forthcoming calculations, observe that our curve Δ is the image of the ramiﬁcation curve 4zw = 1, thus we have a degree 4 rational function x on P1 , which must be branched on 6 points. Three of these will correspond to the images of the 3 cusps, 2 will come from the two points at inﬁnity where the line z = 0 is tangent, whence there will be exactly another branch point corresponding to a line x = x0 tangent at a smooth point. Therefore the factors of the braid monodromy factorization will be four, one half twist, and three cubes of a half twist. In the next section we are going to calculate it for a suitable choice of coordinates.

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4. Making the three cusps real. As we mentioned in the introduction, a three cuspidal quartic has as dual curve a rational irreducible cubic (by Pl¨ ucker’s formulae its degree is 4 × 3 − 3 × 3 = 3) which is by biduality nodal, since the dual of a cuspidal cubic is a cuspidal cubic. Indeed the node is dual to the bitangent line at inﬁnity. Our curve Δ will have two real tangents, and as a consequence three ﬂexes which are not all real, since only one of the three cusps is real. We easily construct however a quartic with three real cusps if we take the dual curve of the aﬃne curve D := {(X, Y )|F (X, Y ) = Y 2 − X 2 (X − 1) = 0}. Since in homogeneous coordinates (X, Y, Z) we have F (X, Y, Z) = Y 2 Z − X 2 (X − Z), hence the gradient of F is given by ∇F = (−3X 2 + 2XZ, 2Y Z, X 2 + Y 2 ), in view of the standard parametrization of D given by X = (t2 + 1), Y = t(t2 + 1), Z = 1 (for which t = ∞ goes to the point at inﬁnity of D), we get a parametrization of the dual curve C as (−3(1 + t2 ) + 2, 2t, (1 + t2 )2 ). and correThe two ﬂexes not at inﬁnity occur for X = 4/3, and t = ±(3)−1/2 , √ spondingly we have the two ﬂexes (−2, ±2 · 3−1/2 , (4/3)2 ) = (− 89 , ± 38 3, 1) (the third ﬂex occurs at the origin). Using the above parametrization or using the computer we may calculate the equation of C as C := {(x, y)|(x2 + y 2 )2 + x3 + 9xy 2 +

27 2 y = 0}. 4

The advantage of this equation is that it is biquadratic in y, so y 2 is solution of the quadratic equation 27 2 2 2 (y ) + 2x + 9x + y 2 + (x3 + x4 ) = 0 4 thus

' 272 27 27 × 9 2 x+ , 2y = − 2x + 9x + ± 32x3 + 108x2 + 4 2 16 2

2 3 4 and, if we set A := (2x2 +9x+ 27 4 ) , the discriminant Θ = A −4(x +x ) appearing in the above square root is clearly positive for x > 0, and clearly vanishes for x = − 98 . On the other side, twice the derivative of Θ equals 192x2 + 432x + 243 = 3(8x + 9)2 ≥ 0, thus Θ is strictly monotone, whence Θ is positive exactly for x > − 98 .

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To complete the picture, observe that Θ > A2 iﬀ x4 +x3 < 0, i.e., iﬀ −1 < x √ < 0, 27 ) is positive exactly outside the interval 34 (−3± 3), and that A(x) = 2((x+ 94 )2 − 16 √ and note that 34 (3 − 3) < 1. We have thus the following picture:

y-axis

r

- 98

r r

-1

r

0

x-axis

r

The real part of the curve C.

Using the previous description, and looking at the above picture, we may now easily describe the motion of the roots y as x moves along the real axis from +∞ to − 98 . For x > 0 we have exactly 4 imaginary roots A1 (x) = iY1 (x), A2 (x) = iY2 (x), B2 (x) = −iY2 (x), B1 (x) = −iY1 (x), where Y1 (x), Y2 (x) ∈ R and Y1 (x) > Y2 (x) > 0. For x = 0, A2 , B2 become = 0, and then for −1 < x < 0 A2 , B2 become real and opposite, B2 is positive and grows as x decreases, while A1 remains imaginary and with decreasing absolute value. For x = −1 A1 , B1 become = 0, while B2 = 14 , A2 = − 41 , ﬁnally in the interval 9 − 8 < x < −1 we have 4 distinct real roots B2 (x) > B1 (x) > 0 > A1 (x) = −B1 (x) > A2 (x) = −B2 (x) and both roots √ B2 (x) > B1 (x) grow as x approaches − 98 : for this value we have B2 = B1 = 38 3.

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5. Braid monodromy of C and fundamental group of the complement We take as projection of the pair (C2 , C) a linear form very close to x, since the projection x has a double critical value x = − 89 : the eﬀect is then that we split the corresponding braid into two factors. √ In order to get a simple picture let us take as base point x0 = 34 ( 3 − 3). Then −1 < x0 < 0, and we have two real roots A2 , B2 and two purely imaginary roots A1 , B1 as in Figure 2, left side. If we move on the real axis to the right to x = 0 then the two real roots meet at zero and we have a horizontal vanishing arc connecting the real roots with a 3/2 -twist around this arc as local monodromy. If we move on the real axis to the left to x = −1 then the two complex roots meet at zero and we get a vertical vanishing arc connecting the imaginary roots with one half-twist around this arc as local monodromy. If instead of going directly to x = −1 we make a half turn around it counterclockwise and continue along the real axis to the left then the imaginary roots turn π/4 counterclockwise around zero, the top root A1 becomes real negative and ”runs after A2 ” and the bottom root B1 becomes real positive and ”runs after B2 ”. The roots meet for the appropriate critical values of x (in a neighbourhood of − 98 ). The corresponding vanishing arcs are just straight intervals connecting A1 with A2 and B1 with B2 . They are disjoint so the order of these two critical values of x is not important. The corresponding monodromy factors are 3/2-twists around these arcs and they commute.We take a cyclic order of the paths counterclockwise around x0 starting ﬁrst with the two critical values near − 89 , then proceeding with the critical value ’x = −1’ (actually , near x = −1: recall in fact that we changed slightly the axis of projection to split the two critical values of x near − 98 , and ﬁnally ending with ’x = 0’. The base point is usually chosen far away from the critical values. We should do this also in our case since we deal with a local picture and we may at a later convenience want to relate it to a global picture where other monodromy factors occur, coming from other critical values. We move then the base point to the right along the real axis, passing x = 0 on the right, making a half-turn clockwise around it. The paths from x0 to the critical values of x will be dragged along. The real roots A2 and B2 get closer to zero and then turn clockwise around zero by a (3/2)π turn (in fact, a full turn of x around x = 0 produces a 3/2-turn of the roots). The whole picture of vanishing arcs moves with them. We get a conﬁguration as on the right side of Figure 2. When x moves further to the right along the real axis the roots remain purely imaginary and just move further away from zero. We have thus obtained a complete determination of the braid monodromy of C, which is illustrated by Figure 2 and summarized in the following

Theorem 5.1. Consider the arcs depicted on the right side of Figure 2.

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Then the braid monodromy of C is given, in this order, by the cube of a halftwist around the arc connecting A1 and A2 , by the cube of a half-twist around the arc connecting B2 and B1 , by the half-twist around the arc connecting A1 and B1 and ﬁnally by the cube of a half-twist around the arc connecting A2 and B2 .

A1 r

A2 r

r B2

r

A1 r $ ' B2 r # r ! A2 % r & " B1

B1

Figure 2. Vanishing arcs Let us now consider again the left part of Figure 2 and let us take as base point for the fundamental group of the ﬁbre C − {A1 , A2 , B2 , B1 } a point y0 with large positive real part and small positive imaginary part. We consider then a geometric basis α1 , α2 , β2 , β1 of π1 (C−{A1 , A2 , B2 , B1 }, y0 ), where for instance α1 is given by a subsegment sA1 on the segment joining y0 with A1 , followed by a full small circle around A1 , and then followed by the inverse path of sA1 (the other loops are deﬁned similarly). By the van Kampen theorem the fundamental group π1 (C2 −C, y0 ) is generated by α1 , α2 , β2 , β1 subject to the relations coming from the braid monodromy, thus we obtain a presentation π1 (C2 − C, y0 ) =< α1 , α2 , β2 , β1 |α1 α2 α1 = α2 α1 α2 , β1 β2 β1 = β2 β1 β2 , α2 β2 α2 = β2 α2 β2 , β2 β1 = α1 β2 > . We need only explain the last relation, coming from the relations σ(γ) = γ, where σ is the half twist on the curve τ , corresponding to the vertical tangency for x = −1, which makes the two roots A1 , B1 become equal. The action of this half twist σ is the following: • α1 → α1 β2 β1 β2−1 α−1 1 • α2 → α1 β2 β1−1 β2−1 α2 β2 β1 β2−1 α−1 1 • β 2 → β2 • β1 → β2−1 α1 β2 .

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We end by describing the monodromy homomorphism μ of the degree 4 covering π1 (C2 − C, y0 ) → S4 . The action of μ must then be, up to conjugation in S4 , the following one: • α1 → (1, 2) • α2 → (2, 3) • β2 → (2, 4) • β1 → (1, 4). In fact, we observe ﬁrst that (**) the generators α1 , α2 , β2 , β1 must map to transpositions, and the image of μ is transitive. One sees then, using the presentation of π1 (C 2 − C), that (***) there is a unique homomorphism of π1 (C 2 −C) into S4 (up to conjugation in S4 ) which satisﬁes (**). If instead we want to calculate the fundamental group of the complement π1 (P2 −C), y0 ) we must add the relation α1 α2 β2 β1 = 1 and we can then simplify the presentation obtaining π1 (P2 − C, y0 ) =< α1 , α2 |α1 α2 α1 = α2 α1 α2 , α2 α1 α1 α2 = 1 >, which is the spherical braid group of three points in P1 , as shown long ago by Zariski, and also in greater generality by Moishezon (cf. [Zar], [Moi1]). This calculation shows that the degree four covering is also branched on the line at inﬁnity, where the local monodromy is the double transposition (1, 3)(2, 4) (in the Galois case we have three branch lines, corresponding to the three nontrivial elements of (Z/2)2 , and we have the standard model for a bidouble cover given by a special projection of the Veronese surface, as described in [Cat4], page 100. Let us brieﬂy recall it: consider the Veronese surface V , i.e., the variety of symmetric matrices of rank 2 ⎞ ⎛ x1 w3 w2 rank ⎝w3 x2 w1 ⎠ = 1. w2 w1 x3 V is isomorphic to P2 with coordinates (y1 , y2 , y3 ) by setting xi = yi2 , w1 = y2 y3 , w2 = y1 y3 , w3 = y1 y2 , and the projection π : V → P2 given by (x1 , x2 , x3 ) corresponds to the (Z/2)2 Galois cover (y1 , y2 , y3 ) → (y12 , y22 , y32 ). The deformed degree 4 covering is then a slightly less special, yet very interesting projection of the Veronese surface.

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6. The tangential developable F of the twisted cubic Γ. This ﬁnal section is devoted to the proof of an interesting characterization of the above surface Theorem 6.1. The tangential developable F of the twisted cubic Γ is the unique irreducible surface of degree 4 in P3 which has the twisted cubic Γ as a cuspidal curve. Remark 6.2. 1) After proving the theorem, we looked again with more care at the book by Conforto and Enriques ”Le superﬁcie razionali” ([SupRaz]), where Chapter V is mostly devoted to the surfaces of degree 4 which contain Γ as a double curve. On page 114 the above theorem is mentioned, and a diﬀerent proof is brieﬂy sketched in a footnote. It is then mentioned in [SupRaz] that the complete classiﬁcation of surfaces of degree 4 ruled by lines, started by Calyley in [Cay] was achieved by Cremona ([Crem]), and a later classiﬁcation was also given by G.Gherardelli ([Gher]). For lack of time (pending deadline), we are not in a position to determine who gave the ﬁrst proof of the above Theorem 6.1. We hope however that our modern description of quartic surfaces having the twisted cubic as double curve may be found simple and useful. 2) Note that, if we view P3 as the space of eﬀective degree 3 divisors on P1 , then Γ, F − Γ, P3 − F are exactly the three PGL(2)-orbits, corresponding to the divisors of respective types 3P , 2P1 + P2 (P1 = P2 ), P1 + P2 + P3 with P1 , P2 , P3 three distinct points. Proof of Theorem 6.1. The twisted cubic curve Γ is the image of P1 under the third Veronese mapping v3 (t0 , t1 ) := (t30 , t20 t1 , t0 t21 , t31 ), and its projective coordinate ideal IΓ is generated by three quadrics, the determinant of the 2×2-minors of the matrix x0 x1 x2 A= . x1 x2 x3 Thus we have a three-dimensional vector space V consisting of the quadrics containing Γ. Thus V := IΓ (2) = H 0 (P3 , IΓ (2)), where IΓ is the ideal sheaf of Γ, is generated by Q0 := x1 x3 − x22 , Q1 := −x0 x3 + x1 x2 , Q2 := x0 x2 − x21 . Observe now that, to each point x ∈ Γ is associated a unique quadric cone Qx containing Γ and with vertex x: since projection with centre x maps Γ to a plane conic. Thus we have a map Γ → P(V ) associating Qx to x (in our notation, P(V ) ˜ denotes the set of 1-dimensional subspaces of the vector space V ). We denote by Γ the image of this map, and observe that there is thus a canonical bijection between ˜ . Γ and Γ

The 3-cuspidal quartic and braid monodromy of degree 4 coverings

125

Lemma 6.3. Consider in the projective plane P(V ) := {Q |Γ ⊂ Q} the quartic ˜ ⊂ P(V ) counted with curve {Q|detQ = 0}. Then this quartic curve is the conic Γ multiplicity 2. Proof of the Lemma. Observe that if a quadric Q contains Γ, then rank (Q) ≥ 3 since Γ is irreducible. CLAIM: If rank (Q) = 3 (then Q is a quadric cone) the vertex x of Q is a point of Γ. Proof of the claim: Otherwise Γ would be a Cartier divisor on Q, whence it is known (but we prove it again below) that its degree should be even. Let in fact F2 be the blow-up of Q at x, thus a basis of P ic(F2 ) is given by the excetional curve σ, and by the strict transform F of a line, which satisfy F 2 = 0, σ 2 = −2, σF = 1. The plane section H is linearly equivalent to 2F + σ, and let Γ ≡ aσ + bF . From the equations σΓ = 0, HΓ = 3 we obtain b − 2a = 0, b = 3, a contradiction (in reality the class of Γ is 3F + σ).

It follows that the quartic curve {Q ∈ P(V )|det Q = 0} is set theoretically the ˜ which is a rational curve. But Γ ˜ is homogeneous by the action of PGL(2) curve Γ, ˜ acting on P(V ), thus Γ is smooth and must be a conic: the assertion follows then right away.

We analyse now the space of quartics which have Γ as a double curve: Lemma 6.4. U := H 0 (P3 , IΓ2 (4)) ∼ = Sym2 (V ). Proof of the lemma. Let F ∈ H 0 (P3 , IΓ2 (4)) : we need to show that F is equal to a quadratic polynomial f (Q0 , Q1 , Q2 ). Let us ﬁrst consider the divisor cut by F on the smooth quadric Q1 : since P ic(Q1 ) has as basis the respective rulings L1 and L2 , and the hyperplane divisor H is linearly equivalent to L1 + L2 , we see by direct calculation that, under the isomorphism Q1 ∼ = P1 × P1 , divQ1 (Q0 ) = Γ + div(u0 ), divQ1 (Q2 ) = Γ+div(u1 ), where (u0 , u1 )(v0 , v1 ) are suitable coordinates on P1 ×P1 . It follows in particular that the quadric cones Qx cut on Q1 the curve Γ plus the line of the ﬁrst ruling passing through x, and more importantly that divQ1 (F ) = 2Γ + div(φ(u0 , u1 )), where φ is a quadratic polynomial. Whence, divQ1 (F ) = div(φ(Q0 , Q2 )), and there is a quadratic form Q3 in P3 such that F = φ(Q0 , Q2 ) + Q1 Q3 . Since however φ(Q0 , Q2 ) ∈ H 0 (P3 , IΓ2 (4)), it follows that Q1 Q3 ∈ H 0 (P3 , IΓ2 (4))

and thus Q3 ∈ H 0 (P3 , IΓ (2)), so that Q3 is a linear combination of Q0 , Q1 , Q2 . For such a surface F as above, Γ is a double curve, and we are going to show that a general such surface possesses 4 pinch points on Γ. As a ﬁrst step in this direction, we describe the conormal bundle to Γ. Lemma 6.5. NΓ∗ (2) ∼ = OP1 (1) ⊕ OP1 (1). Proof of the lemma. We know that Q0 , Q1 , Q2 ∈ H 0 (P3 , IΓ (2)) induce, under the surjection IΓ (2) → IΓ /IΓ2 (2) = NΓ∗ (2), three sections q0 , q1 , q2 which generate the

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Rank 2 bundle NΓ∗ (2). Since det(NΓ∗ (2)) has degree 2 on P1 , as it is easily seen by the cotangent bundle sequence for Γ ⊂ P3 , it follows that NΓ∗ (2) splits either as OP1 (1) ⊕ OP1 (1) or OP1 (0) ⊕ OP1 (2). But we can exclude the second case since any quadric cone Qx ∈ V induces a section q ∈ H 0 (NΓ∗ (2)) whose two components have a simple zero at x, and no other zero.

We have now a discriminant map δ : Sym2 (NΓ∗ (2)) = OP1 (2) ⊕ OP1 (2) ⊕ OP1 (2) → OP1 (4), given by δ(a, b, c) = 4ac − b2 and vanishing on the simple tensors. Assume now that qj = (aj , bj ): then we get an equation for the pinch points of a surface

F = λi,j Qi Qj , i,j

namely, ⎞⎛ ⎞ ⎛ ⎞2 ⎛

λi,j ai aj ⎠ ⎝ λi,j bi bj ⎠−⎝ λi,j [ai bj + bi aj ]⎠ ∈ H 0 (P1 , OP1 (4)). Δ(F ) = 4 ⎝ i,j

i,j

i,j

We see then that a general surface F ∈ P(Sym2 (V )) has 4 pinch points along Γ, and that (***) Γ is a cuspidal curve for F if and only if Δ(F ) ≡ 0. We observe then that (***) is a system of 5 quadratic equations vanishing on the Veronese surface ( the surfaces in P(Sym2 (V )) which are squares Q2 of Q ∈ V ). We can then bet that our problem is equivalent to the problem of the number of conics tangent to 5 ﬁxed lines in the plane. We show that this is indeed the case, because ﬁrst of all 1) P(Sym2 (V )) is the space of conics in P(V ∨ ). 2) Γ is a cuspidal curve for F if and only if Δ(F ) vanishes in 5 ﬁxed distinct points x(1), . . . , x(5) ∈ Γ, i.e., x(1), . . . , x(5) ∈ Γ are pinch points for F . 3) the following holds: Lemma 6.6. x ∈ Γ is a pinch point for F if and only if the corresponding conic ˜ corresponding ˜∈Γ CF in P(V ∨ ) is tangent to the line in P(V ∨ ) dual to the point x to x (i.e., x ˜ is the point given by the quadric cone Qx ). Proof of the lemma. Let us take coordinates (t0 , t1 ) in P1 such that x corresponds ˜ is a smooth conic, we may take a basis of V to the point t1 = 0. Likewise, since Γ corresponding to quadric cones Q0 , Q1 , Q2 with vertices in the respective points of Γ corresponding to t0 = 0, t1 = 0, t2 := t0 − t1 = 0. Observe then for later use that the point x corresponds to the point x ˜ = (0, 1, 0) in P(V ), so its dual line, in the dual basis coordinates, will be the line y1 = 0. The evaluation of the three sections q0 , q1 , q2 in the ﬁbre of the conormal bundle at x : t1 = 0, t0 = 1 yields respective vectors forming a matrix

The 3-cuspidal quartic and braid monodromy of degree 4 coverings

A :=

1 b0

0 0

127

1 . b2

Then a quartic F = i,j λi,j Qi Qj has our point as a pinch point if and only if the following 2 × 2 symmetric matrix has zero determinant: M :=

λ00 + λ02 + λ22 λ00 b0 + 12 λ02 (b0 + b2 ) + λ22 b2

λ00 b0 + 12 λ02 (b0 + b2 ) + λ22 b2 . λ00 b20 + λ02 b0 b2 + λ22 b22

Then we have det(M ) = (b0 − b2 )2 (λ00 λ22 − 4λ200 ). We then observe that (b0 − b2 )2 = 0 because not all the above quartics F have a pinch point in x. Therefore, the condition that x be a pinch point is exactly given 2 = 0, i.e., by the condition that the conic by (λ00 λ22 − 4λ 00 ) CF = {(y0 , y1 , y2 )| i,j λi,j yi yj = 0} be tangent to the line {y1 = 0} dual to the point x ˜.

We are now ready to ﬁnish the proof of Theorem 6.1: assume that F is a quartic surface which has Γ as a double curve, and assume that F is not the square of a quadric Q ∈ V . Then the corresponding conic CF has rank ≥ 2. Assume further that Γ is a cuspidal curve for F : this holds if and only if the ˜ conic CF is tangent to ﬁve ﬁxed lines L1 , . . . , L5 dual to 5 points x ˜1 , . . . , x˜5 ∈ Γ. ∨ ˜ ˜1 , . . . , x ˜5 ∈ Γ, and Then the dual conic CF passes through the ﬁve points x ˜ therefore coincides with Γ. We have thus shown that there is one and only one such quartic F which is not a quadric counted with multiplicity 2, so we conclude that F , which is a union of PGL(2) orbits, is exactly the tangential developable surface of Γ, which is an irreducible surface.

Note. This article is essentially based on classical mathematics, and it is accordingly written in classical style, sometimes referred to by referees as: ”a style which is suitable for Conference Proceedings”. We hope that the style may be suitable for the reader. Ackowledgements. We would like to thank the referee for pointing out a gap in the original proof of Theorem 2.1, and correcting a couple of minor mistakes. We also thank Fabio Tonoli for providing a Macaulay 2 Script verifying that the dual variety of P is indeed a curve. Finally, the ﬁrst named author would like to acknowledge the hospitality of M.S.R.I. in march 2004, where the ﬁrst calculations of the braid monodromy were begun.

Dedication. Finally, the manifold occurrence of words such as ”Veronese surface” or ”Veronese embedding” points out the appropriateness of this article to celebrate the 150-th anniversary of the birth of Giuseppe Veronese.

128

Fabrizio Catanese and Bronislaw Wajnryb

References

[A-B-K-P] J. Amoros, F. Bogomolov, L. Katzarkov, T.Pantev Symplectic Lefschetz ﬁbrations with arbitrary fundamental group, with an appendix by Ivan Smith, J. Diﬀerential Geom. 54 n. 3 (2000), 489–545 [Aur02] D. Auroux, Fiber sums of genus 2 Lefschetz ﬁbrations Turkish J. Math. 27 n. 1 (2003), 1–10. [A-K] D. Auroux, L. Katzarkov, Branched coverings of CP2 and invariants of symplectic 4-manifolds, Inv. Math. 142 (2000), 631-673. [A-D-K-Y] D. Auroux, S. Donaldson, L. Katzarkov and M. Yotov, Fundamental groups of complements of plane curves and symplectic invariants, GT/0203183v1 (2002). [A-D-K] D. Auroux, S. Donaldson, L. Katzarkov Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves. Math. Ann. 326 no. 1 (2003), 185–203 [Cat1] F. Catanese, On the Moduli Spaces of Surfaces of General Type, J. Diﬀ. Geom 19 (1984) 483–515. [Cat2] F. Catanese, Moduli spaces of surfaces and real structures, Annals of Math. 158, no. 2 (2003), 577-592. [Cat3] F. Catanese, “Symplectic structures of algebraic surfaces and deformation”, 14 pages, math.AG/0207254. [Cat4] F. Catanese, “Singular bidouble covers and the construction of interesting algebraic surfaces”, Proceedings of the Warsaw Conference in honour of F. Hirzebruch’s 70th Birthday, A.M.S. Contemp. Math. 241 (1999), 97-120. [C-W] F. Catanese, B. Wajnryb, “Diﬀeomorphism of simply connected algebraic surfaces”, 33 pages , math.AG/0405299. [Cay] A. Cayley, “Second Memoir on skew surfaces, otherwise scrolls ”, Phil. Trans. (1864). [SupRaz] F. Conforto, “Le superﬁcie razionali”, N. Zanichelli, Bologna (1939) XV, 554 p. . [Crem] L. Cremona, “Sulle superﬁcie gobbe di quarto grado”, Memoria dell’ Accademia delle Scienze dell’ Istituto di Bologna, Seire II, Tomo VIII (1868), also Opere, Tomo II, pag. 420. [Don6] S. K. Donaldson, “Lefschetz pencils on symplectic manifolds”, J. Diﬀerential Geom. 53 no. 2 (1999), 205–236. [Don7] S. K. Donaldson, “Lefschetz ﬁbrations in symplectic geometry”, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math. 1998, Extra Vol. II , 309–314. [D-S] S.K. Donaldson, I. Smith, “Lefschetz pencils and the canonical class for symplectic four-manifolds”, Topology 42 n. 4 (2003), 743–785 [Gher] G. Gherardelli, “Sulle superﬁcie rigate di 4◦ grado”, Boll. Unione Mat. Ital. 15 (1936), 17-20 .

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[Jes] C. M. Jessop, “Quartic surfaces with singular points”, Cambridge: University Press,(1916) XXXV u. 197 p. [Kas] A. Kas, “On the handlebody decomposition associated to a Lefschetz ﬁbration”, Paciﬁc J. Math. 89 no. 1 (1980), 89–104. [Lib] A. Libgober, “Fundamental groups of the complements to plane singular curves”, in ’Algebraic geometry, Bowdoin, 1985’ (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., 46, Part 2, Amer. Math. Soc., Providence, RI (1987), 29-45. [Moi1] B. Moishezon, “Stable branch curves and braid monodromies”, in ’Algebraic geometry’ (Chicago, Ill., 1980),Lecture Notes in Math., 862, Springer, Berlin-New York (1981), 107-192. [Moi2] B. Moishezon, “Algebraic surfaces and the arithmetic of braids. I”, in ’Arithmetic and geometry’, Vol. II, Progr. Math., 36, Birkh¨ auser Boston, Boston, MA ( 1983), 199–269. [Moi3] B. Moishezon, “The arithmetics of braids and a statement of Chisini”, in ’Geometric Topology, Haifa 1992’ , Contemp. Math. 164, A.M.S. (1994), 151-175. [Zar] O. Zariski, “On the Poincar´ e group of rational plane curves ”, Amer. J. Math. 58 (1936), 607-619. [Zar2] O. Zariski, “A theorem on the Poincar´ e group of an algebraic hypersurface”, Ann. of Math. (2) 38, n. 1 (1937), 131-141. Fabrizio Catanese Lehrstuhl Mathematik VIII Universit¨ at Bayreuth, NWII D-95440 Bayreuth, Germany Email: [email protected] Bronislaw Wajnryb Department of Mathematics Technion 32000 Haifa, Israel Email: [email protected]

On the classiﬁcation of defective threefolds Luca Chiantini and Ciro Ciliberto

Abstract. In this paper we give the full classiﬁcation of irreducible projective threefolds whose k-secant variety has dimension smaller than the expected, for some k ≥ 2 (see Theorem 0.1 below). As pointed out in the introduction, the case k = 1 was already known before. 2000 Mathematics Subject Classiﬁcation: 14N05

Introduction An irreducible, non–degenerate, projective variety X of dimension n in Pr is called k–defective if its k–secant variety S k (X) has dimension s(k) (X) smaller than the expected σ (k) (X), which is the minimum of r and n(k + 1) + k. The diﬀerence δk (X) = σ (k) (X) − s(k) (X) is called the k–defect of X. The classiﬁcation of defective varieties is important in the study of projective geometry and its applications. The subject goes back to several classical authors, like Terracini ([29]), Palatini ([25]) and Scorza ([26]), to mention a few. More recently the interest on defective varieties has been renewed by Zak’s spectacular results on the classiﬁcation of some important classes of smooth varieties (e.g. Severi varieties, Scorza varieties etc., see [33]). Beside the intrinsic interest of the subject, it turns out that the understanding of defective varieties is relevant also in other ﬁelds of Mathematics: expressions of polynomials as sums of powers and Waring type problems, polynomial interpolation, rank tensor computations and canonical forms, Bayesian networks, algebraic statistics etc. (see [9] as a general reference, [5], [18], [21]). See also [28] and [32] as further examples of results on the subject which have nice applications to number theoretic problems. In the present paper, we give the complete classiﬁcation of complex defective threefolds. One knows that curves are never defective (see [10]). Defective surfaces have been classiﬁed by Terracini in [30]. Terracini’s result has been revisited and extended in [11], [4] and [6]. The case of 1–defective threefolds goes back to Scorza

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[26] and has been revisited in [7]. The case of smooth 1–defective threefolds was also examined by Fujita [16] and Fujita–Roberts [17]. Here we deal with the case of k–defective threefolds, for any k > 1, and we classify minimally k–defective threefolds, i.e. k–defective threefolds that are not h–defective for any h < k. This is of course suﬃcient to describe all defective threefolds. The classiﬁcation is obtained by using the classical tool of tangential projections. The basic invariants are the tangential contact loci (see §1). An important tool is also provided by Castelnuovo’s theory on the growth of Hilbert functions (see [6], §6). After having recalled several basic notations, deﬁnitions and results in §1, we prove some generalities on defective threefolds in §2 and on the contact loci in §3. Then we prove our classiﬁcation results in §§4, 5, 6. Indeed §4 is devoted to the analysis of the case in which the appropriate tangential contact locus is an irreducible divisor, §5 to the case in which the contact locus is reducible, §6 to the case in which the contact locus is an irreducible curve. Now we state our classiﬁcation result, obtained by summarizing Theorems 4.2, 4.5, 6.8 and various corollaries and remarks accompanying them (for the deﬁnition of the invariant nk (X) see section 1.23 below): Theorem 0.1. Let X ⊂ Pr be an irreducible, non–degenerate, projective, minimally k–defective threefold with k ≥ 2. Then X is in the following list: (1) X sits in a cone over the 2–uple embedding of a threefold Y of minimal degree k − 1 in Pk+1 , with vertex either a point, hence r = 4k + 2 and δk (X) = 1, s(k) (X) = r − 1 = 4k + 1, nk (X) = 1, or a line, hence r = 4k + 3 and δk (X) = 1, s(k) (X) = r − 1 = 4k + 2, nk (X) = 2 (see Example 4.3, (1)); (2) k = 3 and either r = 4k + 2 = 14, s(k) (X) = r − 1 = 13, nk (X) = 1, and X is the 2–uple embedding of a hypersurface Y in P4 with deg(Y ) ≥ 3 or r = 4k + 3 = 15, s(k) (X) = r − 1 = 14, nk (X) = 2, and X sits in the cone with vertex a point over the 2–uple embedding of a hypersurface Y as above (see Example 4.3, (2)); (3) either r = 4k + 2, δk (X) = 1, nk (X) = 1, s(k) (Y ) = r − 1 = 4k + 1, and X is the 2–uple embedding of a threefold Y of degree k in Pk+1 with curve sections of arithmetic genus 1 or r = 4k +3, δk (X) = 1, nk (X) = 2, s(k) (Y ) = r −1 = 4k +2, and X sits in the cone with vertex a point over the 2–uple embedding of a threefold Y as above (see Example 4.3, (3)); (4) r = 4k + 3, δk (X) = 1, nk = 2 and s(k) (X) = r − 1 = 4k + 2, and X is the 2–uple embedding of a threefold Y of degree k in Pk+1 with curve sections of genus 0, which is either a cone with vertex a line over a smooth rational curve of degree k in Pk−1 or it has a double line (see Example 4.3, (4)); (5) k = 4, r = 4k + 3 = 19, δ4 (X) = 1, n4 = 2, s(k) (X) = r − 1 = 18, and X is the 2–uple embedding of a threefold Y in P5 with deg(Y ) ≥ 5, contained in a quadric (see Example 4.3, (5)); (6) k ≥ 4, r = 4k + 3, δk (X) = 1, nk = 2, s(k) (X) = r − 1 = 4k + 2, and X is the 2–uple embedding of a threefold Y of degree k + 1 in Pk+1 with curve sections of arithmetic genus 2 (see Example 4.3, (6));

On the classiﬁcation of defective threefolds

133

(7) r = 4k + 3 − i, i = 0, 1, δk (X) = 1, s(k) (X) = r − 1 = 4k + 3 − i, nk = 2 − i, and X sits in a cone with vertex a space of dimension k − i over the 2–uple embedding of a surface Y of minimal degree k in Pk+1 (see Theorem 4.5, case (1)); (8) k = 2, r = 4k + 3 = 11, δk (X) = 1, s(k) (X) = r − 1 = 10, nk = 2, and X sits in a cone with vertex a line over the 2–uple embedding of a surface Y of P3 with deg(Y ) ≥ 3 (see Theorem 4.5, case (2)); (9) k ≥ 3, r = 4k + 3, δk (X) = 1, s(k) (X) = r − 1 = 4k + 2, nk = 2, and X sits in a cone with vertex of dimension k − 1 over the 2–uple embedding of a surface Y of degree k + 1 in Pk+1 with curve sections of arithmetic genus 1 (see Theorem 4.5, case (3)); (10) r ≥ 4k + 3, δk (X) = 1, s(k) (X) = 4k + 2, nk = 2 and X sits in a cone with vertex of dimension k − 1, and not smaller, over a surface which is not k–weakly defective (see Proposition 5.2); (11) r ≥ 4k + 3, δk (X) = 1, s(k) (X) = 4k + 2, nk = 2 and X sits in a cone with vertex of dimension 2k, and not smaller, over a curve (see Proposition 5.3); (12) r ≥ 4k + 2, s(k) (X) = 4k + 1, nk = 1 (hence δk (X) = 1, if r = 4k + 2, whereas δk (X) = 2, if r > 4k + 2) and X sits in a cone with vertex of dimension 2k − 1, and not smaller, over a curve (see Proposition 5.4); (13) 4k + 3 ≤ r ≤ 4k + 5, s(k) (X) = 4k + 2, δk (X) = 1, nk = 1 and X is either the 2–uple embedding of a threefold of minimal degree k in Pk+2 (hence r = 4k + 5), or the projection from a point of P4k+5 of the 2–uple embedding of a threefold Y of minimal degree k in Pk+2 , or the projection from a line ⊂ P4k+5 of the 2–uple embedding Y ⊂ P4k+5 of a threefold Y of minimal degree k in Pk+2 (see Example 6.3); (14) r = 4k + 3, s(k) (X) = r − 1 = 4k + 2, δk (X) = 1, nk (X) = 2 and X is linearly normal, contained in the intersection of a space of dimension 4k + 3 with the Segre 2 embedding of Pk+1 × Pk+1 in Pk +4k+3 , but not lying in the 2–uple embedding of Pk+1 , and such that the two projections of X to Pk+1 span Pk+1 (see Corollary 6.10). Cases (1)–(9) correspond to the situation in which the tangential contact locus is an irreducible surface, case (10) corresponds to the situation in which the tangential contact locus is a reducible curve (a union of lines), cases (11)–(12) correspond to the situation in which the tangential contact locus is a reducible surface, cases (13)–(14) correspond to the situation in which the tangential contact locus is an irreducible curve, namely a rational normal curve of degree 2k. All threefolds in this list are actually minimally k–defective. Threefolds of types (1)–(9) and (14) are not h–defective for any h > k. The same is true for threefolds of type (13) with r < 4k + 5. Threefolds of type (13) with r = 4k + 5 are also (k + 1)–defective. Threefolds of types (10)–(12) can be h–defective for h > k (see Remark 5.5). Joining with Scorza’s result for k = 1, we summarize in the following table the list of all irreducible defective threefolds (see Theorem 1.18 and section 1.23 for the deﬁnition of the invariants k and nk ).

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description X ⊂ cone over 2–uple of 3fold of min. deg in Pk+1 , vertex = point X ⊂ cone over 2–uple of 3fold of min. deg in Pk+1 , vertex = line X = 2–uple of hypersurface of deg ≥ 3 in P4 X ⊂ cone over 2–uple of hypersurface of deg ≥ 3 in P4

k

r

s(k)

δk k nk

≥2

4k + 2

4k+1

1

2

1

≥2

4k + 3

4k+2

1

2

2

3

14

13

1

2

1

3

15

14

1

2

2

X = 2–uple of 3fold of deg k in Pk+1 whose curve sections have pa = 1 X ⊂ cone over 2–uple of 3fold of deg k in Pk+1 whose curve sections have pa = 1

≥4

4k + 2

4k+1

1

2

1

≥4

4k + 3

4k+2

1

2

2

(4)

X = 2–uple of 3fold of deg k in Pk+1 whose curve sections have pa = 0

≥4

4k + 3

4k+2

1

2

2

(5)

X = 2–uple of 3fold of deg ≥ 5 in P5 contained in a quadric

4

19

18

1

2

2

≥4

4k + 3

4k+2

1

2

2

≥1

4k + 3

4k+2

1

2

2

≥1

4k + 2

4k+1

1

2

1

2

11

10

1

2

2

≥3

4k + 3

4k+2

1

2

2

≥1

≥ 4k + 3

4k+2

1

1

2

(1)a (1)b (2)a (2)b (3)a (3)b

(6) (7)a (7)b (8) (9) (10)

X = 2–uple of 3fold of deg k + 1 in Pk+1 whose curve sections have pa = 2 X ⊂ cone over 2–uple of surface of min. deg k in Pk+1 , vertex = Pk X ⊂ cone over 2–uple of surface of min. deg k in Pk+1 , vertex = Pk−1 X ⊂ cone over 2–uple of surface of deg ≥ 3 in P3 , vertex = line X ⊂ cone over 2–uple of surface of deg k + 1 in Pk+1 , whose curve sections have pa = 1, vertex = Pk−1 X ⊂ cone over surface not k–weakly defective, vertex = Pk−1

(11)

X ⊂ cone over curve, vertex = P2k

≥1

≥ 4k + 3

4k+2

1

2

2

(12)a

X ⊂ cone over curve, vertex = P2k−1

≥1

4k + 2

4k+1

1

2

1

(12)b

X ⊂ cone over curve, vertex = P2k−1

≥1

≥ 4k + 3

4k+1

2

2

1

X = 2–uple of 3fold of min. deg in Pk+2 X = projection from point of 2– uple of 3fold of min. deg k in Pk+2 X = projection from line of 2–uple of 3fold of min. deg k in Pk+2

≥1

4k + 5

4k+2

1

1

1

≥1

4k + 4

4k+2

1

1

1

≥1

4k + 3

4k+2

1

1

1

≥1

4k + 3

4k+2

1

1

2

(13)a (13)b (13)c (14)

X ⊂ P4k+3 ∩ (Segre of Pk+1 × Pk+1 ) and X dominates both Pk+1 ’s

On the classiﬁcation of defective threefolds

135

Please note that some of the types in the table may overlap for low k. Brief discussions about the existence of smooth defective threefolds are contained in Example 4.3 and Remarks 4.6 and 5.5 below.

1. Preliminaries and notation 1.1. In this paper we work over the complex ﬁeld C. Let X ⊆ Pr be an irreducible projective scheme over C. We will denote by deg(X) the degree of X and by dim(X) the dimension of X. If X is reducible, by dim(X) we mean the maximum of the dimensions of its irreducible components. We will denote by OX the structure sheaf of X and by IX the ideal sheaf of X in Pr . One says that X is rationally connected if for x, y ∈ X general points, there is a rational curve in X containing them. If Y ⊂ Pr is a subset, we denote by < Y > the span of Y . We will say that Y is non–degenerate if < Y >= Pr . 1.2. If X ⊆ Pr is an irreducible, projective, non-degenerate variety, we will say that it is a cone over a variety Y , if there is a projective subspace V of dimension v, called a vertex of X, and a projective subspace W of dimension r − v − 1, containing Y , such that X = ∪x∈Y,y∈V < x, y >. Let V be a projective subspace of Pr of dimension v, not containing X. We will say that X sits in the cone over a variety Y ⊂ Pr−v−1 with vertex V if and only if the image of the projection of X from V is Y . 1.3. Let X ⊂ Pr be an irreducible, non–degenerate projective variety. One has the following famous bound: deg(X) ≥ r − dim(X) + 1

(1)

which is sharp. The varieties achieving the bound (1) are called varieties of minimal degree and their classiﬁcation is well known (see [15]): curves of minimal degree are the rational normal curves, surfaces of minimal degree are either rational normal scrolls or the Veronese surface in P5 , etc. We will denote by hX the Hilbert function of X, namely for any non–negative integer k, hX (k) is the dimension of the image of the restriction map: ρX,k : H 0 (Pr , OPr (k)) → H 0 (X, OX (k)). Recall that one says that X is k–normal if the map ρX,k is surjective, i.e. if H 1 (Pr , IX (k)) = 0. One says linearly normal, quadratically normal etc. instead of 1-normal, 2–normal, etc. Then X is linearly normal if and only if the hyperplane system H is complete. The variety X is said to be projectively normal if it is k–normal for every k ≥ 1.

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We will need the following result of Castelnuovo’s theory from [6] (see Theorem 6.1): Theorem 1.4. Let X ⊂ Pr be an irreducible, non–degenerate, projective variety of dimension n and degree d. Set ι := ι(X) = min{d + n − r − 1, r − n}. Then: n(n − 1) + 1. 2 If d ≤ 2(r − n) + 1 then the following are equivalent: hX (2) ≥ ι + r(n + 1) −

(2)

(i) the equality holds in (2); (ii) the general curve section of X is linearly normal of genus ι; (iii) the general curve section of X is projectively normal of genus ι. Remark that, if X is non–degenerate, the invariant ι(X) deﬁned in the previous statement is non–negative because of (1). Let us now mention an interesting proposition which follows from results of [3], Corollary 3.3: Proposition 1.5. Let X ⊂ Pr be a smooth isomorphic projection of a variety of minimal degree in Pr+1 . Then X is k–normal for every k ≥ 2. Using it, we can prove the following lemma, which will be useful later: Lemma 1.6. Let Y ⊂ Pr be a non–degenerate threefold of minimal degree r − 2 / Y . Assume that Y is and let Y ⊂ Pr−1 be a projection of Y from a point p ∈ non–singular in codimension 1. Then hY (2) ≥ 4r − 4 and the equality holds if and only if one of the following cases occurs: (i) Y is a cone over a smooth rational curve C of degree r − 2 in Pr−3 with vertex a line; (ii) Y has a line L of double points, and the pull back of L on Y is a conic whose plane contains the center of projection p. Proof. We let S ⊂ Pr−2 be the general surface section of Y and C ⊂ Pr−3 the general curve section of S. Notice that C is a smooth rational curve of degree r − 2. Standard diagram chasing gives: hY (2) − hY (1) = hS (2) + dim(ker{H 1 (Pr−1 , IY (1)) → H 1 (Pr−1 , IY (2))}) hS (2) − hS (1) = hC (2) + dim(ker{H 1 (Pr−2 , IS (1)) → H 1 (Pr−2 , IS (2))}) (see [8], p. 30). Now: hY (1) = r,

hS (1) = r − 1

On the classiﬁcation of defective threefolds

137

and hC (2) = 2r − 3 by Proposition 1.5. Thus: hY (2) = 4r − 4+ + dim(ker{H 1 (Pr−1 , IY (1)) → H 1 (Pr−1 , IY (2))})+

(3)

+ dim(ker{H 1 (Pr−2 , IS (1)) → H 1 (Pr−2 , IS (2))}). This proves the ﬁrst assertion. Moreover hY (2) = 4r − 4 yields: dim(ker{H 1 (Pr−1 , IY (1)) → H 1 (Pr−1 , IY (2))}) = 0 dim(ker{H 1 (Pr−2 , IS (1)) → H 1 (Pr−2 , IS (2))}) = 0. This implies that both Y and S are singular. Suppose in fact that Y is smooth. Then h1 (Pr−1 , IY (1)) = 1 and h1 (Pr−1 , IY (2)) = 0 by Proposition 1.5. The same argument works for S. Let x ∈ S be a singular point. If S is a cone with vertex x we are in case (i). Suppose S is not a cone and let μ be the multiplicity of S at x. The projection of S from x is a non–degenerate surface of degree r − 2 − μ in Pr−3 . This proves that μ = 2. We claim that x is the only singular point of S. In fact S is the projection / T . Since S is not a of a surface T ⊂ Pr−1 of minimal degree from a point p ∈ cone, then T is not a cone, hence it is smooth (see [15]). The singular point x of S arises from a secant (or tangent) line to T passing through p. Suppose there is another singular point y ∈ S. This would correspond to some other secant line to T containing p. The plane Π =< , > would then be 4–secant to T , and this is possible only if Π intersect T along a conic (see again [15]). This would in turn yield a double line for S and codimension 1 singularities for Y , a contradiction. In conclusion we are in case (ii). Conversely, if we are in case (i), one has 0

h (P

r−1

, IY (2)) = h (P 0

r−3

r−1 , IC (2)) = − 2r + 3 2

(see again Proposition 1.5). Hence:

hY (2) =

r+1 r−1 − + 2r − 3 = 4r − 4. 2 2

If we are in case (ii), we have h1 (Pr−1 , IY (1)) = h1 (Pr−2 , IS (1)) = 0, because clearly h0 (Y , OY (1)) = r and h0 (S, OY (1)) = r − 1. So formula (3) yields

hY (2) = 4r − 4.

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It is clear that one can prove a similar more general result for projections of varieties of minimal degree of any dimensions. Since we will not need it, we do not dwell on this here. We also record the following: Proposition 1.7. Let Y ⊂ Pk+1 be an irreducible, non–degenerate, threefold of degree k with smooth curve sections of genus 0 and a singular line. Then Y is the projection in Pk+1 of a threefold Z of minimal degree k in Pk+2 from a point p∈ / Z. In particular: (i) either Y is a cone over a smooth rational curve of degree k in Pk−1 , and therefore Y has a k-tuple line; (ii) or the threefold Z contains a conic sitting in a plane passing through the center of projection, in which case Y has a line of double points. Proof. One knows that the threefold Y is the projection in Pk+1 of a threefold Z / Z. If Z is a cone with vertex a of minimal degree k in Pk+2 from a point p ∈ line, we are in case (i). If Z is a cone with vertex a point, then Y is also a cone. However the point p has to sit on a secant line to Z and we are in case (ii): the conic here is reducible in the two generators of the cone Z passing through the intersection points of with Z. If Z is smooth, then again p has to sit on a secant line to Z. The argument to show that we are in case (ii) then goes as in the proof of Lemma 1.6.

1.8. Let X be an irreducible, projective variety. A prime divisor on X is an irreducible subvariety of codimension 1, not contained in the singular locus. A divisor is a linear combination of prime divisors. If X is normal, this is the deﬁnition of Weil divisors and we will use the symbol ≡ to denote linear equivalence on X. If D is a divisor, we will denote, as usual, by |D| the complete linear system of D. When X is not normal, then we consider the normalization f : X → X. We will identify divisors on X with their pull–backs on X and therefore we will freely use the terminology of linear equivalence and linear systems. Let H be a hyperplane divisor on X. We will denote by H ⊂ |H| the (possibly not complete) hyperplane system, i.e. the linear system cut out on X by the hyperplanes of Pr . If we assume that X is non–degenerate, then H has dimension r. In this case, by abusing notation, we will sometimes identify the divisor H with the unique hyperplane which cuts H on X. If L is a linear system of dimension r of divisors on X, we will denote by: φL : X → Pr the rational map deﬁned by L. If p1 , . . . , pk ∈ X are smooth points and m1 , . . . , mk are positive integers, we will denote by L(−m1 p1 − . . . − mk pk ) the sublinear system of L formed by all divisors in L with multiplicity at least mi at pi , i = 1, . . . , k.

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If L1 and L2 are linear systems of Weil divisors on a X, we deﬁne L1 + L2 as the minimal linear system of Weil divisors on X containing all divisors of the form D1 + D2 where Di ∈ Li , i = 1, 2. The linear system L1 + L2 is called the minimal sum of L1 and L2 . If L is a linear system on X, one writes 2L instead of L + L. Similarly one can consider the linear system hL for all positive integers h. Let L1 and L2 be linear systems on X of dimensions r1 , r2 . Set L = L1 + L2 , and suppose L has dimension r. One can consider the maps: φLi : X → Pri , i = 1, 2 φL : X → Pr determined by the linear systems in question. It is clear that: φL = ψ ◦ (φL1 × φL2 ) where ψ : Pr1 × Pr2 → Pr1 r2 +r1 +r2 is the Segre embedding. Similarly if L is a linear system on V with dimension r, and φL : V → Pr is the corresponding map, then for any positive integer h one has: φhL = ψh ◦ φL where r+m m

ψh : Pr → P(

)−1

is the h-th Veronese embedding. We will need the following lemma: Lemma 1.9. Let C be a smooth, irreducible, projective curve of genus g and let Li , i = 1, 2, be base point free linear systems on C. Set L = L1 + L2 , dim(L) = r, dim(Li ) = ri , i = 1, 2. One has r ≥ r1 + r2 . If L is birational and r = r1 + r2 , then g = 0. If moreover r1 = r2 , then L1 = L2 and this series is complete. Proof. The inequality r ≥ r1 + r2 follows from Hopf’s lemma (see [2], p. 108). If the equality holds, then g = 0 by [2], Exercise B-1, p. 137. Suppose now that, in addition, r1 = r2 . Then by [2], Exercise B-4, p. 138, the two series are equal, and by the birationality assumption on L, also L1 = L2 is birational. If the series is not complete, then its image is a non-normal rational curve Γ in Pr1 . Then, by

Theorem 1.4, hΓ (2) ≥ 2r1 + 2, a contradiction. Let us now recall a deﬁnition: Deﬁnition 1.10. We will say that a linear system L on a smooth, projective variety Z is birational [resp. very big] if the map φL determined by L is birational

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[resp. generically ﬁnite] to its image. The system L is big if the complete system determined by some multiple of L is birational. The following deﬁnition is standard: Deﬁnition 1.11. If W , Z are varieties with W ⊂ Z, Z smooth, and L is a linear system of divisors on Z, one denotes by L|W the restriction of L to W deﬁned as follows. The system L corresponds to a vector subspace V of H 0 (Z, OZ (L)), with L a divisor in L, and L|W corresponds to the image of V via the restriction map H 0 (Z, OZ (L)) → H 0 (W, OW (L)). The following lemma is immediate and the proof can be left to the reader: Lemma 1.12. Let Z be a smooth, projective variety, let L be a linear system on Z and let V be an algebraic family of closed subvarieties of Z such that a general element V of V is smooth and irreducible. Suppose that if z ∈ Z is a general point, there is a variety V in V containing z. Then if L is birational [resp. very big, big] and V is a general element of V, then L|V is also birational [resp. very big, big]. 1.13. Let X be an irreducible projective variety. For any positive integer h we let Symh (X) be the h–fold symmetric product of X. If p1 , . . . , ph are points in X, we denote by [p1 , . . . , ph ] the corresponding point in Symh (X). Namely one has a surjective morphism: πX,h : (p1 , . . . , ph ) ∈ X h → [p1 , . . . , ph ] ∈ Symh (X) which is a ﬁnite covering of degree h! and the monodromy, or Galois, group of this covering is the full symmetric group Sh . 1.14. Let X be an irreducible, projective variety. Let D = {Dy }y∈Y be an algebraic family of divisors on X parameterized by a projective variety Y . We will constantly assume that D is eﬀectively parameterized by Y , i.e. that the corresponding map of Y to the appropriate Hilbert scheme of divisors on X is generically ﬁnite. One says that D is irreducible of dimension m if Y is. An irreducible algebraic family D of dimension m on X is called an involution if there is one single divisor of D containing m general points of X (see §5 of [6]). Involutions have been studied in §5 of [6] to which we defer the reader for details. In particular in [6] a classical theorem of Castelnuovo and Humbert concerning involutions on curves (see [6], Proposition 5.9) has been extended to higher dimensional varieties (see [6], Proposition 5.10). For the reader’s convenience we recall here the result we will need later on: Theorem 1.15. Let X be an irreducible, projective variety of dimension n > 1. Let D be an m–dimensional involution with no ﬁxed divisors and such that its general element is reduced. Then either D is a linear system or it is composed with a pencil, i.e. there is a rational map f : X → C of X to an irreducible curve C such that D is the pull-back, via f , of an involution on C. If m > 1, the general element of D is reducible if and only if D is composed with a pencil.

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1.16. Let X ⊂ Pr an irreducible, non–degenerate projective variety of dimension n. Let k be a non–negative integer and let S k (X) be the k–secant variety of X, i.e. the Zariski closure in Pr of the set: {x ∈ Pr : x lies in the span of k + 1 independent points of X}. Of course S 0 (X) = X, S r (X) = Pr and S k (X) is empty if k ≥ r + 1. Moreover, for any k ≥ 0, one has S k (X) ⊆ S k+1 (X). We will write S(X) instead of S 1 (X) and we will assume k ≤ r from now on. k k of X, i.e. SX ⊆ Symk (X)× One can consider the abstract k–th secant variety SX r P is the Zariski closure of the set of all pairs ([p0 , . . . , pk ], x) such that p0 , . . . , pk ∈ X are linearly independent points and x ∈< p0 , . . . , pk >. One has the surjective k map pkX : SX → S k (X) ⊆ Pr , i.e. the projection to the second factor. Hence: k s(k) (X) := dim(S k (X)) ≤ min{r, dim(SX )} = min{r, n(k + 1) + k}.

(4)

The right-hand side of (4) is called the expected dimension of S k (X) and will be denoted by σ (k) (X). One says that X is k-defective when strict inequality holds in (4). One says that: δk (X) := σ (k) (X) − s(k) (X) is the k–defect of X. The variety X is called defective if it is k–defective for some k ≥ 1. Observe that, if r ≥ n(k + 1) + k, then X is k–defective if and only if there are inﬁnitely many (k +1)–secant Pk ’s passing through the general point of S k (X). Notice also that, if X is h–defective for some h ≥ 1, then it is k–defective for all k such that k ≥ h and s(k) < r. If X is k–defective but not (k − 1)–defective, then we will say that X is minimally k–defective. We will write s(k) , σ (k) , δk etc. instead of s(k) (X), σ (k) (X), δk (X), if there is no danger of confusion. 1.17. If p is a smooth point of X, we denote by TX,p the (projective) tangent space to X at p. If Π is a projective subspace of Pr , we say that Π is tangent to X at p if either: • dim(Π) ≤ dim(X) and p ∈ Π ⊆ TX,p , or • dim(Π) ≥ dim(X) and TX,p ⊆ Π. If Y is a subvariety of X, no component of which is contained in the singular locus of X, we say that Π is tangent to X along Y if Π is tangent to X at the general point of any component of Y . Let k be a positive integer and let p1 , . . . , pk be points of X. We denote by TX,p1 ,...,pk the span of TX,pi , i = 1, . . . , k. If X ⊂ Pr is a projective variety, Terracini’s lemma describes the tangent space to S k (X) at a general point of it and gives interesting information in case X is k–defective (see [29] or, for modern versions, [1], [6], [10], [33]). We may state it as follows:

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Theorem 1.18. (Terracini’s lemma) Let X ⊂ Pr be an irreducible, projective variety. If p0 , . . . , pk ∈ X are general points and x ∈< p0 , . . . , pk > is a general point, then: TS k (X),x = TX,p0 ,...,pk . If X is k–defective, then: (i) TX,p0 ,...,pk is tangent to X along a variety Γ := Γp0 ,...,pk of positive dimension γk := γk (X) containing p0 , . . . , pk ; (ii) the general hyperplane H containing TX,p0 ,...,pk is tangent to X along a variety Σ := Σ(H) := Σp0 ,...,pk (H) of positive dimension k := k (X) containing p 0 , . . . , pk . One has Γ ⊆ Σ and therefore γk ≤ k . Moreover one has: k ≤ dim(< Γ >) ≤ dim(< Σ >) ≤ kk + k + k − δk .

(5)

We may and will assume that all irreducible components of Γ and of Σ contain some of the points p0 , . . . , pk . Otherwise we simply get rid of those components that do not do so. We will call Γ the tangential k–contact locus of X at p0 , . . . , pk . Similarly, for any hyperplane H containing TX,p0 ,...,pk , we will call Σ := Σ(H) the k–contact locus of H at p0 , . . . , pk . We will call k (X) the k-singular defect of X and γk (X) the k-tangential defect of X. The following is a well known, straightforward application of Terracini’s lemma (see [33]): Proposition 1.19. Let X ⊂ Pr be an irreducible, projective, non–degenerate variety. If for some k ≥ 0 one has s(k) (X) = s(k+1) (X), then s(k) (X) = r. In particular, if s(k) (X) = r − 1, then s(k+1) (X) = r. We record also the following result whose easy proof can be left to the reader: Proposition 1.20. Let X ⊂ Pr be an irreducible, projective, non–degenerate variety of dimension n. Let Π ⊂ Pr be a projective subspace of dimension s and let π be the projection of Pr from Π to Pr−s−1 . Let Y = π(X) and let m be its dimension. Then: (i) for every positive integer k one has π(S k (X)) = S k (Y ), hence s(k) (Y ) ≤ s(k) (X) ≤ s(k) (Y ) + s + 1; (ii) if n = m and s(k) (Y ) = (k + 1)n + k then also s(k) (X) = (k + 1)n + k; (iii) if n = m, s(k) (Y ) = s(k) (X) and Y is k–defective [resp. minimally k– defective] then also X is k–defective [resp. minimally k–defective] and π maps the tangential k–contact locus [resp. the k–contact locus] of X to the tangential k–contact locus [resp. the k–contact locus] of Y .

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1.21. We recall from [6] the deﬁnition of a k–weakly defective variety, i.e. a variety X ⊂ Pr such that if p0 , . . . , pk ∈ X are general points, then the general hyperplane H containing TX,p0 ,...,pk is tangent to X along a variety Σ := Σ(H) := Σp0 ,...,pk (H) of positive dimension k := k (X) containing p0 , . . . , pk . Similarly we can say that a variety X ⊂ Pr is k–weakly tangentially defective if whenever p0 , . . . , pk ∈ X are general points, then TX,p0 ,...,pk is tangent to X along a variety Γ := Γp0 ,...,pk of positive dimension γk : γk (X) containing p0 , . . . , pk . Of course k– weakly tangentially defectiveness implies k–weakly defectiveness, but the converse does not hold in general. Moreover, by Terracini’s lemma, a k–defective variety is also k–weakly defective but again the converse does not hold in general (see [6]). Remark 1.22. A curve which is not a line is never k–weakly (tangentially) defective for any k. Hence a curve is never k–defective. For a variety X being 0–weakly tangentially defective means that it is developable, i.e. the Gauss map of X has positive dimensional ﬁbres, in particular, according to Zak’s theorem on tangencies (see [33]), X must be singular, unless X is a linear space. Instead, 0–weakly defective means that the dual variety of X is not a hypersurface. In the surface case this happens if and only if the surface is developable, i.e. if and only if the surface is either a cone or the tangent developable to a curve (see [19]). However this is no longer the case if dim(X) > 2 (see [13]). Finally no variety is 0–defective. 1.23. Let X ⊂ Pr be, as above, an irreducible, non–degenerate, projective variety of dimension n. For every non–negative integer k, we set: rk = rk (X) := r − dim(TX,p1 ,...,pk ) − 1 = r − s(k−1) (X) − 1. Consider the projection of X with center TX,p1 ,...,pk . We call this a general k– tangential projection of X, and we will denote it by τX,p1 ,...,pk or simply by τX,k . We will denote by Xk its image. Notice that Xk is non–degenerate in Prk . We deﬁne τX,0 as the identity so that X0 = X. We set nk := nk (X) := dim(Xk ) and mk := mk (X) = n − nk . Notice that mk is the dimension of the general ﬁbre of the map τX,k : X → Xk . Lemma 1.24. Let X ⊂ Pr be an irreducible, projective variety of dimension n ≥ 2. Then mk ≤ γk . Proof. Consider the general k–tangential projection from TX,p1 ,...,pk , τX,k : X → Xk . Let p0 be a general point of X. The pull–back via τX,k of the tangent space to Xk at τX,k (p0 ) is TX,p0 ,...,pk . Hence TX,p0 ,...,pk is tangent to X along the whole −1 (τX,k (p0 )), which has dimension mk . By the deﬁnition of γk we have the ﬁbre τX,k assertion.

The following result is an immediate consequence of Terracini’s lemma (see [6], §3): Proposition 1.25. Let X ⊂ Pr be an irreducible, projective variety of dimension n ≥ 2 which is minimally k–defective. Then:

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(i) nh = n for 1 ≤ h ≤ k − 1, whereas nk ≤ n − δk < n, thus mh = 0 for 1 ≤ h ≤ k − 1, whereas mk ≥ δk ; (ii) 0 < nk < rk , i.e. Xk is a proper subvariety of positive dimension of Prk ; (iii) if r ≥ n(k + 1) + k then nk = n − δk , i.e. mk = δk ; (iv) δk ≤ n − 1 and r ≥ (n + 1)k + 2; (v) if r = (n + 1)k + 2 then δk = 1, mk = n − 1 and Xk is a plane curve. Proof. Let us prove part (i). Consider the general h–tangential projection τX,h : X → Xh ⊆ Prh from TX,p1 ,...,ph . Let p0 be a general point of X. For all h, the pull–back via τX,h of the tangent space to Xh at τX,h (p0 ) is TX,p0 ,...,ph , hence: s(h) = dim(TX,p0 ,...,ph ) = nh + dim(TX,p1 ,...,ph ) + 1 = nh + s(h−1) + 1. Since X is minimally k–defective, then for all i < k one has s Hence formula (6) gives, for all h < k, nh = n. On the other hand, we know that:

(i)

dim(TX,p0 ,...,pk ) = s(k) < σ (k) ≤ n(k + 1) + k.

(6)

= (i + 1)n + i.

(7)

Hence by formula (6), applied for h = k, we have that nk ≤ n − δk < n. This proves (i). Since X is k-defective, then dim(S k (X)) < r, hence dim(TX,p0 ,...,pk ) = s(k) < r, i.e. TX,p0 ,...,pk cannot coincide with the whole space. Therefore Xk is a proper subvariety of Prk . Since X is non–degenerate, one has nk > 0. This proves part (ii). If r ≥ n(k + 1) + k then σ (k) = n(k + 1) + k. Therefore (6) and (7) yield nk = n − δk , i.e. part (iii). Since nk > 0 one has δk < n. Furthermore r − (n + 1)k = rk > nk ≥ 1. Hence r ≥ (n + 1)k + 2. This proves part (iv). If r = (n + 1)k + 2, then σ (k) = min{n(k + 1) + k, r} = r = (n + 1)k + 2 and formula (6), applied for h = k, gives nk = 2 − δk . On the other hand

2 = r − (n + 1)k = rk > nk ≥ 1, thus nk = 1 and δk = 1. This proves part (v). The following example is well known: Example 1.26. Let k ≥ 1 be an integer and let X be the Segre embedding 2 of Pk+1 × Pk+1 in Pk +4k+3 . We claim that X is 1–defective with δ1 (X) = 2, n1 (X) = 2k. Indeed one sees that X1 is nothing but the Segre embedding of Pk × Pk in k2 +2k . The assertion immediately follows. P 1.27. We recall the following elementary criterion, needed in the sequel, which tells us when a variety sits in a cone (see [6], Proposition 4.1):

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Proposition 1.28. Let X ⊂ Pr be an irreducible, non–degenerate projective variety of dimension n. Let Π ⊂ Pr be a projective subspace of dimension s not containing X. Then X projects from Π to a variety Y of dimension m < n, i.e. X sits in the cone with vertex Π over Y , if and only if the general tangent space to X intersects Π along a subspace of dimension n − m − 1. In particular X is a cone with vertex Π if and only if the general tangent space to X contains Π and X sits in a Ps+1 containing Π if and only if the general tangent space to X meets Π along a subspace of dimension n − 1. As a consequence we have: Proposition 1.29. Let X ⊂ Pr be an irreducible, non–degenerate projective variety of dimension n. Let Π ⊂ Pr be a projective subspace of dimension s ≥ 0 not containing X, let Π be a complementary subspace of dimension r − s − 1, and assume that X projects from Π to a variety Y ⊂ Π of dimension n − 1. Then: (i) for every k ≥ s, S k (X) is the cone with vertex Π over S k (Y ), and therefore s(k) (X) = s(k) (Y ) + s + 1 and γk (X) = γk (Y ) + 1. In particular X is k– defective if k > s [and γk (X) = 1 if and only if Y is not k–weakly defective] and X is s–defective if and only if Y is s–defective; (ii) for every positive integer k < s, one has s(k) (X) ≥ s(k) (Y ) + k + 1. In particular if Y is not k–defective, then the equality holds and X is also not k–defective. As a consequence, if Y is not s–defective, then X is minimally (s+1)–defective, whereas if Y is minimally s–defective, then also X is minimally s–defective. Finally then γk (X) = 1 if and only if Y is not (s + 1)–weakly defective. Proof. Let p ∈ X be a general point. By Proposition 1.28, TX,p meets Π in one point. Hence there is a map: f : p ∈ X → TX,p ∩ Π ∈ Π and the image of f is non–degenerate in Π since X is non–degenerate in Pr . Let k ≥ s. Let p0 , . . . , pk be general points of X and let q0 , . . . , qk be the corresponding projections on Y . Then TX,p0 ,...,pk contains Π and projects from Π to TY,q0 ,...,qk . Hence the general tangential k–contact locus of S pulls–back to X to the general tangential k–contact locus of X. Moreover, by Terracini’s lemma and by Proposition 1.28, we have that S (k) (X) is a cone with vertex Π. By Proposition 1.20, S (k) (X) is the cone with vertex Π over S (k) (Y ). Thus (i) follows. Let k ≤ s be a positive integer. The above argument shows that TX,p0 ,...,pk intersects Π along a Pl , with l ≥ k. Since S k (X) projects from Π to S k (Y ), part (ii) follows from Proposition 1.28. The rest of the assertion is trivial.

In a similar way one proves the following:

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Proposition 1.30. Let X ⊂ Pr be an irreducible, non–degenerate projective variety of dimension n. Let Π ⊂ Pr be a projective subspace of dimension 2s ≥ 0 not containing X and assume that X projects from Π to a variety Y ⊂ Pr−2s−1 of dimension n − 2. Then: (i) for every k ≥ s, S (k) (X) is the cone with vertex Π over S (k) (Y ), hence s(k) (X) = s(k) (Y ) + 2s + 1 and therefore X is k defective; (ii) for every positive integer k < s, one has s(k) (X) ≥ s(k) (Y ) + 2k + 2. In particular if Y is not k–defective, then the equality holds and X is also not k–defective. As a consequence, if Y is not (s−1)–defective, then X is minimally s–defective. Proposition 1.31. Let X ⊂ Pr be an irreducible, projective variety of dimension n. Let Π ⊂ Pr be a projective subspace of dimension 2s − 1 ≥ 0 not containing X and assume that X projects from Π to a variety Y ⊂ Pr−2s of dimension n − 2. Then: (i) for every k ≥ s − 1, S (k) (X) is the cone with vertex Π over S (k) (Y ), hence s(k) (X) = s(k) (Y ) + 2s. In particular X is k defective if k > s − 1 and X is (s − 1)–defective if and only if Y is (s − 1)–defective; (ii) for every positive integer k < s − 1, one has s(k) (X) ≥ s(k) (Y ) + 2k + 2. In particular if Y is not k–defective, then the equality holds and X is also not k–defective. As a consequence, if Y is not (s−1)–defective, then X is minimally s–defective. 1.32. Now we give a deﬁnition which extends a deﬁnition given in [22]. Deﬁnition 1.33. Let Z ⊆ Pr be an irreducible, non–degenerate, projective variety of dimension n. Let h be a non–negative integer such that h ≤ n. Let z ∈ Z be a general point and suppose that TZ,z ∩ Z − {z} has an irreducible component of dimension h. In this case we will say that Z is h–tangentially degenerate. We will simply say that Z is tangentially degenerate if it is h–tangentially degenerate for some h. Remark that, if h > 0, to say that Z is h–tangentially degenerate is equivalent to say that TZ,z ∩ Z has an irreducible component of dimension h. We notice the following results: Proposition 1.34. Let Z ⊆ Pr be an irreducible, non–degenerate, projective variety of dimension n. One has: (a) Z is n–tangentially degenerate if and only if r = n and Z = Pn ; (b) if n ≥ 2, then Z is (n − 1)–tangentially degenerate if and only if either r = n + 1 or Z is a scroll over a curve.

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Proof. Part (a) is obvious. Part (b) is classical, and it can be easily deduced from [13], Theorem 2.1 and Theorem 3.2.

Lemma 1.35. Let X ⊂ Pr be an irreducible, projective variety, with r > 2n + 1. Let x ∈ X be a general point and suppose that the projection of X from x is a variety X ⊂ Pr−1 which is not tangentially degenerate. Then a general tangential projection of X is a birational map to its image. Proof. If the conclusion does not hold, then for x, y ∈ X general points, the (n+1)– dimensional space spanned by x and TX,y meets X at another point x . But then, by projecting X from x, we get a variety X ⊂ Pr−1 such that the tangent space

to X at a general point z meets X in some other point z . 1.36. By Terracini’s lemma and elementary considerations (see Proposition 1.34), there are no defective curves. The study of defective surfaces goes back to Palatini [24], Scorza [27], Terracini [30], whose classiﬁcation result is the ﬁrst complete one. In modern times, we mention Dale [11] and Catalano–Johnson [4], whereas in [6] one ﬁnds the full classiﬁcation of k–weakly defective surfaces for any k, which includes Terracini’s classiﬁcation of defective surfaces. The classiﬁcation of 1–defective threefolds was taken up by Scorza in [26]. The case of smooth 1–defective threefolds was also examined by Fujita [16] and Fujita– Roberts [17]. Scorza’s classiﬁcation has been revisited in [7]. We recall here the result: Theorem 1.37. Let X ⊂ Pr be an irreducible, non–degenerate, projective, 1– defective threefold. Then r ≥ 6 and X is of one of the following types: (1) X is a cone over a surface S; (2) X sits in a 4–dimensional cone over a curve C; (3) r = 7 and X sits in a 4–dimensional cone over the Veronese surface in P5 ; (4) X is either the 2–Veronese embedding of P3 in P9 or a projection of it in Pr , r = 7, 8; (5) r = 7 and X is a hyperplane section of the Segre embedding of P2 × P2 in P8 . Conversely, all the irreducible threefolds in the above list are 1-defective. Furthermore δ1 ≤ 2 and m1 ≤ 2 and actually m1 = δ1 = 1 unless we are in case (1) and X is either a cone over a curve or a cone over a Veronese surface in P5 , in which case m1 = δ1 = 2. Remark 1.38. It is useful to remark that, for a 1–defective threefold X as in the various cases listed in Theorem 1.37 above, the general tangential 1–contact locus is:

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(1) a reducible curve consisting of two general rulings of X, unless S is a 1– weakly defective surface; (2) a reducible surface consisting of two general ﬁbres of the projection of X to C; (3) an irreducible surface, the pull–back on X of a general conic of the Veronese surface in P5 ; (4) an irreducible conic, the image of a general line in P3 ; (5) an irreducible conic ([7], Example 2.5). Notice that: • if the general tangential 1–contact locus is a reducible curve, then we are in case (1) and X, being a cone over a non–defective surface, lies in Pr , with r ≥ 7; • if the general tangential 1–contact locus is a reducible surface, then we are either in case (2) or in case (1) and X is the cone over a 1–weakly defective surface with reducible general tangential 1–contact locus. By looking at the classiﬁcation of weakly defective surfaces (see [6], Theorem 1.3), we see that also in this latter case X sits in a cone of dimension 4 over a curve and that X sits in Pr , with r ≥ 7; • if the general tangential 1–contact locus is an irreducible curve, then we are in case (4) or (5). Let G the family of tangential 1–contact loci. This is a 4–dimensional family of conics. If x ∈ X is a general point, we have therefore a 2–dimensional family Gx of conics in G passing through x. It is important to notice, for future purposes, that the tangent lines to the conics in Gx ﬁll up the whole tangent space TX,x . This is trivial in case (4), and in case (5) it immediately follows by the discussion in [7], Example 2.5. In connection with the previous classiﬁcation results, it is worth pointing out the following: Proposition 1.39. Let X ⊂ Pr , r ≥ (k + 1)n + k, be an irreducible, non– degenerate projective variety of dimension n. Suppose that X enjoys the property that k + 1 general points of X lie on a curve Γ ⊂ X such that dim(< Γ >) ≤ 2k. Then X is k–defective. Proof. Notice that Γ is not defective and therefore S k (Γ) =< Γ >. Furthermore our assumption on dim(< Γ >) implies that the map from the abstract secant variety of Γ and S k (Γ) has positive dimensional ﬁbers, i.e. there are inﬁnitely many (k + 1)–secant Pk ’s to Γ and therefore to X, containing the general point of < Γ >. This shows that there are inﬁnitely many (k + 1)–secant Pk ’s to X

containing the general point of S k (X), thus X is k–defective.

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2. Basic properties of defective threefolds From now on we will concentrate on defective threefolds. In this section we make a remark which, though not really needed in the sequel, is, in our opinion, of some interest. We start by noticing that the basic tool for the proof of Theorem 1.37 is the study of the ﬁrst general tangential projection τX,1 : X → X1 . One studies separately the cases in which X1 is either a curve or a weakly–defective surface. In these situations, one has interesting information about the general hyperplane section of X which are important steps toward the classiﬁcation. When k > 1, the corresponding analysis is not conclusive. However, one has the following results: Proposition 2.1. Let X ⊂ Pr be an irreducible, non–degenerate, projective, minimally k–defective threefold. Assume that mk = 2, i.e. Xk is a curve. Let S be a general hyperplane section of X. Then the general k–tangential projection Sk of S sits in a cone with a vertex of dimension at most k − 2 over the curve Xk . Moreover: (i) when r ≥ 6k, then S is (2k − 1)–defective; (ii) when r ≥ 6k − 2, then S is (2k − 1)–weakly defective. Proof. Since X is k–defective, we have that S k (X) is a proper subvariety of Pr , hence k < r. Let p0 , . . . , pk be general points on X. Then < p0 , . . . , pk > is a proper subspace of Pr . Let H be a general hyperplane containing < p0 , . . . , pk >. By varying the points p0 , . . . , pk on X, then H also varies and it turns out to be a general hyperplane in Pr . Let S be the surface cut out by H on X. Then: dim(H ∩ TX,P1 ,...,Pk ) − dim(TS,P1 ,...,Pk ) ≤ k − 1.

(8)

Notice that Sk , which is the projection of S from TS,P1 ,...,Pk , is contained in a cone W over the projection S of S from H ∩ TX,P1 ,...,Pk . We may assume that W is of minimal dimension with the property of containing Sk . The vertex Π of W is contained in the projection of H ∩ TX,P1 ,...,Pk from TS,P1 ,...,Pk , hence, by (8), we have dim(Π) ≤ k − 2. Remark that S is also the projection of S from TX,p1 ,...,pk . Hence S is contained in Xk . Since Xk is a curve, we have S = Xk , proving the ﬁrst part of the assertion. The tangent space to Sk at a point q intersects in a point the vertex Π of the cone W , because the projection of Sk from Π is a curve (see Proposition 1.28). Let q1 , . . . , qk−1 be general points of Sk . Then, by the minimality assumption on W , the space TSk ,Q1 ,...,Qk−1 contains Π. Thus the general (k − 1)–tangential projection of Sk , which is the general (2k − 1)–tangential projection of S, certainly has dimension at most 1. Assume that S is not k–defective, otherwise there is nothing to prove. Then Sk sits in Ps , s = r − 3k − 1. From the previous argument, we get that the span of k general tangent planes to Sk has dimension at most 3k − 2. Thus, if s ≥ 3k − 1, i.e. if r ≥ 6k, then Sk is (k − 1)–defective, hence S is (2k − 1)–defective. This proves part (i).

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If the span of k − 1 general tangent planes to Sk has dimension at most 3k − 5 and if s ≥ 3k − 4, i.e. if r ≥ 6k − 3, then Sk is (k − 2)–defective, hence S is (2k − 2)–defective and therefore also (2k − 1)–defective. Assume that the span of k − 1 general tangent planes to Sk has dimension 3k − 4. Then the projection of Sk from TSk ,q1 ,...,qk−2 is a surface and therefore TSk ,q1 ,...,qk−2 cannot contain the vertex Π of the cone W . Since, as we saw, all spaces TSk ,qi intersect Π in a point, by the minimality assumption on W we have that TSk ,q1 ,...,qk−2 ∩Π has codimension 1 in Π. Now the projection S2k−2 of Sk from TSk ,q1 ,...,qk−2 is a surface which sits in the cone Z over Xk with vertex the projection of Π from TSk ,Q1 ,...,Qk−2 ∩Π, which is a point. Hence S2k−2 coincides with the cone Z. Moreover S2k−2 spans a Ps where s = s − 3(k − 2) = r − 3k − 1 − 3(k − 2) = r − 6k + 5. If s ≥ 3, i.e. if r ≥ 6k − 2, then S2k−2 is 1-weakly defective. Then S is (2k − 1)-weakly defective (see [6], Proposition 3.6).

Similar arguments lead to the following: Proposition 2.2. Let X ⊂ Pr be an irreducible, non–degenerate, projective, minimally k–defective threefold. Assume that mk = 1 and that Xk is a developable surface. Let S be a general hyperplane section of X. Then the general k-tangential projection Sk of S sits in a cone of dimension k + 1 over the developable surface Xk . When r ≥ 6k + 1, then S is (2k − 1)-weakly defective.

3. Basic properties of the contact loci In this section we study, mostly in the case of defective threefolds, some basic properties of the contact loci Γ and Σ deﬁned in §1. We start with the following: Proposition 3.1. Let X ⊂ Pr be an irreducible, non–degenerate, projective, k– defective variety. For a general choice of p0 , . . . , pk ∈ X and a general choice of the hyperplane H containing TX,p0 ,...,pk , the contact loci Γ = Γp0 ,...,pk and Σ = Σ(H) = Σp0 ,...,pk (H) are equidimensional and smooth at each of the points p0 , . . . , pk . Furthermore either they are irreducible or they consist of k + 1 irreducible components, one through each of the points p0 , . . . , pk . Proof. We prove the assertion for Γ. The proof for Σ is quite similar. First of all, let us move slightly the points pi ’s on some component of Γ to a new set of points {q0 , . . . , qk }. Then q0 , . . . , qk are also general points on X. Furthermore TX,p0 ,...,pk contains the tangent spaces to X at the points qi ’s, so for dimension reasons, it coincides with TX,q0 ,...,qk . Hence TX,q0 ,...,qk is also tangent to X along Γ = Γp0 ,...,pk . This tells us that, by the generality of the points p0 , . . . , pk , Γ is smooth at p0 , . . . , pk , and therefore there is only one irreducible component of Γ through each of the points p0 , . . . , pk . Since [p0 , . . . , pk ] is a general point of

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Symk+1 (X), there is the monodromy action of the full symmetric group Sk+1 on p0 , . . . , pk as recalled in §1.13. Hence we can permute the points pi as we like, and this implies that all components of Γ have the same dimension. Assume now that there is a component of Γ which contains more than one of the points pi ’s, say p0 and p1 . Again by monodromy, we can let p0 stay ﬁxed and we can move p1 to any one of the points pi , i > 1. Then we see that also p0 and pi , i > 1, stay on an irreducible component of Γ. Since p0 sits on only one irreducible component of Γ, then this component has to contain all the points p0 , . . . , pk and therefore it has to coincide with Γ. This proves the proposition.

Since Γ ⊆ Σ, the next corollary immediately follows: Corollary 3.2. Let X ⊂ Pr be an irreducible, non–degenerate, projective, k– defective variety of dimension n. If k = γk , e.g. if either k = 1 or γk = n − 1, then the general contact locus Σ coincides with the general tangential contact locus Γ. The next proposition, which will be useful later, tells us how tangential contact loci behave under a general tangential projection: Proposition 3.3. Let X ⊂ Pr be an irreducible, non–degenerate, projective, k– defective, but not 1–defective variety of dimension n, hence k ≥ 2. Let X1 be the general tangential projection of X. Then the general tangential k–contact locus Γ of X and the general tangential (k − 1)–contact locus Γ1 of X1 have the same dimension and Γ is reducible if and only if Γ1 is. Proof. Since X is not 1–defective, we have n1 = n. Hence, if p0 ∈ X is a general point, the projection τ := τX,1 : X → X1 from TX,p0 is generically ﬁnite. If p1 , . . . , pk are general points of X, so that τ (p1 ), . . . , τ (pk ) are general points of X1 , then the image of the span TX,p0 ,p1 ,...,pk via τ is the span TX1 ,τ (p1 ),...,τ (pk ) . Hence Γ maps onto Γ1 via τ . The generic ﬁniteness of τ implies dim(Γ) = dim(Γ1 ). Moreover if Γ1 is reducible then also Γ is. Conversely, suppose Γ = Γp0 ,...,pk is reducible. Then Γ = ∪ki=0 Γ(i) , where Γ(i) is (i) (i) the irreducible component containing the point pi . Set Γ1 = τ (Γ(i) ), so that Γ1 is the component of Γ1 through the point τ (pi ). Since p1 , . . . , pk are general points of X, then τ (p1 ), . . . , τ (pk ) are general points of X1 . Therefore, once p1 , and thus (1) (1) (1) (2) Γ1 , has been ﬁxed, we can choose p2 so that τ (p2 ) ∈ / Γ1 , and so Γ1 = Γ1 . Proceeding in this way we see that for a general choice of p1 , . . . , pk , the varieties (1) (k)

Γ1 , . . . , Γ1 are all distinct varieties, proving that Γ1 is reducible. Now we restrict to the threefold case. One can easily detect when Γ, and therefore Σ, has codimension 1 from the behavior of the tangential projection of X: Proposition 3.4. Let X ⊂ Pr be an irreducible, non–degenerate, projective, minimally k–defective threefold. Then γk = 2 if and only if either one of the following holds:

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(i) Xk is a curve; (ii) Xk is a developable surface. Proof. We know that 0 < nk < 3. Suppose that γk = 2. Choose general points p0 , . . . , pk ∈ X. Let Γ0 be the irreducible component of Γp0 ,...,pk containing p0 . Consider the projection τX,k from the space TX,p1 ,...,pk . Then τX,k (Γ0 ) is contained in τX,k (TX,p0 ,...,pk ), which is a projective space of dimension nk . This shows that τX,k (Γ0 ) is a proper subvariety of Xk , otherwise Xk would be equal to the linear space τX,k (TX,p0 ,...,pk ), whereas we know that Xk is a proper, non–degenerate subvariety of Prk (see Proposition 1.25). Hence τX,k (Γ0 ) is either a point or a curve. In the former case the ﬁber of τX,k at p0 , which is a general point of X, has codimension 1, hence nk = 1. In the latter case, the tangent plane to Xk at its general point τX,k (p0 ) meets Xk along the curve τX,k (Γ0 ), which passes through τX,k (p0 ). Hence Xk is a developable surface. Conversely if either nk = 1 or Xk is a developable surface, then dim(Γ) = 2, because Γ contains the pull–back, via τX,k , of the contact locus of Xk with its general tangent space.

Next, we look at the families G and S respectively describing the general contact loci Γ = Γp0 ,...,pk when p0 , . . . , pk vary and by Σ = Σ(H)p0 ,...,pk when p0 , . . . , pk and H vary. Recalling the deﬁnition of involution from §1.14, we have: Lemma 3.5. In the above setting, if γk = 2 [resp. k = 2], then the family of divisors G [resp. S] is an involution of dimension k + 1. Hence, if the general member of G [resp. of S] is irreducible, then G [resp. S] is a linear system. Proof. To prove the ﬁrst part of the assertion one argues as in [6], pp. 172–173: though the surface case is treated there, the argument applies, word by word, to our situation. The ﬁnal part of the assertion follows by Theorem 1.15.

The following proposition gives more precise information about the situation described in Proposition 3.4: Proposition 3.6. Let X ⊂ Pr be an irreducible, non–degenerate, projective, minimally k–defective threefold. Assume that γk = 2. Then: (i) if Xk is a curve and r > 4k + 2, then the involution G is composed with a pencil; (ii) if Xk is a developable surface, the same conclusion holds provided r > 4k + 3. Proof. Assume that r > 4k + 2 and Xk is a curve. By Proposition 1.25, we know that Xk−1 is a threefold in Prk−1 , whose general tangential projection is a curve. Notice that rk−1 = r + 4 − 4k > 6. Then by Theorem 1.37 we have that Xk−1 is a cone over a curve with vertex a line and therefore two general tangent spaces to Xk−1 have contact with Xk−1 along two planes. By pulling back to X, we see

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that Γp0 ,...,pk has at least two irreducible components. It follows from Theorem 1.15 that G is composed with a pencil. This ﬁnishes case (i). Similarly, assume that Xk is a developable surface and r > 4k + 3. By Lemma 3.6 of [7] a general hyperplane section S of Xk−1 is 1–weakly defective but not 1–defective. By the classiﬁcation of weakly defective surfaces (see [6], Theorem (1.3)), one gets that either: (a) S is developable or (b) S sits in a cone over a curve, with vertex along a line or (c) S sits in P6 . Case (c) is excluded because S is non–degenerate in Prk−1 −1 and rk−1 − 1 = r + 3 − 4k > 6. Then only cases (a) and (b) may occur and, in both of them the tangential contact variety Γ of S with the span of two general tangent planes is reducible. Now we claim that the general tangential contact locus of Xk−1 at two general points is reducible. This, as in case (i), leads to the assertion. As for the claim, let q1 , q2 be two general points on Xk−1 . Then TXk−1 ,q1 ,q2 is a P6 , because δk (X) = δ1 (Xk−1 ) = 1. Taking a general hyperplane H through q1 , q2 and its section S with Xk−1 , we have that TS ,q1 ,q2 is a P5 , because S is not 1–defective. Hence TS ,q1 ,q2 = H ∩ TXk−1 ,q1 ,q2 . Let Γ be the contact variety of TXk−1 ,q1 ,q2 with Xk−1 . Of course the tangential contact locus Γ for S is the intersection of Γ with H . Since Γ is reducible and by the genericity of H , we see that Γ is reducible. This proves the claim and therefore the proposition.

4. The irreducible divisorial case Let X ⊂ Pr be an irreducible, non–degenerate, projective, minimally k–defective threefold with k ≥ 2. In this section we examine the case in which γk (X) = 2 and the general tangential k–contact locus is irreducible. By Proposition 3.4 we know that either Xk is a curve or Xk is a developable surface. Furthermore Proposition 3.6 and Proposition 1.25 imply that 4k + 2 ≤ r ≤ 4k + 3 and if r = 4k + 2 then Xk is a curve. Recall that, by Lemma 3.5, the family of divisors G of tangential k–contact loci is a linear system of dimension k + 1 which is not composed with a pencil. Observe that, by construction, the linear system 2G (see §1.8) is contained in the hyperplane linear system H of X. In particular we have the linear equivalence relation H ≡ 2Γ + Δ, with Δ eﬀective. We recall that this equivalence takes place on the normalization of X. We let φG : X → Y ⊂ Pk+1

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be the rational map deﬁned by G. Since G is not composed with a pencil, then n := dim(Y ) > 1. We set d := deg(Y ). The classiﬁcation is based on the classiﬁcation of Y ⊂ Pk+1 according to its dimension and degree. A straightforward application of Theorem 1.4 gives the following useful information: Lemma 4.1. In the above setting, one has: n(n − 3) +2 (9) 2 where ι = ι(Y ) = min{d − k + n − 2, k + 1 − n}. Moreover if d < 2(k − n) + 3, the equality holds in (9) if and only if the general curve section of Y is projectively normal. 4k + 4 ≥ r + 1 ≥ hY (2) ≥ ι + k(n + 1) −

We will discuss separately the two cases n = 2, 3. Theorem 4.2. In the above setting, assume that n = 3. Then ι < 3, Δ = 0 and we are in one of the following cases: (1) ι = 0 and Y is a threefold of minimal degree in Pk+1 ; then X sits in a cone over the 2–uple embedding of Y , with vertex either a point if r = 4k + 2 or a line if r = 4k + 3; (2) ι = 1, k = 3, and Y is a hypersurface of degree d ≥ 3 in P4 ; then either r = 4k + 2 = 14 and X is the 2-uple embedding of Y or r = 4k + 3 = 15 and X sits in the cone with vertex a point over the 2-uple embedding of Y ; (3) ι = 1 and Y is a threefold of degree k in Pk+1 with curve sections of arithmetic genus 1; then either r = 4k + 2 and X is the 2-uple embedding of Y or r = 4k + 3 and X sits in the cone with vertex a point over the 2-uple embedding of Y ; (4) ι = 1 and Y is a threefold of degree k in Pk+1 with curve sections of genus 0 which is either a cone with vertex a line over a smooth rational curve of degree k in Pk−1 or it has a double line; then r = 4k + 3 and X is the 2-uple embedding of Y ; (5) ι = 2, k = 4 and Y is a threefold of degree d ≥ 5 in P5 , contained in a quadric; then r = 4k + 3 = 19 and X is the 2-uple embedding of Y ; (6) ι = 2, k ≥ 4 and Y is a threefold of degree k + 1 in Pk+1 with curve sections of arithmetic genus 2; then r = 4k + 3 and X is the 2-uple embedding of Y . Proof. In the present case equation (9) reads: 4k + 4 ≥ r + 1 ≥ hY (2) ≥ ι + 4k + 2 hence 0 ≤ ι ≤ 2. Then 2G is a subsystem of H of codimension ≤ 2. It follows that the image of Δ on X imposes at most two conditions to the hyperplanes of Pk+1 . As the map X → X is ﬁnite, the only possibility is that Δ = 0.

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If ι = 0, then d = k − 1, i.e. Y is a threefold of minimal degree in Pk+1 . Since 2G is a subsystem of codimension at most 2 in H, we are in case (1). If ι = 1 then either d = k or k = 3 and d ≥ 3. In the latter case we are in case (2). In the former case if hY (2) = 4k + 3, then the curve section of Y is linearly normal of degree k and arithmetic genus 1 (see Theorem 1.4) and we are in case (3). If hY (2) = 4k + 4 then, again by Theorem 1.4, the curve section of Y cannot be linearly normal of degree k and arithmetic genus 1, hence it has arithmetic genus 0 and it is therefore smooth. By Lemma 1.6 we are in case (4). If ι = 2, then all the inequalities in (9) are equalities, hence X is the 2–uple embedding of Y . We have either k = 4 and d ≥ 5 or d = k + 1 and k ≥ 4. If k = 4, the equalities in (9) imply that Y is contained in a quadric of P5 and we are in case (5). If d = k + 1, then Y has degree k + 1. Since equalities hold in (9), by Theorem 1.4, a general curve section of Y is linearly normal of degree k + 1, with arithmetic genus 2 and we are in case (6).

Example 4.3. We show that if X is as in any one of the previous cases, then it is minimally k–defective. We may assume k ≥ 2. (1) Assume that X ⊂ Pr sits in a cone W over the 2-uple embedding Y ⊂ P4k+1 of a threefold Y of minimal degree in Pk+1 with vertex L a line or a point. Accordingly one has 4k + 3 ≥ r ≥ 4k + 2. Remark that by deﬁnition X projects onto Y from L. In Example 6.3 below it is proved that Y is k–defective and s(k) (Y ) = 4k. By Proposition 1.20, part (i), one has s(k) (X) ≤ s(k) (Y ) + dim(L) + 1 = 4k + dim(L) + 1 < σ (k) (X), thus X is k–defective. Let us prove that X is minimally k–defective. In Example 6.3 below we will see that Y is minimally (k − 1)–defective and that s(k−1) (Y ) = 4k − 2. Moreover the general (k − 1)–contact locus of Y is an irreducible curve. Suppose, by contradiction, that X is (k − 1)–defective. Then by Proposition 1.20, part (i), we have 4k − 2 = s(k−1) (Y ) ≤ s(k−1) (X) < σ (k−1) (X) = 4k − 1, thus s(k−1) (X) = 4k − 2. By Proposition 1.20, part (iii), X it would be, as well as Y , minimally (k − 1)– defective. Moreover the general (k − 1)–contact locus of X projects from L to the general (k − 1)–contact locus of Y , hence it is an irreducible curve. However we will see at the beginning of §6 below that the maximum for the embedding dimension r of a minimally (k − 1)–defective threefolds whose general (k − 1)–contact locus is an irreducible curve is r ≤ 4k + 1. This gives a contradiction, which proves that X is not (k − 1)–defective. If L is a point, then r = 4k + 2 and by Proposition 1.25, part (v), we have δk (X) = 1, Xk is a plane curve and s(k) (X) = 4k + 1. If L is a line, then r = 4k+3. By Proposition 3.6, part (i), we have nk = 2. Then Proposition 1.25, part (iii), yields δk (X) = 1 hence s(k) (X) = 4k + 2. Actually we see that in this case Xk is a cone in P3 . Indeed, by projecting from a general point of L one has to go back to the previous situation. (2) Let Y be a threefold in P4 of degree d ≥ 3. Let Y be the 2–uple embedding of Y in P14 . If p is a point in Y we abuse notation and denote by p also the corresponding point on Y .

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We have σ (3) (Y ) = 14. Let p0 , . . . , p3 be general points on Y . Since there is a quadric in P4 singular at p0 , . . . , p3 , namely the hyperplane < p0 , . . . , p3 > counted twice, then TY ,p0 ,...,p3 is contained in a hyperplane. This proves that Y is 3–defective. Let us prove that Y is minimally 3–defective. Certainly Y is not 1–defective. Indeed let P, P be general points on Y and call L, L the tangent hyperplanes to Y at P, P . Then L, L are just two general hyperplanes in P4 and one computes very easily that the family of quadrics in P4 which are tangent to L, L has the expected dimension 6 = 14 − 8. Suppose that Y is 2–defective and therefore minimally 2– defective. Let p0 , p1 , p2 be general points on Y . There are hyperplane sections of Y tangent at p0 , p1 , p2 and having an irreducible singular locus of dimension 1, namely the hyperplane sections of Y corresponding to the quadrics in P4 singular along the plane < p0 , p1 , p2 >. Then, as we will see at the beginning of §6 below, the embedding dimension of Y should be bounded above by 13 = 4k + 5, a contradiction. By Proposition 1.25, part (v), we have δ3 (Y ) = 1, s(3) (Y ) = 13, n3 (Y ) = 1 and Y3 is a plane curve. Assume now that X sits in a cone with vertex a point v over Y , so that r = 15 and that X maps to Y from v. By Proposition 1.20, part (i), we have s(3) (X) ≤ s(3) (Y ) + 1 = 14, hence X is 3–defective. By the same Proposition 1.20, part (ii), since Y is minimally 3–defective, also X is minimally 3–defective. By Proposition 3.6, part (i), we have nk = 2. Then Proposition 1.25, part (iii), yields δk (X) = 1 hence s(k) (X) = 14. (3) Assume that X sits in a cone with vertex a point (or the empty set) over the 2-uple embedding Y of a threefold Y of degree k in Pk+1 with linearly normal curve section of arithmetic genus 1. Then Y sits in P4k+2 (see Theorem 1.4). In [6], Example 4.7, it is proved that Y is k–defective and s(k) (Y ) ≤ 4k + 1. An argument completely similar to the one of the previous Example (2) shows that Y is minimally k–defective. If X = Y then Proposition 1.25, part (v), yields δk (X) = 1, nk (X) = 1, (k) s (Y ) = 4k + 1. Assume X = Y , hence X ⊂ P4k+3 . Then Proposition 1.20, part (i), yields (k) s (X) ≤ s(k) (Y ) + 1 = 4k + 2, thus X is k–defective. Again Proposition 1.20 implies it is minimally k–defective. Then by Proposition 3.6, part (i), we have nk = 2. Finally Proposition 1.25, part (iii), yields δk (X) = 1, hence s(k) (X) = 4k + 2. (4) There are threefolds Y of degree k in Pk+1 with smooth curve sections of genus 0 and a singular line. They are described in Proposition 1.7. By Lemma 1.6, the 2–uple embedding X of Y sits in P4k+3 . By arguing as in Example 4.5 or 4.7 of [6], one sees that X is k–defective. Then an argument completely similar to the one of the previous Example (2) shows that X is minimally k–defective. As usual we ﬁnd δk (X) = 1, nk = 2 and s(k) (X) = 4k + 2.

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(5) Let X ⊂ P19 be the 2-uple embedding of a threefold Y contained in a unique quadric of P5 . If p is a point in Y we abuse notation and denote by p also the corresponding point on X. One has σ (4) (X) = 19. However consider ﬁve general points p0 , . . . , p4 of Y . Since there is a quadric singular at any ﬁve points of P5 , i.e. the double hyperplane < p0 , . . . , p4 >, then there is a hyperplane section of X tangent at p0 , . . . , p4 , hence X is 4–defective. The usual kind of arguments show that X is minimally 4–defective and that δ4 (X) = 1, n4 = 2 and s(k) (X) = 18. (6) Let X be the 2–uple embedding of a threefold Y of degree k + 1 in Pk+1 with curve sections of arithmetic genus 2. Then X sits in P4k+3 by Theorem 1.4. Moreover X is k–defective. This follows by arguing as in [6], Example 4.7 in the usual way. Again one has δk (X) = 1, nk = 2 and s(k) (X) = 4k + 2. Notice that for all the above examples X ⊂ Pr , one has s(k) (X) = r − 1. Then, by Proposition 1.19, s(k+1) (X) = r, i.e. X is not h–defective for any h = k. It is worth adding a few words about the existence of smooth threefolds of the above types. It is obvious that this can happen for threefolds of types (1), (2) and (5). Threefolds of types (3) and (6) can be smooth, but only for ﬁnitely many values of k (this follows, for instance, from the results in [20]; more speciﬁcally, for the genus 1 case, see [23] for the genus 1 case). Threefolds of type (4) can never be smooth. Next let us turn to the case n = dim(Y ) = 2. First we prove a lemma on the dimension of the system 2G. Lemma 4.4. In the above setting, if n = 2 then dim(2G) ≤ r − 2. If the equality holds, then Δ = 0. Proof. Notice that X is not a cone because it is k–minimally defective, with k > 1. The system 2G maps X to a surface Y , then it cannot coincide with H. Since H has no ﬁxed parts, then dim(2G) < r. If dim(2G) = r − 1, then, as in Theorem 4.2, we see that X sits in the cone, with vertex a point, over Y , which is a surface. Then X would be the cone over Y , a contradiction. If dim(2G) = r − 2, then X sits in a cone over Y with vertex a line . The image of Δ has to be contained in ∩ X, hence, as above, Δ = 0.

Now we are ready for the proof of the classiﬁcation theorem in the case n = 2: Theorem 4.5. In the above setting, assume n = 2. Then r = 4k + 3 − i, i = 0, 1 and one of the following cases occurs: (1) X sits in a cone with vertex a space of dimension k − i over the 2–uple embedding of a surface Y of minimal degree in Pk+1 ; (2) k = 2, i = 0 and X sits in a cone with vertex a line over the 2–uple embedding of a surface Y of P3 with deg(Y ) ≥ 3;

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(3) k ≥ 3, i = 0 and X sits in a cone with vertex of dimension k − 1 over the 2–uple embedding of a surface Y of degree k + 1 in Pk+1 with curve sections of arithmetic genus 1. All threefolds in the above list are minimally k–defective with δk (X) = 1, s(k) (X) = r − 1 = 4k + 3 − i, nk = 2 − i, and are not h–defective for any h > k. Proof. Recall that r = 4k + 3 − i with i = 0, 1. Formula (9) and Lemma 4.4 give: 4k + 2 − i ≥ (r + 1) − 2 = r − 1 ≥ hY (2) ≥ 3k + 3 + ι where ι = min(d − k, k − 1). Then, by writing hY (2) = 3k + 3 + q, we see that X sits in a cone W with vertex of dimension s = r − 3k − q − 3 = k − q − i over the 2-uple embedding Y of a non degenerate surface Y ⊂ Pk+1 , such that hY (2) = 3k + 3 + q. By Proposition 1.29, X is certainly (s + 1)–defective. By the minimality assumption we must have k ≤ s + 1 = k − q − i + 1, thus q + i ≤ 1. If q = 0, by Theorem 1.4, Y is a surface of minimal degree in Pk+1 . In this case Y is minimally k–defective (see Theorem (1.3) of [6]) and, Proposition 1.29 implies that X is minimally k–defective too. One has s(k) (X) = s(k) (Y ) + k − i + 1 = 4k + 2 − i, hence δk (X) = 1 and nk (X) = 2 − i. If q = 1, then i = 0. Taking into account Theorem 1.4, we see that only the following cases may occur: • k = 2 and Y is a surface in P3 of degree d ≥ 3 and we are in case (2); • k ≥ 3 and Y is a surface of degree k + 1 in Pk+1 with curve sections of arithmetic genus 1, and we are in case (3). In both the above cases Y is not k–defective (see Theorem (1.3) of [6]), and therefore, by Proposition 1.29, X is minimally k–defective.

Remark 4.6. (1) Observe that in the last statement, in cases (1) and (3) we cannot specify whether Δ ≡ H − 2Σ is zero or not. When it is not, then X meets the vertex of the cone in codimension 1. In case (2) instead, Δ = 0 by Lemma 4.4. (2) A few words about the existence of smooth threefolds in the list of Theorem 4.5. For example, consider case (1). Give an embedding whatsoever of a smooth Y surface of minimal degree in Pk+1 in a Pk−i . This is certainly possible if k − i > 4. Call Y the surface we get in this way. Then join any point of Y with the corresponding point of Y . The resulting ruled threefold is smooth. With the same idea one produces smooth threefolds of type (3) in Theorem 4.5 as soon as k ≥ 6. However note here that such a construction works only for k ≤ 8, since, as it is well known, there are no smooth surfaces of degree k + 1 in Pk+1 with curve sections of arithmetic genus 1 as soon as k ≥ 9.

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5. The reducible case Let X ⊂ Pr be an irreducible, non–degenerate, projective, minimally k–defective threefold with k ≥ 2. In this section we examine the case in which the general element of the family G is reducible. By Proposition 3.1, we know that in this case Γ = Γp0 ,...,pk has exactly k + 1 components, one passing through each of the points pi , i = 0, . . . , k. The following proposition immediately follows by Proposition 3.3, Remark 1.38: Proposition 5.1. Let X ⊂ Pr , be an irreducible, non–degenerate, projective, minimally k–defective threefold with k ≥ 2. Suppose that the general tangential k– contact locus of X is reducible. One has: (1) if γk = 1, then r ≥ 4k + 3, δk = 1, nk = 2 and the general (k − 1)–tangential projection Xk−1 of X is a cone over a surface which is not 1–weakly defective; (2) if γk = 2 and nk = 2, then r ≥ 4k + 3, δk = 1 and the general (k − 1)– tangential projection Xk−1 of X is a threefold contained in a 4-dimensional cone over a curve; (3) if γk = 2 and nk = 1 then r ≥ 4k + 2 and the general (k − 1)–tangential projection Xk−1 of X is a cone with vertex a line over a curve. Conversely, threefolds like in case (1), (2) and (3) above are minimally k– defective with reducible general tangential k–contact locus. We can now specify case (1) of Proposition 5.1: Proposition 5.2. Let X ⊂ Pr be an irreducible, non–degenerate projective threefold. Then the following are equivalent: (i) the general (k − 1)–tangential projection Xk−1 of X is a cone over a surface which is not 1–weakly defective; (ii) X sits in a cone of dimension k + 2, and not smaller, over a surface which is not k–weakly defective. Proof. Suppose the general (k − 1)–tangential projection Xk−1 of X is a cone over a non–developable surface S, hence X is minimally k–defective. Part (i) is trivially true for k = 1. So we assume k ≥ 2 and proceed by induction on k. Let p0 ∈ X be a general point and let τ := τX,1 : X → X1 be the tangential projection from TX,p0 . By Proposition 5.1 and Proposition 3.3, γk−1 (X1 ) = 1 and the general tangential (k − 1)–contact locus of X1 is a reducible curve. By induction, we know that X1 sits in a cone over a surface S1 with vertex Π := Πp0 of dimension k − 2. Notice that the general tangent space to X1 meets Π at one point and projects from Π to the general tangent space to S1 (see Proposition 1.28).

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Call π : X → S1 the composition of τ with the projection of X1 from Π. For a general point p ∈ X, call Zp the intersection of X1 with < Π, π(p) >, which is the general ﬁbre of the projection of X1 from Π. Hence Zp is a curve. Take general points p1 , . . . , pk ∈ X. Terracini’s lemma and Proposition 1.29 imply that TX1 ,τ (p1 ),...,τ (pk ) =< Π, TS,π(p1 ),...,π(pk ) >, hence TX1 ,τ (p1 ),...,τ (pk ) is tangent to X1 along the curves Zpi , i = 1, . . . , k. Furthermore TX1 ,τ (p1 ),...,τ (pk ) is tangent to X1 along the pull–back of the general tangential (k − 1)–contact locus to S1 . Since, as we saw, the general tangential (k − 1)–contact locus of X1 is a reducible curve, we have that S1 is not (k − 1)–weakly defective. Notice that, if we move p1 to a new general point p1 , then TX1 ,τ (p1 ),...,τ (pk ) =< Π, TS,π(p1 ),...,π(pk ) > is again tangent to X1 along the curves Zpi , i = 2, . . . , k. Call Ci the pull–back to X of Zpi via τ . One has that TX,p0 ,p1 ,...,pk is tangent to X along a curve which contains all the Ci ’s. In other words Γp0 ,p1 ,...,pk ⊃ C1 ∪ · · · ∪ Ck . By what we saw above, if we move the point p1 to a new general point p1 , then Tp0 ,p1 ,p2 ...,pk is again tangent along the curves C2 , . . . , Ck . Therefore also by moving p0 to some other point p0 ∈ X, again Γp0 ,p1 ,...,pk contains the curves Ci , i = 1, . . . , k. By equation (5), we know that h(Γ) ≤ 2(1+k)−1 = 2k+1, so dim(< Γ >) ≤ 2k. On the other hand we claim that: dim(< Γ >) ≥ k − 1 + dim(< Ci >).

(10)

Otherwise there is an h < k such that Λ :=< C1 , . . . , Ch > contains all curves Ci , with h < i ≤ k. Then, by the genericity of the points p1 , . . . , pk , the space Λ would contain the whole of X, but this is impossible, since: dim(< Λ >) ≤ dim(< Γ >) ≤ 2k < 4k + 2 ≤ r. By (10) it follows that dim(< Ci >) ≤ k + 1. Set L := Lp0 =< TX,p0 , Π >. By Proposition 1.29, X cannot lie on a cone on a surface with vertex Π. Thus dim(L) = k + 2. We have: dim(< Zpi >) = dim(< Ci >) − dim(< Ci > ∩TX,p0 ) − 1; dim(< Zpi > ∩Π) = dim(< Ci > ∩L) − dim(< Ci > ∩TX,p0 ) − 1. Since dim(< Zpi > ∩Π) = dim(< Zpi >) − 1, because Zpi projects from Π to a point in S, we have that also Hi :=< Ci > ∩L is a hyperplane in < Ci >, i = 1, . . . , k. Now move the point p0 to a new point p0 ∈ X, which gives us a new vertex Π := Πp0 and a new space L := Lp0 =< TX,p0 , Π >. However, since Γp0 ,p1 ,...,pk contains the curves Ci , i = 1, . . . , k, we have that also < Ci > ∩L is a hyperplane in < Ci >, i = 1, . . . , k. Now dim(< L, L >) ≤ 2k + 5 < 4k + 2 ≤ r, so that Ci cannot be contained in < L, L >, for a general choice of pi , otherwise X would be contained in < L, L >. Hence the hyperplane Hi of < Ci > is ﬁxed, as p0 moves. Then the space < H1 , . . . , Hk > does not depend on p0 , i.e. it is contained in Lp for a general point p ∈ X.

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Consider now the linear space H = ∩P ∈U LP , where U is a suitable dense open subset of X. Notice that Lp has to vary when p varies in X, because TX,p ⊂ Lp . Hence dim(H) ≤ dim(Lp ) − 1 = k + 1. Moreover H contains all the hyperplanes Hi of < Ci > so it meets the tangent line of Ci at Pi , which is a general point of X. Thus H intersects all the tangent spaces to X. We conclude by Proposition 1.28 that X projects from H either to a curve or to a surface. If dim(H) ≤ k − 1, then by Proposition 1.29 and since X is minimally k– defective, we have that dim(H) = k − 1 and X projects from H to a surface S which is not k–weakly defective, proving (ii). Assume dim(H) ≥ k. Then since for a general point p ∈ X we have H ∪ TX,p ⊂ Lp and dim(Lp ) = k + 2 then H would meet TX,p . If dim(H) = k then dim(H ∩ TX,p ) ≥ 1 and, by Proposition 1.28, we would have that X sits in a cone with vertex H over a curve. Then, by applying Proposition 1.29, we see that X would be (k − 1)–defective, a contradiction. If dim(H) = k + 1 then dim(H ∩ TX,P0 ) ≥ 2 and, by Proposition 1.28, X would be contained in a subspace of dimension dim(H) + 1 = k + 2 a contradiction. We have thus completed the proof of the fact that (i) implies (ii). Let us assume now that (ii) holds. Then, by Proposition 1.29, X is minimally k–defective and, since S is not k–weakly defective, then γk (X) = 1 and the general tangential k–contact locus is a reducible curve. Then, by Proposition 5.1, (i) holds.

The following proposition is proved with a similar argument, but with a slight modiﬁcation: Proposition 5.3. Let X ⊂ Pr be an irreducible, non–degenerate projective threefold. Let k ≥ 2 be an integer. Then the following are equivalent: (i) the general (k − 1)–tangential projection Xk−1 of X sits in a 4–dimensional cone over a curve; (ii) X sits in a cone of dimension 2k + 2, and not smaller, over a curve. Proof. Let us prove that (i) implies (ii). The assertion trivially holds for k = 2. So we assume k ≥ 3 and proceed by induction on k. Let p0 ∈ X be a general point and let τ := τ1 : X → X1 be the tangential projection from TX,p0 . By induction, we know that X1 sits in a cone over a curve C with vertex Π := Πp0 of dimension 2k − 2. Call π : X → C the composition of τ with the projection of X1 from Π. For a general point p ∈ X, call Zp the intersection of X1 with the span < Π, τ (p) >, which is the ﬁbre of the projection of X1 from Π. Hence ZQ is a surface. Take general points p1 , . . . , pk ∈ X. One has: dim(TC,π(p1 ),...,π(pk ) ) ≤ 2k − 1 hence: dim(< Π, TC,π(p1 ),...,π(pk ) >) ≤ 4k − 2

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and TX1 ,τ (p1 ),...,τ (pk ) ⊆< Π, TC,π(p1 ),...,π(pk ) >, so that: dim(TX1 ,τ (P1 ),...,τ (Pk ) ) ≤ 4k − 2. Since δk (X) = 1 (see Proposition 5.1 ) the equality has to hold in the previous inequality, which implies TX1 ,τ (p1 ),...,τ (pk ) =< Π, TC,π(p1 ),...,π(pk ) >. Furthermore TX1 ,τ (P1 ),...,τ (Pk ) is tangent to X1 along the surface ZPi , i = 1, . . . , k. Call Si the pull–back to X of ZPi via τ . It follows that TX,p0 ,p1 ,...,pk is tangent to X along a surface which contains all the Si ’s. In other words Γp0 ,p1 ,...,pk ⊃ S1 , . . . , Sk . In particular the tangential k–contact locus of X is reducible. As in the proof of Proposition 5.2 we see that by moving p0 to some other point p0 ∈ X, then Γp0 ,p1 ,...,pk also contains the surfaces Si , i = 1, . . . , k. By equation (5), one has dim(< Γ >) ≤ 3k + 1. On the other hand, as in the proof of proposition 5.2, one has dim(< Γ >) ≥ k − 1 + dim(< Si >). Hence dim(< Si >) ≤ 2k + 2. Set L := LP0 =< TX,p0 , Π >, which is a linear space of dimension 2k +2. Again one sees that Hi :=< Si > ∩L is a hyperplane in < Si >, i = 1, . . . , k. Now move the point p0 to a new point p0 ∈ X, which gives us a new vertex Π := Πp0 and a new space L := Lp0 =< TX,P0 , Π >. However, since Γp0 ,p1 ,...,pk contains Si , i = 1, . . . , k, we have that also < Si > ∩L is a hyperplane in < Si >, i = 1, . . . , k. Now we claim that dim(< L, L >) < r. In fact, < TX,p0 , TX,p0 > contains TX1 ,τ (p0 ) and Π intersects this space along a line. Hence dim(< TX,p0 , TX,p0 , Π >)≤ 2k + 4. For analogous reasons, Π intersects < TX,p0 , TX,p0 > at least along a line. Hence dim(< L, L >) ≤ 4k + 1 < r. Since < L, L > is a proper subspace of Pr , the general surface Si cannot be contained in it. Hence the hyperplane Hi of < Si > is ﬁxed, as p0 moves. Then the space < H1 , . . . , Hk > does not depend on p0 , i.e. it is contained in Lp for all points p ∈ X. Now one concludes exactly as in the proof of Proposition 5.2. Let us prove now that (ii) implies (i). Suppose X lies in a cone with vertex Π of dimension 2k over a curve C. By Proposition 1.30, X is minimally k–defective, hence Xk−1 is a threefold. Since the general tangent space to X meets Π along a line, in the (k − 1)–tangential Π projects to a line L, and Xk−1 sits in the cone

with vertex L over Ck−1 . With similar arguments, using Proposition 1.30, one proves the following result: Proposition 5.4. Let X ⊂ Pr be an irreducible, non–degenerate projective threefold. Then the following are equivalent: (i) the general (k − 1)–tangential projection Xk−1 of X is a cone over a curve; (ii) X sits in a cone of dimension k + 2 over a k–defective surface with reducible general tangential k–contact locus, hence X sits in a cone of dimension 2k+1, and not smaller, over a curve. The previous results ﬁnish the classiﬁcation in this case.

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Remark 5.5. (1) The minimally k–defective threefolds occurring in Propositions 5.2, 5.3 and 5.4 can be h–defective for some h > k. To be more precise, consider a threefold as in Proposition 5.2 and let h > k. We know that X ⊂ Pr lies on a cone with vertex a subspace Π of dimension k − 1 over a surface S. By applying Proposition 1.29, we see that s(h) (X) = s(h) (Y ) + k. For example, if S is not h–defective, then s(h) (X) = min{r, 3h + k + 2}, so that X is h–defective if and only if 3h + k + 2 < r. If S is defective, X is defective even for higher values of h. We leave the details to the reader. Similarly, if X is as in Proposition 5.3 [resp. Proposition 5.4] a similar argument shows that, for any h > k, one has s(h) (X) = min{r, 2h + 2k + 2} [resp. s(h) (X) = min{r, 2h + 2k + 1}], so that X is h–defective if and only if 2h + 2k + 2 < r [resp. 2h + 2k + 1 < r]. (2) A threefold as in Proposition 5.2 can be smooth. This can be proved with a construction analogous to the one proposed in Remark 4.6, part (2). Similarly one can prove the existence of smooth threefolds as in Proposition 5.3. Take indeed a smooth scroll surface Y over a curve C spanning a P2k . Embed C in a Pr−2k−1 . Then join every point of C with the corresponding line of Y . The resulting scroll threefold X is smooth. An analogous construction works for producing smooth threefolds as in Proposition 5.4.

6. The irreducible curvilinear case Let X ⊂ Pr be an irreducible, non–degenerate, projective, minimally k–defective threefold with k ≥ 2. In this section we examine the case γk (X) = 1 and the general element of the family G is irreducible. By Proposition 3.3 we know that the general tangential projection Xk−1 is a 1–defective threefold whose general contact locus is an irreducible curve. From Remark 1.38 it follows that Xk−1 is as in cases (4) or (5) of Theorem 1.37. In any event, we have that Xk−1 sits in Ps , s = 7, 8, 9, hence: r ∈ {4k + 3, 4k + 4, 4k + 5}. Proposition 6.1. In our setting, the dimension k of a general k–contact locus is 1. Proof. Since γk = 1 and r ≥ 4k + 3, we know by Lemma 1.24 and Proposition 1.25 (iii) that a general k–tangential projection Xk of X is a surface. If k = 2, then any hyperplane tangent to Xk is indeed tangent along a curve (i.e. Xk is developable). But then, it is well known that also the general tangent plane to Xk has contact locus of dimension 1. This means that γk > 1, a contradiction.

Now we can start our classiﬁcation. The ﬁrst step is to show that X is as described in Proposition 1.39.

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Proposition 6.2. Let X ⊂ Pr be an irreducible, non–degenerate, projective, minimally k–defective threefold with k ≥ 1. Assume γk = 1 and the general element Γ of the family G irreducible. Then Γ is a rational normal curve in P2k . Hence X is rationally connected and therefore regular, i.e. any desingularization X of X has h1 (X , OX ) = 0. Proof. The assertion holds for k = 1 (see Remark 1.38). So we may assume k > 1 and proceed by induction on k. Let p ∈ X be a general point, let τ := τ1 : X → X1 be the tangential projection from TX,p and let Γ1 be the general tangential (k − 1)–contact locus of X1 . By induction, Γ1 is a rational normal curve in P2k−2 and Γ maps onto Γ1 via τ (see the proof of Proposition 3.3). One has dim(< Γ >) = 2k. Indeed by equation (5) of Theorem 1.18 and by the previous Proposition, we have dim < Γ >≤ 2k + 1, moreover the center of the projection τ , that is the tangent space TX,p , meets < Γ > at least in the tangent line to Γ at p and the image of Γ via τ spans a P2k−2 . It follows also that, in particular, < Γ > meets TX,p exactly along the aforementioned tangent line, so that τ|Γ is a general tangential projection of Γ. Now we claim that: (i) Γ is not tangentially degenerate; (ii) if x ∈ Γ is a general point, the projection Γ of Γ from x is also not tangentially degenerate. The assertions in the statement about Γ follow from (i) and (ii). Indeed, by (ii) and Lemma 1.35, τ|Γ is birational to its image Γ1 , hence Γ is rational. Furthermore, since deg(Γ1 ) = 2k − 2 by induction, < Γ > meets TX,p exactly along TC,p , as we saw, and TC,p ∩ C = {p} by (i), we have deg(Γ) = 2k. To prove (i) and (ii), we notice that G is a family of curves of dimension 2(k+1). This is an easy count of parameters which follows from the fact that there is a unique curve of G containing k + 1 general points of X. Let p1 , . . . , pk be general points of X and let G be the 2–dimensional family of curves of G passing through p1 , . . . , pk . We claim that: (iii) the tangent lines to the curves of G at Pi ﬁll up the whole tangent space TX,pi , for all i = 1, . . . , k. Indeed, this is true for k = 1 (see Remark 1.38). Then proceed by induction. The curves in G are mapped via τ1 = τX,p1 to the contact curves on X1 through τ1 (pi ), i = 2, . . . , k. Since the diﬀerential of τ1 is an isomorphism at p2 , . . . , pk , which are general points, by induction the tangent lines to the curves of G at pi ﬁll up the tangent space TX,pi , for all i = 2, . . . , k. Arguing by monodromy (see §1.13 and the proof of Proposition 3.1) the same is true for i = 1. Now we claim that (iii) implies (i). Indeed, if Γ were tangentially degenerate, then, by (iii), X itself would be 2-tangentially degenerate. By Proposition 1.34,

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X, which does not lie in P4 , would be a scroll on a curve. Then also Xk−1 would be a scroll, which is not the case (see Remark 1.38). Furthermore (iii) also implies (ii). In fact if we apply the same argument as above to the projection X of X from a general point x ∈ X, we conclude that X would be a scroll. Let Π be a general plane of X . Then either Π pulls back to a plane on X or it pulls back to a quadric Q containing x. In the former case X would be a scroll, which, as we saw, is impossible. In the latter case X is swept out by a family Q of quadrics such that through two general points of X there is a quadric in Q containing them. In this case X would be 1-defective (see Proposition 1.39), which is against the minimality assumption.

To go on with the classiﬁcation, we need the following: Example 6.3. Let Y ⊂ Pk+2 be a threefold of minimal degree k and let X ⊂ P4k+5 be its double embedding. In [6], Example 4.5, it is proved that X is (k + 1)– defective and s(k+1) (X) ≤ 4k + 4. One can prove that X is actually minimally k–defective, that s(k) (X) = 4k + 2 and s(k+1) (X) = 4k + 4, so that δk (X) = 1, nk = 1. Actually we will see that γk (X) = 1 and the general tangential k–contact locus is an irreducible curve. First we observe that X is not h–weakly defective, hence not h–defective, for any h ≤ k − 1. Indeed, if p0 , . . . , ph ∈ X are general points, there are hyperplane sections of X tangent at p0 , . . . , ph and having isolated singularities at p0 , . . . , ph . Take, for instance the hyperplane sections corresponding to sections of Y with a general quadric cone with vertex along the span of the points corresponding to p0 , . . . , ph . By the way, the same argument applied for h = k, shows that there are hyperplane sections of X tangent at p0 , . . . , pk and having an irreducible curve of singular points containing p0 , . . . , pk , i.e. the rational normal curve of degree 2k which is the image of the intersection of Y with the span of the points corresponding to p0 , . . . , pk . Now suppose X is not k–defective. Then, if p0 , . . . , pk ∈ X are general points, we would have dim(TX,p0 ,...,pk ) ) = 4k + 3. However, if p ∈ X is another general point, one has dim(TX,p0 ,...,pk ,p ) ≤ 4k + 4. This would imply that the general tangent space TX,p to X intersects TX,p0 ,...,pk in dimension at least 2. But this would force X to span a P4k+4 (see Proposition 1.28), a contradiction. Hence X is minimally k–defective and therefore s(k) (X) = 4k + 2, δk (X) = 1, nk = 1 (see Proposition 1.25). Then the same argument implies that s(k+1) (X) ≥ 4k + 4 and therefore s(k+1) (X) = 4k + 4. Of course also a projection of X to P4k+3+i , i = 0, 1, is minimally k–defective. To apply an inductive argument we will need the following: Proposition 6.4. Let X ⊂ Pr be an irreducible, non–degenerate, projective, regular threefold. Let k ≥ 2 and assume that a general tangential projection X1 of X is the 2-uple embedding of a minimal threefold in Pk+1 . Then X is the 2-uple embedding of a minimal threefold in Pk+2 .

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Proof. Let τ := τ1 : X → X1 be the general tangential projection from TX,p . Let Z [resp. Z1 ] be a desingularization of X [resp. X1 ] and let H [resp. H1 ] be the pull–back on Z [resp. Z1 ] of the hyperplane linear system of X [resp. X1 ]. The tangential projection τ induces a rational map φ : Z → Z1 . Notice that: φ∗ (H1 ) = H(−2p).

(11)

, so r = 4k + 5. Furthermore X1 is linearly By the hypothesis X1 spans a P normal, namely H1 is a complete linear system. By (11) also H is complete, i.e. X is also linearly normal. By assumption, H1 is the double of a linear system L1 and the associated map φL1 : Z1 → Y1 ⊂ Pk+1 is such that Y1 is a threefold of minimal degree. The linear system L1 pulls back via φ to a linear system Lp on Z. Of course φLp = φL1 ◦ φ : Z → Y1 ⊂ Pk+1 and therefore dim(Lp ) = k + 1. The system Lp depends on p, but, since X is regular, as p moves on X all the systems Lp ’s are subsystems of the same complete linear system L on Z. We notice that: 4k+1

2Lp = φ∗ (H1 ) = H(−2p).

(12)

Hence all divisors in Lp contain p and therefore we have: Lp ⊆ L(−p).

(13)

As an immediate consequence of (12) and (13) we have the linear equivalence relation 2L ≡ H. Since H is complete, we have 2L ⊆ H. Moreover, by (13), one has m := dim(L) ≥ k + 2. We call Y ⊂ Pm the image of Z via the map φL associated to L. If m > k + 2, then dim(H) ≥ dim(2L) > 4k + 5 (see Theorem 1.4), a contradiction. Similarly we ﬁnd a contradiction if dim(H) > dim(2L). Hence H = 2L and m = k + 2. Moreover for p general, Lp = L(−p). It follows that X is the 2-uple embedding of Y and Y ⊂ Pk+2 projects from a general point on it to a minimal

threefold of Pk+1 . Hence Y is also of minimal degree. Remark 6.5. In the previous setting, assume that a general tangential projection X1 of X is a projection of the 2-uple embedding Y of a threefold Y of minimal degree in Pk+1 , from a space not intersecting Y . Then X is the projection of the 2-uple embedding of a threefold of minimal degree in Pk+2 , from a space not intersecting it. Indeed, by replacing, if necessary, X1 by its linearly normal embedding, the previous argument shows that the linearly normal embedding of X is the 2–uple embedding of a minimal threefold in Pk+2 . A completely similar argument holds when X1 is a projection of the 2-uple embedding of a minimal threefold from a point on it. We skip the details. Proposition 6.6. Let X ⊂ Pr be an irreducible, non–degenerate, projective, regular threefold. Let k ≥ 2 and assume that a general tangential projection X1 of

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X is the projection of the 2-uple embedding Y of a threefold Y of minimal degree in Pk+1 from a space which meets Y at a point. Then X is the projection of the 2–uple embedding of a threefold of minimal degree in Pk+2 from a space which meets it at a point. The case of projections from more than one point on X is contained in the analysis below. Again, in order to apply an inductive argument, we will need the following: Proposition 6.7. Let X ⊂ Pr be an irreducible, non–degenerate, projective, regular threefold. Let k ≥ 2 and assume that a general tangential projection X1 of X is linearly normal and contained in the Segre embedding of Pk × Pk . Suppose that each of the two projections of X1 to Pk spans Pk . Then X is linearly normal and contained in the Segre embedding of Pk+1 × Pk+1 . Moreover each of the two projections of X spans Pk+1 . Proof. Let τ := τ1 : X → X1 be the general tangential projection from TX,P . Let Z [resp. Z1 ] be a desingularization of X [resp. X1 ] and let H [resp. H1 ] be the pull–back on Z [resp. Z1 ] of the hyperplane linear system of X [resp. X1 ]. The tangential projection τ induces a rational map φ : Z → Z1 . Equation (11) still holds. Then, as in the proof of Proposition 6.4, one sees that X is linearly normal, i.e. H is complete. We have two linear systems A1 , B1 on Z1 which come by pulling back the linear system |OPk (1)| via the projections of Pk × Pk to the two factors. One has H1 = A1 + B1 (see §1.8). The linear systems A1 , B1 pull–back via φ to linear systems Ap , Bp , on Z. By the regularity assumption on X, as p varies Ap , Bp vary inside two complete linear systems A, B on Z. From the relation: Ap + Bp = φ∗ (H1 ) = H(−2p)

(14)

we see that every divisor in Ap + Bp vanishes doubly at p, and therefore either: Ap ⊆ A(−p), Bp ⊆ B(−p)

(15)

Bp ⊆ B(−2p).

(16)

or, say:

In any case we have the equivalence relation A + B ≡ H. Since H is complete, then A + B ⊆ H.

(17)

Ap + Bp ⊆ A(−p) + B(−p) ⊆ H(−2p).

(18)

In case (15) holds, we have:

Then, by (14), equality has to hold everywhere in (18) and therefore in (15) and (17). This yields dim(A) = dim(B) = k + 1, proving the assertion.

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If (16) holds, one has: Ap + Bp ⊆ A + B(−2p) ⊆ H(−2p).

(19)

Again, by (14), equality has to hold everywhere in (19). From (17) we also have A(−p) + B(−p) ⊆ H(−2p) which is clearly incompatible with H(−2p) = A + B(−2p).

We can now get the following partial classiﬁcation result: Theorem 6.8. Let X ⊂ Pr be an irreducible, non–degenerate, projective, minimally k–defective threefold with k ≥ 2. Assume γk = 1 and irreducible general tangential k–contact locus. Then one of the following holds: (1) r = 4k + 5 and X is the 2-uple embedding of a threefold of minimal degree in Pk+2 ; (2) r = 4k + 4 and X is the projection of the 2-uple embedding of a threefold of minimal degree in Pk+2 from a point p ∈ P4k+5 ; (3) r = 4k + 3 and either X is the projection of the 2-uple embedding of a threefold of minimal degree in Pk+2 from a line ⊂ P4k+5 , or X is linearly normal, it is contained in the intersection of a space of dimension 4k + 3 2 with the Segre embedding of Pk+1 × Pk+1 in Pk +4k+3 and each of the two projections of X on the two factors of Pk+1 × Pk+1 spans Pk+1 . In this former case each of the linear systems Ai , i = 1, 2, on X corresponding to the two projections φi : X → Pk+1 , is base point free and the general surface Ai ∈ Ai is irreducible. Proof. We know that the possibilities for r are the ones listed in statement. Assume that r = 4k+5. Then, by Theorem 1.37, Xk−1 is the 2-uple embedding of P3 . The conclusion follows by induction from Proposition 6.4. If r = 4k + 4, by Theorem 1.37, Xk−1 is a projection of the 2-uple embedding of P3 . The conclusion follows by induction from Remark 6.5 and Proposition 6.6. If r = 4k+3, then by Proposition 3.3, by Theorem 1.37 and Remark 1.38, either Xk−1 is some projection of the 2-uple embedding of P3 or Xk−1 is a hyperplane section of P2 × P2 . In the former case the conclusion follows again by induction from Remark 6.5 and Proposition 6.6. In the latter case, Xk−1 is linearly normal and the two projections of Xk−1 to P2 are both surjective. The conclusion follows by induction from Proposition 6.7.

The last part of case (3) of Theorem 6.8 deserves more attention. The analysis is based on the following proposition: Proposition 6.9. Let Z be an irreducible, smooth, regular threefold and let Ai , i = 1, 2, be two distinct base point free linear systems of dimension k + 1 ≥ 3 on Z such that a general surface Ai ∈ Ai is smooth and irreducible and the minimal

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sum A1 + A2 is birational on Z. Then the minimal sum A1 + A2 has dimension at least 4k + 3. Furthermore if the minimum is attained then a general curve C in the class A1 ·A2 is smooth, irreducible and rational and the linear system Ai|C is the complete gkk on C. Moreover, either: (a) Ai , i = 1, 2, is complete, or (b) A1 ≡ A2 , and dim(|Ai |) = k + 2, i = 1, 2, or (c) A1 is complete, whereas dim(|A2 |) = k +2 and there is an eﬀective, non–zero divisor E on X, such that h0 (Z, OZ (E)) = 1, A2 − A1 ≡ E and OA2 (E) = OA2 . If, in addition, the linear systems Ai , i = 1, 2, are very big, then also the general curves Ci in the classes A2i are smooth, irreducible and rational. Moreover: k−1 (a’) in case (a) above the linear system Ai|Ci is the complete gk−1 on Ci , i = 1, 2;

(b’) in case (b) above the linear system Ai|Ci is the complete gkk on Ci , i = 1, 2; k−1 (c’) in case (c) above the linear system A1|C1 is the complete gk−1 on C1 whereas k the linear system A2|C2 is the complete gk on C2 .

Proof. The curve C is smooth by Bertini’s theorem. We claim that C is also irreducible. Otherwise, by Bertini’s theorem, the linear system A1|A2 would be composite with a pencil P and, if P ∈ P is a general curve, we would have P · A1 = 0. Similarly we would have P · A2 = 0 and therefore each curve P would be contracted by the map φA1 +A2 , against the hypothesis that A1 + A2 is birational. Consider the linear systems Ai|C , i = 1, 2. Since A1 = A2 , one has dim(Ai|Aj ) = k + 1, i, j = 1, 2, i = j

(20)

dim(Ai|C ) = k, i = 1, 2.

(21)

and therefore By Lemma 1.9 we have dim(A1|C + A2|C ) ≥ 2k. Let dim(A1|A1 + A2|A1 ) = s. By looking at the exact sequence 0 → H 0 (Z, OZ (A2 )) → H 0 (Z, OZ (A1 + A2 )) → H 0 (A1 , OA1 (A1 + A2 )) we see that the image of the rightmost map contains the vector space corresponding to A1|A1 + A2|A1 . This implies that: dim(A1 + A2 ) ≥ s + 1 + dim(A2 ) = s + k + 2. Similarly, by looking at the sequence 0 → H 0 (A1 , OA1 (A1 )) → H 0 (A1 , OA1 (A1 + A2 )) → H 0 (C, OC (A1 + A2 ))

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one obtains s + 1 ≥ dim(A1|C + A2|C ) + dim(A1|A1 ) + 2 ≥ 3k + 2 whence dim(A1 + A2 ) ≥ 4k + 3 follows. If the equality holds, then Lemma 1.9 implies that C is rational. Since, by the hypotheses and by Lemma 1.12, the linear series A1|C + A2|C is birational, then Lemma 1.9 again implies that A1|C = A2|C and that this is a complete linear series. Since C is rational, by (21) we have Ai · C = k. Since A1|C = A2|C are complete, it follows that the systems A1|A2 and A2|A1 are also complete. If h0 (Z, OZ (A1 − A2 )) = h0 (Z, OZ (A2 − A1 )) = 0, then, by looking at the exact sequence: 0 → H 0 (Z, OZ (Ai − Aj )) → H 0 (Z, OZ (Ai )) → H 0 (Aj , OAj (Ai )), i, j = 1, 2, i = j (22) we see we are in case (a). Suppose we are not in case (a) and assume that h0 (Z, OZ (A2 − A1 )) > 0, so that there is an eﬀective divisor E which is linearly equivalent to A2 −A1 . Suppose E is zero. Then A1 ≡ A2 . Moreover since A2|A1 is complete, from the sequence (22) with i = 2, j = 1, we see that h0 (Z, OZ (A2 )) = dim(A2|A1 ) + 2 = k + 3, namely we are in case (b). If E is non zero, then h0 (Z, OZ (A1 − A2 )) = 0. Since A1|A2 is complete, the completeness of A1 follows from the exact sequence (22) with i = 1, j = 2. Moreover E · C = 0. Since the linear system |C| on A2 is base point free, with positive self–intersection, we see that h0 (A2 , OA2 (E)) = 1. From the exact sequence: 0 → H 0 (Z, OZ (−A1 )) → H 0 (Z, OZ (E)) → H 0 (A2 , OA2 (E)) we deduce that h0 (Z, OZ (E)) = 1. Now look at the sequence (22) with i = 2, j = 1. Since, as we saw, h0 (A1 , OA1 (A2 )) = dim(A2|A1 ) + 1 = k + 2, we have h0 (Z, OZ (A2 )) = k + 3, i.e. dim(|A2 |) = k + 2. Now, inside |A2 | we have the two linear systems E + A1 and A2 . They intersect along a k–dimensional linear system. Hence every section of H 0 (Z, OZ (A2 )) determining a divisor of A2 is of the form f e + g, with f ∈ H 0 (Z, OZ (A1 )) variable, e ∈ H 0 (Z, OZ (E)) and g ∈ H 0 (Z, OZ (A2 )) ﬁxed deﬁning a divisor BA2 not containing E. Hence every solution of the system e = g = 0, i.e. any point in E ∩ B, should be a base point of A2 . Since A2 is base point free, we have that E ∩ B = ∅. Thus we are in case (c). Suppose now Ai is very big. Arguing as at the beginning of the proof, we see that the general curve Ci in the class A2i is smooth and irreducible. By what we proved already, we have the exact sequence 0 → OA1 (−A1 ) → OA1 (A2 − A1 ) → OC (A2 − A1 ) OC → 0 from which we deduce h0 (A1 , OA1 (A2 − A1 )) = 1, h1 (A1 , OA1 (A2 − A1 )) = 0. Indeed, by Lemma 1.12, OA1 (A1 ) is big and nef and therefore Kawamata-Viehweg vanishing theorem says that hi (A1 , OA1 (−A1 )) = 0, 0 ≤ i ≤ 2.

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Now look at the sequence 0 → OA1 (A2 − A1 ) → OA1 (C) → OC1 (C) → 0. Since A2|A1 ⊂ |OA1 (C)|, by (20) we have h0 (A1 , OA1 (C)) ≥ k + 2. It follows that h0 (C1 , OC1 (C)) ≥ k + 1. Since deg(OC1 (C)) = k, we deduce that C1 is rational. Suppose we are in case (a). Since Z is regular and A1 is complete, we have: h0 (A1 , OA1 (C1 )) = h0 (A1 , OA1 (A1 )) = dim(A1 ) = k + 1. Finally look at the sequence 0 → OA1 → OA1 (C1 ) → OC1 (C1 ) → 0. Clearly A1 is a rational surface, so we have h1 (A1 , OA1 ) = 0. Thus one has h (C1 , OC1 (C1 )) = k. This proves the assertion for A1|C1 . The same for C2 . Thus we are in case (a’). The analysis in case (b) and (c) is the same, leading to (b’) and (c’), respectively.

0

Corollary 6.10. Let X be an irreducible, non–degenerate threefold in the Segre embedding of Pk+1 × Pk+1 , k ≥ 2, which does not lie in the 2–uple embedding of Pk+1 . Assume that each of the two projections of X to Pk+1 spans Pk+1 . Then X spans a space of dimension at least 4k + 3. Furthermore, if X spans a P4k+3 , then given k + 1 general points of X, there is a rational normal curve C of degree 2k on X containing the given points, and X is minimally k–defective and s(k) (X) = 4k + 2, δk (X) = 1, nk (X) = 2. Finally, if the two projections of X to Pk+1 are generically ﬁnite to their images, then either: (a) they both map birationally X to rational normal scrolls, and then the degree of X is 8k − 2, or (b) they both map birationally X to projections of rational normal scrolls in Pk+2 from a point, and then the degree of X is 8k, or (c) one of them maps birationally X to a rational normal scroll and the other maps birationally X to the projection of a rational normal scroll in Pk+2 from a point, and then the degree of X is 8k − 1. Proof. The ﬁrst assertion follows from Proposition 6.9 applied to a desingularization Z of X. Consider the two linear systems Ai , i = 1, 2, on Z corresponding to the two projections X → Pk+1 . Notice that the general surface of Ai , i = 1, 2, is irreducible (see Theorem 6.8) and smooth by Bertini’s theorem, since Ai is base point free. Suppose X spans a P4k+3 . Given k + 1 general points p0 , . . . , pk on Z, let Ai ∈ Ai be the surface containing p0 , . . . , pk . Then, according to Proposition 6.9, the image in X of the curve C = A1 · A2 is a rational normal curve of degree 2k. The k–defectivity of X follows from Proposition 1.39.

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Let us prove that X is minimally k–defective. We ﬁrst claim that X is not 1–defective. Assume, by contradiction, that X is 1-defective. Then, by Theorem 1.37, the only possibilities are that X is either a cone, or X sits in a 4–dimensional cone over a curve. Notice that the Segre embedding of Pk+1 × Pk+1 is swept out by two (k + 1)–dimensional families of Pk+1 ’s and that each line on the Segre embedding of Pk+1 × Pk+1 is contained in a Pk+1 of either one of these two families. As a consequence, each irreducible cone contained in the Segre embedding of Pk+1 × Pk+1 is contained in a Pk+1 . This implies that X cannot be a cone, since it spans a P4k+3 . Suppose X sits in a 4–dimensional cone over a curve. Let p ∈ X be a general point and consider the general tangential projection τ of Pk+1 × Pk+1 from p. As we saw in Example 1.26, Pk+1 × Pk+1 projects onto Pk × Pk . Let X be the image of X via τ . Notice that X is a projection of the image X1 of the tangential projection of X from TX,p . Thus X is a cone over a surface. By the above argument, X would span at most a Pk . Thus 4k + 3 = dim(< X >) = dim(< X >) + 2k + 3 ≤ 3k + 3, a contradiction. Let us consider again a general point p ∈ X and the general projection τp from TX,p . Since X is not 1–defective, its image is a threefold spanning a P4k−1 . Consider again a desingularization Z of X. We abuse notation and denote by p the point of Z corresponding to p ∈ X. Let H be the linear system on Z corresponding to the hyperplane section system on X. We have A1 (−p) + A2 (−p) ⊆ H(−2p). On the other hand, since dim(Ai (−p)) = k, i = 1, 2, by Proposition 6.9, we have 4k − 1 = dim(H(−2p)) ≥ dim(A1 (−p) + A2 (−p)) ≥ 4k − 1. This proves that A1 (−p) + A2 (−p) = H(−2p). On the other hand φH(−2p) just maps Z to X1 . By what we saw in §1.8, we have that X1 is a threefold which sits in Pk × Pk , spans a P4k−1 and each of the two projections of X1 to Pk spans Pk . Thus, by arguing as above, we see that X1 is not 1–defective. By iterating this argument, one proves that Xi is not 1–defective for any i = 1, . . . , k − 2, hence that X is minimally k–defective. Now, notice that the above argument proves that Xk−1 is the hyperplane section of the Segre embedding of P2 × P2 , hence s(1) (Xk−1 ) = 6, δ1 (Xk−1 ) = 1, n1 (Xk−1 ) = 2. Thus s(k) (X) = 4k + 2, δk (X) = 1, nk (X) = 2. Finally, the cases (a), (b) and (c) in the statement, correspond to the homologous case in Proposition 6.9.

The following proposition takes care of the cases left out in the statement of Corollary 6.10. Proposition 6.11. Let X be an irreducible, non–degenerate threefold in the Segre embedding of Pk+1 × Pk+1 , k ≥ 2, which lies in the 2-uple embedding of Pk+1 . Suppose the two projections of X to Pk+1 span Pk+1 . Then these projections coincide and are immersions of X in Pk+1 . Furthermore the dimension s of the linear span of X satisﬁes s > 4k + 3 unless: (a) s = 4k + 1 and the image of X in Pk+1 is a threefold Y of minimal degree; (b) s = 4k + 2 and the image of X in Pk+1 is a threefold Y of degree k with curve sections of arithmetic genus 1;

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(c) s = 4k + 3 and the image of X in Pk+1 is: (c1 ) either a threefold Y of degree k + 1 with curve sections of arithmetic genus 2 or (c2 ) a threefold Y of degree k which is the projection in Pk+1 of a threefold of minimal degree in Pk+2 . In the latter case Y is described in Lemma 1.6. The threefolds in case (a) are (k − 1)–defective, whereas the threefolds in cases (b), (c) are minimally k–defective. Proof. In the present situation it is clear that the two projections of X to Pk+1 coincide and are immersions φ : X → Y ⊂ Pk+1 of X in Pk+1 . Let d be the degree of Y . Set d = k − 1 + ι. Now we make free and iterate use of Theorem 1.4. So we have s ≥ 4k+1+ι. If ι = 0 we are in case (a). If ι = 1 and s = 4k+2 we are in case (b). If ι = 1 and s = 4k + 3, the curve sections of Y cannot have arithmetic genus 1, otherwise s = 4k + 2. Then the curve sections of Y have arithmetic genus 0, and we are in case (c2 ), as described in Lemma 1.6. Finally if ι = 2 and s = 4k + 3 we are in case (c1 ). In all other cases s ≥ 4k + 4. The defectivity of the varieties in the list follows by Example 6.3, (6), Example 4.3 (3), (4), (6).

Example 6.12. (1) It is clear that there are examples of smooth threefolds as in cases (1) and (2) of Theorem 6.8. It is also not diﬃcult to ﬁnd examples of smooth threefolds as in case (3) of the same theorem, and speciﬁcally as in cases (a), (b), (c) of Corollary 6.10. For instance, take a threefold Y of minimal degree in Pk+2 and call H its hyperplane linear system. Put Ai = H(−pi ), i = 1, 2, with p1 , p2 ∈ Pk+2 distinct points. Then A1 + A2 embeds Y in the product Pk+1 × Pk+1 and also in P4k+3 , since it corresponds to a projection of the 2–uple embedding of Y from the line =< p1 , p2 >. Notice that these projections have degree d with d = 8k, 8k−1, 8k− 2, according to the fact that does not intersect Y , meets Y at one point, or it is a secant of Y . (2) There are examples of minimally k-defective threefolds, contained in the intersection of the Segre embedding of Pk+1 × Pk+1 with a P4k+3 , for which one the linear systems Ai of Proposition 6.9 is not very big. This is the case of the threefold X obtained by embedding P1 × P2 in P19 with the complete system A of divisors of type (1, 3). Terracini pointed out in [31] that X is 4–defective. Terracini’s paper was reconsidered, from a modern point of view, by Dionisi and Fontanari ([12]), who also proved that the X is not 3–defective. Let us show how this threefold ﬁts in our classiﬁcation. The linear system A can be decomposed as the sum A1 + A2 with A1 of type (1, 1), A2 of type (0, 2). Both A1 , A2 send P1 × P2 to P5 , so X also sits in the product P5 × P5 , and the projections to the two factors are both non–degenerate. Therefore X ﬁts in the last case (15) of our classiﬁcation. According to Proposition

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6.9, a general curve in the class A1 · A2 is rational: indeed it is embedded as a rational normal quartic in the natural Segre embedding of P1 × P2 . Observe, however, that A2 is not big: the second projection of P5 × P5 sends X to a surface. So we cannot apply to X the last part of Proposition 6.9 as well as the last part of Corollary 6.10. In particular the degree of X is 27 < 30 = 8k − 2 and X is not the projection of the 2–uple embedding of a quartic threefold in P6 from a line. Other similar examples of this sort can be considered and are related to the so–called phenomenon of Grassmann defectivity (see [12]). We hope to come back on this subject in a future paper.

References ˚dlansdvik B., Joins and Higher secant varieties, Math. Scand. 61 (1987), [1] A 213–222. [2] Arbarello E., Cornalba M., Griﬃths P., Harris J. Geometry of Algebraic Curves, Grundlehren der Math. 267 Springer Verlag (1984). [3] Alzati A., Russo F., On the k–normality of projected algebraic varieties, Bull. Braz. Math. Soc. 33 (2002), 1–22. [4] Catalano–Johnson M., When do k general double points impose independent conditions on degree d plane curves?, Curves Seminar of Queen’s Vol. X, Queen’s Papers Pure Appl. Math. (1995), 166–181. [5] Catalisano M. V., Geramita A. V., Gimigliano A., Ranks of tensors, secant varieties of Segre varieties and fat points, Linear Algebra Appl. 355 (2002), 263–285. [6] Chiantini L., Ciliberto C., Weakly defective varieties, Trans. Amer. Math. Soc. 354 (2002), 151–178. [7] Chiantini L., Ciliberto C., Threefolds with degenerate secant variety: on a theorem of G. Scorza, Dekker Lect. Notes Pure Appl. Math. 217 (2001), 111– 124. [8] Ciliberto C., Hilbert functions of ﬁnite sets of points and the genus of a curve in a projective space, Springer Lecture Notes in Math. 1266 (1987), 24–73. [9] Ciliberto C., Geometric aspects of polynomial interpolation in more variables and of Waring’s problem, European Congress of Mathematics (Barcelona, 2000), Progr. Math. 201 (2001), 289–316. [10] Dale M., Terracini’s lemma and the secant variety of a curve, Proc. London Math. Soc. 49 (3) (1984), 329–339. [11] Dale M., On the secant variety of an algebraic surface, Univ. Bergen, Dept. of Math. preprint 33 (1984). [12] Dionisi C., Fontanari C. Grassmann defectivity ´ a la Terracini, Le Matematiche 56 (2001), 245–255.

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[13] Ein L., Varieties with small dual variety I, Inventiones Math. 86 (1989), 783– 800. [14] Eisenbud D., Harris J., Curves in projective spaces, Montreal Univ. Press (1982). [15] Eisenbud D., Harris J., On varieties of minimal degree, Algebraic Geometry (Bowdoin 1985), Proc. Symp. in Pure Math. 46 (1987), 3–13. [16] Fujita T., Projective threefolds wityh small secant varieties, Scientiﬁc Papers of the College of General Education, Univ. of Tokyo 32 (1982), 33–46. [17] Fujita T., Roberts J., Varieties with small secant varieties: the extremal case, Amer. J. of Math. 103 (1981), 953–976. [18] Garcia L. D., Stillman M., Sturmfels B., Algebraic geometry of bayesian networks, preprint math.AG/0301255 (2003). [19] Griﬃths P., Harris J., Algebraic geometry and local diﬀerential geometry, Ann. Scient. Ec. Norm. Sup. 12 (1979), 335–432. [20] Hartshorne R., Curves with high self–intersection on algebraic surfaces, Publ. Math. I.H.E.S. 36 (1969). [21] Iarrobino A., Kanev V., Power sums, Gorenstein algebras, and determinantal loci, Springer Lecture Notes in Math. 1721 (1999). [22] Kaji H., On the tangentially degenerate curves, J. London Math. Soc. 33 (2) (1986), 430–440. [23] Murre J., Classiﬁcation of Fano threefolds according to Fano and Iskovskih, Springer Lecture Notes in Math. 947 (1982), 35–92. [24] Palatini F., Sulle superﬁcie algebriche i cui Sh (h + 1)–seganti non riempiono lo spazio ambiente, Atti Accad. Torino 41 (1906), 634–640. [25] Palatini F., Sulle variet` a algebriche per le quali sono di dimensione minore dell’ordinario, senza riempire lo spazio ambiente, una o alcune delle variet` a formate da spazi seganti, Atti Accad. Torino 44 (1909), 362–374. [26] Scorza G., Determinazione delle variet` a a tre dimensioni di Sr , r ≥ 7, i cui S3 tangenti si intersecano a due a due, Rend. Circ. Mat. Palermo 25 (1908), 193–204. [27] Scorza G., Un problema sui sistemi lineari di curve appartenenti a una superﬁcie algebrica, Rend. R. Ist. Lombardo 41 (2) (1908), 913–920. [28] Soul´e C., Secant varieties and successive minima, preprint math.AG/0301255 (2003). [29] Terracini A., Sulle Vk per cui la variet` a degli Sh , (h+1)-seganti ha dimensione minore dell’ordinario, Rend. Circ. Mat. Palermo 31 (1911), 392–396. [30] Terracini A., Su due problemi concernenti la determinazione di alcune classi di superﬁcie, considerate da G. Scorza e F. Palatini, Atti Soc. Natur. e Matem. Modena 6 (1921-22), 3–16. [31] Terracini A., Sulla rappresentazione di coppie di forme ternarie mediante somme di potenze, Ann. Mat. Pura e Applic. XXIV-III (1915), 91–100.

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Luca Chiantini and Ciro Ciliberto [32] Voisin C., On linear subspaces contained in the secant varieties of a projective curve, preprint math.AG/0110256 (2001). [33] Zak F. L., Tangents and secants of varieties, Transl. Math. Monog. 127 (1993).

Luca Chiantini Dipartimento di Matematica Universit` a di Siena Pian dei Mantellini, 44 53100 Siena, Italia Email: [email protected] Ciro Ciliberto Dipartimento di Matematica Universit` a di Roma Tor Vergata Via della Ricerca Scientiﬁca 00133 Roma, Italia Email: [email protected]

Matching conditions for degenerating plane curves and applications Ciro Ciliberto and Rick Miranda

Abstract. In previous papers the Authors used a degeneration technique in order to compute the dimension of linear systems of plane curves of given degree with general base points of assigned multiplicities. This technique involved a reduction to studying two related systems on two components of a reducible degeneration of the plane. The analysis failed in case either one of the systems had a multiple base component. In this article we partially recover from this by introducing a more reﬁned analysis of the situation. We then present various applications, including a description of the limit scheme of fat points coming together at a single point in various cases and an analysis of the limiting system of a linear system of curves when three, four, or ﬁve multiple base points approach a line. The speciﬁc interesting case of the linear system of curves of degree 158 with 10 general base points of multiplicity 50 is considered and the system is proved to be empty. 2000 Mathematics Subject Classiﬁcation: 14C20

1. Introduction The computation of the dimension of the space of plane curves of degree d with prescribed multiple base points has drawn great interest for some time. Several conjectures, notably the Segre Conjecture [17] and [5], the Nagata Conjecture [16], and the Harbourne-Hirschowitz Conjecture [13], [9], all deal with aspects of this problem. A number of methods have been developed, each having some success. For an arbitrary number of points, there are several results dealing with small multiplicities (at most M ): Hirschowitz (via the Horace Method) for M = 3 [12], Mignon (also via the Horace Method) for M = 4 [15], and lately Yang (using a variety of methods) for M = 7. The authors have previously developed a technique based on a degeneration of the plane [3],[4] which treated the case of uniform multiplicities m ≤ 20. Less is known about larger multiplicities, even in the case of a ﬁxed number n > 9 of points. Up to now there are only three approaches, each having some limited success. One is that of Evain [6], [7]; another is that of Harbourne-Ro´e [10];

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and a third is that of Buckley and Zompatori [2], which is based on our original method. This degeneration technique involved a reduction to studying two related systems on two components of a reducible degeneration of the plane. In order to prove that the original linear system in question had the expected dimension, it proved to be suﬃcient for the linear systems on the components to have the expected dimension. This required that these systems had reduced members in general. The analysis failed in case either one of the systems had a multiple base component. In this article we attempt to recover from this by introducing a more reﬁned analysis of the situation. We are able to apply this more reﬁned analysis to make several computations which were previously impossible, both with our prior methods, or with any other of the other methods; examples are presented in the ﬁnal section. The analysis permits an understanding of the limit scheme when fat points come together at a single point in various cases; this we present in Section 3. These ideas are not unrelated to those of Evain, see [6] and [7]. When fat points approach a line, the method also provides in certain situations an analysis of the limiting system of curves, and we develop this for three, four, and ﬁve fat points in Sections 5 and 6. These lemmas have been exploited in [18] to prove the Harbourne-Hirschowitz Conjecture for multiplicities at most 7. These two threads (fat points approaching a point and approaching a line) are illustrated in the ﬁnal section where we present examples utilizing both techniques. The authors would like to acknowledge the contributions of the referee, who made many excellent suggestions. We would also like to thank the organizers of the conference in Siena held in June 2004 where this work was presented. We would like to especially acknowledge the eﬀorts of Prof. Chiantini who provided a wonderful opportunity both for mathematics and for fellowship during that week.

2. Matching conditions Let p : X → Δ be a proper ﬂat family of surfaces with smooth total space X . For t = 0, we assume that the ﬁber Xt = p−1 (t) is smooth and connected. We assume that the central ﬁber X0 is reduced but reducible, consisting of a smooth surface V and a union of smooth surfaces W = ∪Wi meeting transversally along a (possibly reducible) curve R = ∪Ri , which has all smooth components Ri = V ∩ Wi . Fix a smooth irreducible curve E ⊂ V which meets R transversally in τ = (E · R)V points. Let C ⊂ X be an irreducible surface which is, via p, a proper ﬂat family of curves Ct ⊂ Xt . We further assume that σ = −(C · E) > 0. The central ﬁber C0 of the curve family C consists of a curve CV ⊂ V and CW ⊂ W . Note that (C · E) = (CV · E), and therefore the above assumption implies that E ⊂ CV ; hence since C0 is a divisor on X0 , the curve CW must pass through the

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points r1 , . . . , rτ where E meets R and W . It is our intention to understand better how the curve CW behaves at these τ points. Let π : X → X be the blowup of X along the curve E. This will create the exceptional divisor T ⊂ X which will be a ruled surface; E becomes a section of the ruling on T . For t = 0, Xt = Xt , but the central ﬁber X0 consists now of three parts, namely T and the proper transforms V and W of V and W respectively. Note that V ∼ = V , but W is isomorphic to the blowup of W at the τ points on R where E intersects R. We will denote by F1 , . . . , Fτ the exceptional curves on W , all of which are ﬁbers of the ruling of T . Denote by C the strict transform of C. There is an integer α such that C = π ∗ (C) − αT ; α is the multiplicity with which C contains E. The assumption that σ > 0 implies that E ⊂ C, and so α > 0. Note that by restricting to V , we see that CV must contain E as a component with multiplicity μ ≥ α; by restricting to W , we conclude that CW passes through the points r1 , . . . , rτ with multiplicity at least α. This condition, that CW must have multiplicity at these points (or in some cases as noted later an even more complicated singularity) will be referred to as a matching condition, justifying the title of the article. Proposition 2.1. With the above notation, (a) if τ + (E 2 )V ≥ 0, then α ≥ σ/τ . (b) if τ + (E 2 )V ≤ 0, then α ≥ −σ/(E 2 )V . Proof. We ﬁrst note that T |T = −E −

Fi ∼ −E − τ F

i

(where ∼ denotes numerical equivalence) and that π ∗ (C)|T = π ∗ (C)|π∗ (E) = π ∗ (C|E ) ∼ −σF where F is the class of a ﬁber of T . Hence C |T ∼ −σF − α(−E − τ F ) = αE + (ατ − σ)F.

(2.2)

Also note that by the triple point formula [14], (E 2 )T = −τ − (E 2 )V . Note that C |T is eﬀective. To prove (a), we then have that (E 2 )T ≤ 0; in this case if ατ − σ < 0, we could not have C |T being eﬀective, and so ατ − σ ≥ 0 as claimed. To prove (b), we then have (E 2 )T ≥ 0; then E is nef on T , and so (E·C |T )T ≥ 0. But

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(E · C |T )T = α(E 2 )T + ατ − σ = = α(−τ − (E 2 )V ) + ατ − σ = −σ − α(E 2 )V which, being nonnegative, gives the result.

Corollary 2.3. With the above notation, suppose that E is a (−1)-curve on V , i.e., E is smooth, rational, and E 2 = −1. Then α ≥ σ/τ . Proof. In this case τ + (E 2 )V = τ − 1 ≥ 0 so case (a) of the Proposition applies.

When E is a (−1)-curve on V , the relation (CV ·E) = (C ·E) = −σ implies that the linear system |CV | on V in which the restricted curve must lie must contain E as a base curve with multiplicity σ: |CV | = σE + |M |, where M = CV − σE is the residual system. The above corollary is most commonly used in the situation when the (−1)curve E meets R in one point (so that τ = 1). We conclude in this case that α ≥ σ, which means that the curve CW on the other surface must have a point of multiplicity at least σ where it meets E. We note for reference that this imposes σ(σ + 1)/2 conditions on the curve CW . The second most common use of the above considerations is when the (−1)curve E meets R in two points (τ = 2). If we write σ = 2 − e, where e = 0, 1, the above result implies that at each of the two points where E meets R, the corresponding curve CW must have multiplicity . However this is not the whole story in this case. Note that since E is a (−1)curve, and τ = 2, the ruled surface T obtained by blowing E up is a ruled surface isomorphic to the minimal ruled surface F1 , and by the triple point formula the curve E (where T meets V ) is the (−1)-curve on T also. By (2.2), the restriction of the proper transform of the family of curves to T is C |T ≡ αE + (ατ − σ)F (where ≡ denotes linear equivalence). Since α ≥ , we see that |C |T | is contained in the linear system |E + (2 − (2 − e))F | = |E + eF |. If e = 0, we see that E is still a base curve for the family C , with multiplicity on T . If e = 1, the multiplicity of E in this system on T is − 1. If = e = 1 (the σ = 1 case), there is in fact no more to do, and the matching condition on CW is simply that the curve pass through the two points. We will therefore assume in what follows that σ ≥ 2. In this case we blow up E again, creating a new ruled surface S, which is in fact a product surface F0 = P1 × P1 (this follows from using the triple point formula to conclude that the self-intersection of E on S is zero). The same type of analysis as before shows that if C is the proper transform of the family C of curves on this second blowup, then C |S ≡ ( − e)E.

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Indeed, using similar notation, we see that C = π ∗ C − ( − e)S and since S|S ≡ −E − F , C |S = (π ∗ C − ( − e)S)|S ≡ −( − e)F − ( − e)(−E − F ) = ( − e)E. From this we conclude two things. First, the original curve CW on W not only has to meet R in the two points of multiplicity , but these are in fact tacnodal, in the sense that there are inﬁnitely near points of multiplicity − e required at both of these points. We see this because the proper transform as noted above meets the second exceptional curve at − e points. Secondly, we note that the divisor (of degree − e) cut out on the two inﬁnitely near exceptional curves on W must correspond, via the isomorphism aﬀorded by the horizontal sections on the surface S: these two exceptional curves are both ﬁbers of S, and the linear system on S consists of − e sections. The zero-sections of S give a natural correspondence between these two exceptional curves, and under this correspondence the two divisors of degree −e must agree. We will refer to this pair of tacnodal singularities as a pair of tacnodal (, − e)-points, with corresponding second-order tangents. We note ﬁnally that these two singularities impose, a priori, ( + 1) + ( − e)( − e + 1) + − e = 22 + (3 − 2e) + e2 − 2e = 22 + (3 − 2e) − e ( (2 + 3) if e = 0 = 2 2 + − 1 if e = 1 conditions on the matching curve CW , if σ ≥ 2. We have proved the following. Corollary 2.4. With the above notation, suppose that E is a (−1)-curve on V . (a) If E meets the curve R in one point p, then the matching condition for the curve CW is that it must have a point of multiplicity at least σ at p. (b) If E meets the curve R in two points p1 , p2 , and σ = 2 − e with e = 0, 1, then the matching condition for the curve CW is that it must have a pair of tacnodal (, − e)-points, with corresponding second-order tangents, at the two points pi . Of course, the (−1)-curve E may meet the curve R in more than two points. We will not analyze this situation here, since these cases do not occur in the applications presented below.

3. Collisions of fat points As a ﬁrst application of the above analyses, let us use the above considerations to analyze degenerations of schemes deﬁned by a collection of fat point s, i.e., points

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with multiplicity, on a surface X. The analysis is valid for arbitrary collections of multiplicities, but we will assume for simplicity that all the multiplicities are the same, equal to m. Suppose that we have an ample divisor H on X, and we want to analyze the divisors in the linear system |dH| for large d when the fat points come together. Our approach will be the following. Suppose that k distinct fat points of multiplicity m degenerate in a general way to a point p on a surface X. Let us explain what we mean here by this. Construct the trivial family X × Δ, where Δ is a disc, and in the central ﬁber blow up the point (p, {0}), to a plane P ∼ = P2 . The new central ﬁber then consists of two surfaces, the blowup X of X at p, and the plane P , meeting along a smooth rational curve R (which is the exceptional curve in X and is a line in P ). The assumption that the points are degenerating generally means that the k limit points are k general points on P . If we denote by H the pullback bundle on X , one limiting line bundle on the central ﬁber is the bundle which is dH on X and is trivial on P . Other limiting line bundles to dH on the general ﬁber are obtained by twisting this basic limiting line bundle by the divisor −tP , for any integer t; this gives the limiting line bundle which is dH − tR on X , and O(t) on the plane P . Any general curve in dH on the general X with multiplicity m at the k points will degenerate to a curve in the central ﬁber, which must be a union of two curves, one in X and one in P , satisfying the matching conditions as given in Section 2; these two curves must be divisors in the two limiting linear systems for some t. One can make an analysis for every t, obtaining diﬀerent limiting schemes; the constraint is that we must have both line bundles eﬀective, so that the limiting curves in each of the surfaces exist. Note that for such a t, the curve in X must be a divisor in |dH − tR|, and so the limiting curve on X must have a point of multiplicity t. We choose t so that it is minimal with respect to the property that O(t) has nonzero sections with multiplicity m at the k general limit points on P ; this is in some sense the most general limit, since it gives rise to the lowest possible multiplicity of the limit curve on X . The extra conditions on the limiting curve, which deﬁne the limit scheme of the k fat points, are obtained by analyzing the matching conditions as given in Section 2. One must make a separate analysis for each k. This we will execute for k ≤ 5 in the following. Proposition 3.1. Suppose that k fat points of multiplicity m degenerate together on a surface X, to a point p ∈ X. Then the general limiting scheme (for k ≤ 5) is given as follows. (a) If k = 2, the limit is a point of multiplicity m with an inﬁnitely near point of multiplicity m. (b) If k = 3 and m = 2 − e with e = 0, 1, the limit is a point of multiplicity 3 − e, with three inﬁnitely near points of multiplicity − e.

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(c) If k = 4, there is an involution ι on the P1 which is the ﬁrst-order neighborhood of the point p, and the limit is a point of multiplicity 2m, with the extra condition that the tangent cone divisor D of degree 2m on the P1 is of the form D = D1 + D2 with D2 = ι(D1 ). (d) If k = 5, and m = 2 − e with e = 0, 1, the limit is a point of multiplicity 2m with a pair of inﬁnitely near tacnodal (, − e)-points, with corresponding second-order tangents. Proof. If k = 2, we have that the minimal degree on the plane P is t = m, and the only divisor in the linear system on P is m times the line L joining the two points of multiplicity m. Indeed, when we blow up the two points, L becomes a (−1)-curve on the blow up of P , with σ = m (using the notation of Section 2), and τ = 1. Hence the matching condition on the curves of X are that they must have a point of multiplicity m at the point of intersection L ∩ R, by Corollary 2.4(a). This means that on X we have a point of multiplicity m with an inﬁnitely near point of multiplicity m, as stated in part (a). Suppose that k = 3 and m = 2 − e with e = 0, 1. Then the minimal degree t on P is t = 3 − e. The three lines joining pairs of the multiple points on P become (−1)-curves on the blow up of P at the three points, and their intersection with the system is (3 − e) − 2(2 − e) = e − , so that σ = − e and τ = 1 for each of these three lines. Again, by Corollary 2.4(a), this implies that the matching curves on X must have three points of multiplicity − e. Hence on X we have a point of multiplicity 3 − e, with three inﬁnitely near points of multiplicity − e as claimed in (b). Suppose that k = 4; then the minimal degree on P is t = 2m, and the linear system is composed with the pencil of conics through the four points. This pencil gives an involution on the curve R, by sending a point on R to the second intersection with R of the unique conic in the pencil through that point. The matching condition on X is simply that the divisor of degree 2m on the curve R must be matched by one of the divisors obtained by restricting m members of the pencil of conics to R. This is the involution condition as stated in (c). Suppose that k = 5; then the minimal degree on P is again t = 2m, and this time the unique member of the linear system on P is the conic C with multiplicity m. This conic C becomes a (−1)-curve on the blow up of P at the ﬁve points, and this time σ = m = 2 − e and τ = 2; hence Corollary 2.4(b) applies, and the

matching condition on X is that claimed in part (d). It is instructive to compare the lengths of these limit schemes to the length of the scheme consisting of the k fat points of multiplicity m (which is km(m + 1)/2). For k = 2, the limit scheme consists of one point of multiplicity m and one inﬁnitely near point of multiplicity m (an (m, m) tacnode) and so the length is equal to 2 · m(m + 1)/2 as required. For k = 3, the limit scheme consists of one point of multiplicity 3 − e and three inﬁnitely near points of multiplicity − e; thus the total length is

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(3 − e)(3 − e + 1)/2 + 3( − e)( − e + 1)/2 = 62 + (3 − 6e) which is the same as 3m(m + 1)/2. For k = 4, the limit scheme has length equal to the length of a fat point of multiplicity 2m, plus the length of the involution condition (which is m linear conditions): this is therefore (2m)(2m + 1)/2 + m = 2m2 + 2m which is the same as 4 · m(m + 1)/2 as needed. Finally for k = 5, the limit scheme has length equal to the length of the fat point of multiplicity 2m, plus the length of the pair of tacnodal singularities with the correspondence condition. If m = 1, we have a point of multiplicity 2 and two ﬁxed tangents, which has length 5 as required. If m ≥ 2, and m = 2 − e as above, then the computation of the number of conditions for the tacnodal pair of singularities was made just before Corollary 2.4; it is 22 + (3 − 2e) − e. Thus we have a total length of (4 − 2e)(4 − 2e + 1)/2 + 22 + (3 − 2e) − e = 102 + (5 − 10e) which is the same as 5 · m(m + 1)/2 as it should be. We remark that these limit schemes (for k = 4 and k = 5) are not simply deﬁned by their base loci. One could continue in the same spirit and make this type of analysis for larger k. However this would require the knowledge of the minimal t such that the system on P is nonempty, and also some detailed information about that system. As it is classically known, this is available for k up to 9, (see for instance [16]), but it becomes signiﬁcantly more involved. For larger k even the minimal t is not known in general (although there are precise conjectures, see [9], [13]).

4. Proving linear systems are empty via degeneration In this section we want to explain how to use a degeneration technique, combined with the matching conditions explained above, to prove that a complete linear system of curves on a surface is empty under suitable hypotheses. The degeneration technique is modeled after that used in [3], and will be useful in proving that certain linear systems of plane curves with prescribed multiple base points is empty. The matching lemmas developed in Section 2 permit a more reﬁned analysis. To be speciﬁc, suppose that one wants to prove that the linear system L on a surface P is empty. Take a smooth ﬂat family P → Δ, where Δ is a disc, whose general ﬁber is P , and whose special ﬁber is the union of two smooth surfaces P and F, meeting transversally along the smooth curve R.

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Denote by L a line bundle on the threefold P whose restriction to the general ﬁber P is the desired bundle L. L will restrict to line bundles LP on P and to LF on F. One can also take the bundle L(k) = L ⊗ OP (kP) to produce a limit of the line bundle L; note that these give the only limits of L on the central ﬁber. The bundle L(k) restricts to the pairs of bundles LF,k = LF (kR) on F and LP,k = LP (−kR) on P. Suppose now that we have a family of curves C on the general ﬁber as considered above. Closing this family up to the special ﬁber, one obtains two curves CF in F and CP in P. There will be a unique k such that CP is a divisor in the system LP,k , and CF is a divisor in the system LF,k . Suppose that one has a set of (−1)-curves E ⊂ F, meeting R transversally in at most two points, such that E · LF,k < 0; each of these curves gives a matching condition on the sections of LP,k according to the analysis of Section 2. Similarly, there may be (−1)-curves E ⊂ P, meeting the double curve transversally in at most two points, with E · LP,k < 0; these give matching conditions on the sections of LF,k in the same way. Deﬁne Lm P,k to be the system on P deﬁned by these matching conditions; this is a subsystem of LP,k . Similarly deﬁne the subsystem Lm F,k . We refer to these subsystems as the “matching systems”. ˆ P,k to be the subsystem of LP,k consisting of those elements vanishing Deﬁne L ˆ F,k . These form the kernels of the natural restriction map on R; similarly deﬁne L to the systems on the double curve R, and we will therefore refer to them as the ˆ m of the matching ˆ m and L kernel systems. We also have the intersections L P,k F,k systems with the kernel systems. Sometimes we will abuse notation and we will ˆ m with their residual systems with respect to R. ˆ m and L identify L P,k F,k The content of the matching lemmas developed in the previous section is exactly the following. If the system L on the general ﬁber P is not empty, we can choose a family C of curves in this system, ﬂat over Δ. For this family there will be a unique k such that the limit is a curve C0 = CP ∪ CF with CP a member of Lm P,k and CF a member of Lm ; C = C ∩ P and C = C ∩ F scheme-theoretically. Furthermore, P F F,k the restriction of CP and CF to the curve R must be the same. Note that the curve C0 contains the double curve R if and only if both CP and CF are members of the kernel systems. With these observations, the following Proposition is clear.

Proposition 4.1. With the above notation and assumptions, suppose that for every k, there are no pairs of curves (CP , CF ) in the corresponding matching sysm tems Lm P,k and LF,k whose restrictions to the curve R agree. Then the system L on the general surface P is empty.

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Note that the assumption implies that, for every k, at least one of the two ˆ m or L ˆ m is empty. Indeed, if neither is empty, then a kernel matching systems L P,k F,k limit curve containing the curve R is possible. We can reﬁne this Proposition by making the following observation: Lemma 4.2. With the above notations and assumptions, we have ˆm Lm P,k+1 = LP,k and ˆm Lm F,k−1 = LF,k Proof. The content of the lemma is that, in these situations, the matching conditions with one less twist by the double curve R gives the same result as considering the kernel system for the matching system. The considerations are the same for the two surfaces; let us just consider the P system. Suppose that E is a (−1)-curve on F with (E · LF,k ) = −σ, and (E · R) = 1. The matching condition in this case is that the divisors in LP,k must have a point of multiplicity σ at p = E ∩ R. The kernel system consists of divisors which contain R, and so the residual system consists of divisors in LP,k+1 with a point of multiplicity σ − 1 at p. Since (E · LP,k+1 ) = (E · LP,k ) − (E · R) = σ − 1 this is exactly the same as the matching conditions imposed by E on LP,k+1 as claimed. If instead (E · R) = 2, the corresponding σ will decrease by two, but the multiplicity of the matching conditions (in this case given by the pair of tacnodal points with corresponding second-order tangents) goes down by one, exactly the contribution of the double curve R. This proves the result.

This allows us to reﬁne Proposition 4.1 as follows: Proposition 4.3. Suppose that there exists a k0 , such the system Lm P,k0 on P is ˆ m on F is also empty. Then the system L on the empty and the kernel system L F,k0 general surface P is empty. Proof. Suppose on the contrary that the system L on the general surface is nonempty. Then there will exist a twist k and two curves in the matching systems m Lm P,k on P and LF,k on F which is a limit of the general curve. This k cannot be greater than or equal to k0 , since the matching system Lm P,k0 is empty, and this is also empty for k ≥ k , since they are subsystems. implies that Lm 0 P,k ˆ m is empty, and therefore as above all sub= L By Proposition 4.2, Lm F,k0 −1 F,k0 systems Lm F,k with k < k0 are empty. m We conclude that for every k, one of the two systems Lm P,k or LF,k is empty, and hence by Proposition 4.1, L is empty.

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In the next sections we will apply the results above to make some interesting reductions for analyzing linear systems of plane curves with prescribed multiple base points.

5. The Three Point Lemma We begin with a preliminary computation which is necessary to execute the ﬁrst of our reductions. We denote by Ld (n0 , . . . , nr ) the linear system of plane curves of degree d with r + 1 general base points pi having multiplicities ni . Consider L = Ld (m0 , m1 , m2 , m3 ) and denote by i the line joining p0 to pi . Note that each i becomes a (−1)-curve on the blowup of the pi ’s. Lemma 5.1. Fix a degree d and non-negative integers m1 , m2 , m3 . Let s = m1 + m2 + m3 . (a) If s ≤ d then the smallest m0 such that L is empty is m0 = d + 1. Moreover Ld (d, m1 , m2 , m3 ) = i mi i + Ld−s (d − s) and has dimension d − s. (b) If s > d then L is empty if m0 ≥ 2d − s + 1. (c) Assume s > d and mi ≥ s − d for each i = 1, 2, 3. Then

(d − s + mi )i + L2s−2d (s − d, s − d, s − d, s − d) Ld (2d − s, m1 , m2 , m3 ) = i

and has dimension s − d (and the general member consists of s − d conics in the pencil through the four points). (d) Assume s > d, r ≥ 0, and 2d ≥ s + r ≥ d. Then i splits oﬀ Ld (2d − s − r, m1 , m2 , m3 ) exactly max(0, d − s + mi − r) times. Proof. Statements (a), (c), and (d) are trivial, and (b) is just the fact that the excess intersection of L with a smooth conic through the four points cannot be negative.

Let’s try to systematically apply the degeneration method as in [3]. This method uses a degeneration of the plane to a plane P union a ruled surface F ∼ = F1 , meeting along a line R in the plane attached to the (−1)-curve in F. This is achieved by blowing up a line in the central ﬁber of the trivial family of planes. We divide the base points by putting three points on the F side as follows. Suppose that we are considering the original system L = Ld (m1 , m2 , m3 , . . . , mn ). Let s = m1 + m2 + m3 , and assume that s > d and mi ≥ s − d for each i = 1, 2, 3. Apply the degeneration method putting the ﬁrst three points on F and twist by k0 = s − d, i.e., subtract (s − d)P, so that the relevant four systems are:

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ˆ F,k0 : Ld (2d − s + 1, m1 , m2 , m3 ), L LF,k0 : Ld (2d − s, m1 , m2 , m3 ), LP,k0 : L2d−s (m4 , m5 , . . . , mn ), ˆ P,k0 : L2d−s−1 (m4 , m5 , . . . , mn ). L (We are identifying F implicitly with a plane blown up at one point, and therefore systems on F with r base points correspond in an obvious way to systems on the plane with r + 1 base points.) ˆ F,k0 is empty, and dim(LF,k0 ) = By the previous Lemma, the kernel system L s − d. The matching conditions give three extra points on the P surface, lying on the line R, having multiplicities d − s + mi , i = 1, 2, 3 (using (c) of the previous Lemma). In other words, we have ˆ F,k0 : Ld (2d − s + 1, m1 , m2 , m3 ) = ∅, L Lm F,k0 : Ld (2d − s, m1 , m2 , m3 ), Lm P,k0 : L2d−s ([d − m2 − m3 , d − m1 − m3 , d − m1 − m2 ], m4 , . . . , mn ), ˆm L P,k0 : L2d−s−1 ([d − m2 − m3 − 1, d − m1 − m3 − 1, d − m1 − m2 − 1], m4 , . . . , mn ), where the brackets indicate that these points are collinear, and otherwise general. Note that the virtual dimension of Lm P is the same as the virtual dimension of the original system L on the plane. Suppose that this virtual dimension is negative, so that we are trying to use the method to show that L is empty. ˆ F,k0 is certainly empty, the following is Since we have that the kernel system L immediate by Proposition 4.3: Proposition 5.2. If Lm P,k0 = L2d−s ([d − m2 − m3 , d − m1 − m3 , d − m1 − m2 ], m4 , . . . , mn ) is empty, then L is empty. This is a signiﬁcant reduction, modulo the hypothesis that the extra three multiplicities created by the matching condition all are with points that lie on a line. We note that the multiplicity numbers that arise with the three-point lemma are exactly those that would arise from performing a quadratic Cremona transformation centered at the three points. Remark 5.3. The above Proposition was formulated to consider the linear system L of degree d with general multiple base points. The result is a reduction to a system for which three of the points are collinear. It is not necessary in fact for the original system to have all general base points: one can apply the same construction in case there are constraints to the positions of the points. The only condition that is needed to apply the construction is that the three base points

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that one wants to make collinear are general enough to permit them to become collinear, and that this specialization does not aﬀect whatever constraints there may be on the other points. In any case, the reduction algorithm to prove in this way that a system L as above is empty is: 1. Choose three multiplicities m1 , m2 , m3 so that s = m1 + m2 + m3 ≥ d + 1 and mi ≥ s − d for each i = 1, 2, 3. 2. Reduce the degree of the system from d to 2d − s. 3. Reduce these three multiplicities to d − m2 − m3 , d − m1 − m3 , d − m1 − m2 and put these points on a line. 4. Show that the resulting system is empty.

6. The Four- and Five-Point Lemmas Let L = Ld (m0 , m1 , m2 , m3 , m4 ) denote the linear system of plane curves of degree d with ﬁve general multiple points p0 , . . . , p4 with multiplicities m0 , . . . , m4 . Denote by i the line joining p0 to pi , and by C the conic through the ﬁve points. These all become (−1)-curves on the blowup of the points. Lemma 6.1. Fix a degree d and non-negative integers m1 , . . . , m4 . Let s = m1 + m2 + m3 + m4 . (a) If s ≤ d then the smallestm0 such that L is empty is m0 = d + 1. One has Ld (d, m1 , m2 , m3 , m4 ) = i mi i + Ld−s (d − s) and has dimension d − s. (b) If s > d write s − d = 2t + e with t ≥ 0 and 0 ≤ e ≤ 1. Then L is empty if m0 = d + 1 − t − e. (c) Assume s > d, s − d = 2t + e as above, and t + e ≤ d and t + e ≤ mi for each i. Then Ld (d − t − e, m1 , m2 , m3 , m4 ) = i (mi − t − e)i + tC + L3e (2e, e4 ) and has dimension 2e. (d) Assume s > d, s − d = 2t + e as above, and r ≥ 0. If Ld (d − t − e − r, m1 , m2 , m3 , m4 ) is not empty, then i splits oﬀ Ld (d − t − e − r, m1 , m2 , m3 , m4 ) exactly max(0, mi −t−e−r) times, and the conic C splits oﬀ exactly max(0, t − r) times. Proof. Statement (a) is obvious; since Ld (d, m1 , m2 , m3 , m4 ) · i = −mi , the line i splits oﬀ mi times leaving the residual system Ld−s (d − s). Since this residual system is non-empty, the smallest m0 such that L is empty is at least d + 1. Since m0 = d + 1 gives an empty system, we are done.

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To prove (b), consider a general cubic C passing through the ﬁve points, with a double point at the ﬁrst point. Its proper transform on the ﬁve-fold blowup of the plane is nef. However (C · L) = 3d − 2(d + 1 − t − e) − s = d − s + 2t + 2e − 2 = e − 2 < 0 which is a contradiction if L is not empty. To prove (c), let L = Ld (d − t − e, m1 , m2 , m3 , m4 ), and note that L · i = t + e − mi which is nonpositive by assumption; therefore i splits oﬀ the system mi − t − e times. Since i (mi − t − e) = s − 4t − 4e, the residual system has degree d − (s − 4t − 4e) = 4t + 4e − (2t + e) = 2t + 3e, and the multiplicity of the residual system at p0 is d − t − e − (s − 4t − 4e) = 3t + 3e − (2t + e) = t + 2e. The multiplicity of the residual system at pi for i > 0 is t + e. The intersection of this residual system with the conic C is 2(2t + 3e) − (t + 2e) − 4(t + e) = 4t + 6e − t − 2e − 4t − 4e = −t so that C splits oﬀ the system t times. After splitting oﬀ C, this further residual system is L3e (2e, e4 ) which has dimension 0 if e = 0 and has dimension 2 if e = 1. Statement (d) follows by a similar analysis. QED Let’s try to systematically apply the method outlined above using four points on the F side as follows. Suppose that we are considering the original system L = Ld (m1 , . . . , mn ) (with n ≥ 4). Let s = m1 + m2 + m3 + m4 , assume that s > d, and write s − d = 2t + e as above. Suppose further that t + e ≤ d and t + e ≤ mi for 1 ≤ i ≤ 4. Apply the degeneration method putting the ﬁrst four points on F and twist by k0 = t + e, so that the relevant four systems are: ˆ F,k0 : Ld (d − t − e + 1, m1 , m2 , m3 , m4 ), L LF,k0 : Ld (d − t − e, m1 , m2 , m3 , m4 ), LP,k0 : Ld−t−e (m5 , m6 , . . . , mn ), ˆ P,k0 : Ld−t−e−1 (m5 , m6 , . . . , mn ). L ˆ F,k0 is empty, and dim(LF,k0 ) = 2e. By the previous Lemma, the kernel system L The matching conditions give ﬁve extra points on the P surface, lying on a line, having multiplicities m1 − t − e, m2 − t − e, m3 − t − e, m4 − t − e, and t (using (c) of the Lemma). The sum of these ﬁve extra multiplicities is s − 3t − 4e, which is equal to d− (3t+ 4e − s+ d) = d− t− 3e and is therefore at most the degree d− t− e of the P system. In particular the matching system Lm P,k0 has a moving divisor of degree 2e restricted to the line. This matches perfectly with the dimension of the system on F. We note that a small miracle occurs now: the virtual dimension of the P system Lm P,k0 is identical to the virtual dimension of the original system L: in fact if one computes the diﬀerence algebraically and simpliﬁes, one obtains v(L) − v(LP,k0 ) = e(e − 1)/2 and since e = 0 or 1, this is zero.

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Proposition 6.2. If Lm P,k0 = Ld−t−e ([m1 − t − e, m2 − t − e, m3 − t − 3m4 − t − e, t], m5 , . . . , mn ) is empty, then L is empty. As in the Three-Point Lemma, the proof is simply to apply Proposition 4.3: ˆ m and Lm are empty. both L F,k0 P,k0 Again, this is a signiﬁcant reduction, modulo the hypothesis that the extra ﬁve multiplicities created by the matching condition all are with points that lie on a line. In addition, the same considerations as for the Three-Point Lemma apply (see Remark (5.3)) with regard to having any constraints on the points: as long as the four points are general enough to become collinear, the set of points may enjoy some other constraints and this does not aﬀect the reduction process. In any case, the reduction algorithm to prove L is empty is: 1. Choose four multiplicities m1 , . . . , m4 so that s = m1 +m2 +m3 +m4 ≥ d+1. 2. Write s − d = 2t + e as above and check that t + e ≤ d and t + e ≤ mi for 1 ≤ i ≤ 4. 3. Reduce the degree of the system by t + e, from d to d − t − e. 4. Reduce these four multiplicities by t + e, from mi to mi − t − e. 5. Add one additional multiplicity of t. 6. Assume that the ﬁve ‘new’ points having multiplicities mi −t−e (i = 1, 2, 3, 4) and t are collinear, and that the points are otherwise general. 7. Prove that the resulting system is empty. We have developed also a Five-Point Lemma in a similar fashion, which we present below without proof. The details are quite parallel to those of the Threeand Four-Point Lemmas. We will not be applying the Five-Point Lemma in the applications presented below. Proposition 6.3. Let L be the linear system L = Ld (m1 , . . . , mn ). Suppose that the virtual dimension v(L) < 0. If Lm P = Ld−t−e ([m1 − t − e, m2 − t − e, m3 − t − e, m4 − t − e, m5 − t − e], m6, . . . , mn ) has the expected dimension, then L is empty. Again, Remark (5.3) applies in this case also: constraints on the position of the points in the original system L are permitted, as long as they do not forbid the ﬁve points to become collinear.

7. Applications We begin by presenting two computations which prove that certain linear systems are empty. Both computations use the ideas of Section 3 on collisions of fat points. The ﬁrst case we consider is the famous linear system L38 (1210 ) of virtual dimension −1 (see [8] for the ﬁrst proof of its emptyness).

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7.1. The system L38 (1210 ) via collisions. Take ﬁve of the ten 12-tuple points and bring them together. The limit is, according to Proposition 3.1(d), a point of multiplicity 24, with two inﬁnitely near tacnodal (6, 6)-point singularities having corresponding second-order tangents. The resulting system therefore has this one singularity, and ﬁve other general 12-tuple points. We now perform four Cremona transformations. The ﬁrst, centered at the point of multiplicity 24 and two of the 12-tuple points, results in a system of degree 28, with one point of multiplicity 14, three points of multiplicity 12, two points of multiplicity 2, and two tacnodal (6, 6)-points having corresponding second-order tangents. The two double points and the two tacnodal points lie on a line (the common tangent to the tacnodes). The second, centered at the point of multiplicity 14 and two of the 12-tuple points, results in a system of degree 18, with one point of multiplicity 12, one point of multiplicity 4, four points of multiplicity 2, and we still have the two tacnodal (6, 6)-points having corresponding second-order tangents. Now all four of the double points, the multiplicity 4 point, and the two tacnodes all lie on a conic (the Cremona image of the tangent line noted above); this conic is tangent to the tacnodes. The third Cremona transformation is centered at the point of multiplicity 12 and the two tacnodes: the result is a system of degree 12, with three points of multiplicity 6, one 4-tuple point, and four double points; two of the points of multiplicity 6 have corresponding tangents. These two points of multiplicity 6, the 4-tuple point, and the four double points all lie on a conic. The fourth Cremona transformation is centered at the three points of multiplicity 6, and results in a system of degree 6, with one 4-tuple point and four double points. The given system of sextics has dimension ﬁve (it is in fact the Cremona image of a complete system of conics). There are now also two lines which the sextics must meet in a corresponding way. This correspondence between the two lines is in fact given by projection from a point Q in this plane, not on either of the two lines. The two lines L and M and the ﬁve multiple points are now general; there is a conic containing the ﬁve multiple points, but there is always a conic through ﬁve general points. It is not diﬃcult to see that, under these assumptions, no sextic in this ﬁvedimensional subsystem V satisﬁes the correspondence given by projection from Q; this is a priori six conditions on the ﬁve-dimensional system, and we must check that this gives an empty system. Indeed, the techniques of the collision method presented in Section 3 show that we can bring the ﬁve points together: the result is a point P of multiplicity six, whose tangent cone must lie in a speciﬁc ﬁve-dimensional subspace of the six-dimensional ambient space. Upon doing this, the system of sextics becomes a system of six lines through the multiplicity six point, with a speciﬁc codimension one condition. Moreover this codimension one subspace V is general, which can be achieved by acting with projective transformations with ﬁxed point P .

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To see that the system is now empty, note that projection from P is exactly what gives the correspondence relation between the divisors cut out by the sextics on the two lines. Consider then the automorphism φ on one of the lines, say L, given by projection from P (from L to M ) followed by projection from Q (from M back to L). This automorphism can also be considered to be general, modulo the condition that it ﬁxes the point of intersection L ∩ M . In order for us to have a sextic C satisfying the correspondence condition on the two lines, the divisor that C cuts on L must be ﬁxed by this automorphism. The reader can easily become convinced that, since φ is general, there are only ﬁnitely many ﬁxed divisors of degree six for φ. By the generality of V , we can assume that none of these ﬁxed divisors are cut out by curves C in V . This ﬁnishes the proof.

7.2. The system L158 (5010 ) via collisions. We will now consider in a similar way the system of curves of degree 158 with ten multiple points of order 50. The interest in this system was communicated to us by B. Harbourne and J. Ro´e (see [11]); it is in some sense an extremal case that appears in their analysis of the Nagata Conjecture. The system has virtual dimension equal to −31, and so should be empty. This we will show. Again collide ﬁve of the singularities together, and make an identical set of four Cremona transformations as above. We obtain a system of degree 48, with points of multiplicity 25, 25, 24, 16, 8, 8, 8, and 8; the two points of multiplicity 25 have corresponding tangents. At this point the line joining the two 25-tuple points splits twice from the system, and the two lines joining the 25-tuple points to the 24-tuple point splits once, resulting in a system of degree 44, with three points of multiplicity 22, one point of multiplicity 16, and four points of multiplicity 8; two of the 22-tuple points have corresponding tangents. Now a Cremona transformation at the three 22-tuple points results in a system of degree 22, with one point of multiplicity 16 and four points of multiplicity 8; in addition (as above), there are two lines L and M for which the intersection with the curves must correspond. The four lines joining the 16-tuple point to the four 8-tuple points each split twice, and the conic through all ﬁve of the points splits four times, giving a residual system of degree 6, with one 4-tuple point and four double points. The correspondence condition with the two lines is still to be satisﬁed. This is the exact same situation as in the previous analysis! We conclude again that the system is empty. We note that this immediately implies that the system L79 (2510 ) is also empty, since it is exactly half of the above system. It is in fact this system that was found to be of interest in [11]. According to this article, the emptyness of this system implies that if d and m are positive integers with d/m < 177/56 ≈ 3.160714, then there are no curves of degree d with 10 general m-tuple points. It is our understanding that this constant is the best known √ at this time. Nagata’s Conjecture for this case would improve this constant to 10 ≈ 3.162278.

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7.3. The system L158 (5010 ) via Four-Point Lemmas. The above proof that the system L158 (5010 ) is empty relied on the results of Section 3 on collisions of fat points. Let us remake the proof, this time exploiting the results of Section 6 where four points move onto a line. The ﬁrst step is to use the four-point lemma with four of the multiplicity 50 points. In this case (using the notation of Proposition 6.2), d = 158, s = 200, so that s − d = 42, and t = 21, with e = 0. We reduce to a system of degree 137, with six points of multiplicity 50, four points of multiplicity 29, and one of multiplicity 21; the four 29-tuple points and the 21-tuple point are collinear. We will now perform several Cremona transformations. In the course of our analysis, the Cremona transformations carry the lines that become part of the constraints on the positions of the points to other rational curves of higher degree. In order to keep track of this, we introduce a matrix notation which encodes the linear system in question, and the constraints, as follows. For a linear system having n fat points, and involving k constraint curves, we use a matrix with k + 2 rows and n + 1 columns. The columns (after the ﬁrst) are indexed by the points. The ﬁrst row is simply an indexing row that gives indexes to the n fat points. The second row indicates the degree (in the ﬁrst column) and the multiplicities (in the succeeding colums) of the linear system under analysis. The third and following rows indicate the degree and multiplicity of the constraint curves (in the same manner as above) that the points must satisfy. For example, the result of applying one Four-Point Lemma to the ﬁrst four points of the system L158 (5010 ) is denoted by the matrix: 1 2 137 29 29 1 1 1

3 4 5 6 7 8 9 10 11 29 29 50 50 50 50 50 50 21 1 1 0 0 0 0 0 0 1

This indicates that the linear system has degree 137; that points 1, 2, 3, and 4 now have multiplicity 29, while points 5–10 remain with multiplicity 50, and that the extra point (numbered 11) added in the Four-Point Lemma has multiplicity 29. The last row indicates that points 1, 2, 3, 4, and 11 lie on a line (each having multiplicity one of course). We now perform four quadratic Cremona transformations. The ﬁrst is centered at points 5, 6, and 7. The second is centered at points 8, 9, and 10. The third is centered at points 5, 6, and 7 again. The fourth is centered at points 1, 2, and 3. The result is a linear system of degree 83, indicated by the matrix: 1 2 3 4 5 6 7 8 9 10 11 83 27 27 27 29 24 24 24 24 24 24 21 7 3 3 3 1 2 2 2 2 2 2 1 The line has been transformed to a septic, triple at the ﬁrst three points, passing through the fourth, double at the next six, and passing through the eleventh. Now use the Four-Point Lemma again, with points 4, 5, 6, and 7. This reduces the system to one of degree 74, and adds an additional point on the new line of

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multiplicity 9: 1 2 3 4 5 6 7 8 9 10 11 12 74 27 27 27 20 15 15 15 24 24 24 21 9 7 3 3 3 1 2 2 2 2 2 2 1 0 1 0 0 0 1 1 1 1 0 0 0 0 1 We now perform two quadratic Cremona transformations; The ﬁrst is centered at points 1, 2, and 3, while the second is centered at points 8, 9, and 10. The result is a linear system of degree 62, indicated by the matrix: 1 2 3 4 5 6 7 8 9 10 11 12 62 20 20 20 20 15 15 15 19 19 19 21 9 4 1 1 1 1 2 2 2 1 1 1 1 0 4 1 1 1 1 1 1 1 2 2 2 0 1 Both constraint curves are now quartics. At this point use another Four-Point Lemma, with points 1, 2, 3, and 11, leading to: 1 2 3 4 5 6 7 8 9 10 11 12 13 52 10 10 10 20 15 15 15 19 19 19 11 9 9 4 1 1 1 1 2 2 2 1 1 1 1 0 0 4 1 1 1 1 1 1 1 2 2 2 0 1 0 1 1 1 1 0 0 0 0 0 0 0 1 0 1 At this point the second of the two quartics meets the system negatively (the intersection number is −10), and must be in the ﬁxed part of the system. In fact it splits three times. After splitting the quartic three times, one may check that the line (represented by the last row) splits once; the residual system is:

39 4 4 1

1 6 1 1 1

2 6 1 1 1

3 4 6 17 1 1 1 1 1 0

5 6 7 8 9 10 11 12 13 12 12 12 13 13 13 10 6 8 2 2 2 1 1 1 1 0 0 1 1 1 2 2 2 0 1 0 0 0 0 0 0 0 1 0 1

We now perform three quadratic Cremona transformations; the ﬁrst is centered at points 4, 8, and 9, the second is centered at points 4, 5, and 10, and the third is centered at points 4, 6, and 7. The result is a linear system of degree 30, indicated by the matrix:

30 4 4 4

1 6 1 1 1

2 6 1 1 1

3 6 1 1 1

4 8 1 1 3

5 9 2 1 1

6 7 8 9 10 10 9 9 1 1 2 2 2 2 1 1 1 1 1 1

10 11 12 13 10 10 6 8 1 1 0 0 2 0 1 0 1 1 0 1

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All three constraint curves have been transformed into quartics at this point. Now use a Four-Point Lemma with points 10, 11, 12, and 13: 28 4 4 4 1

1 6 1 1 1 0

2 6 1 1 1 0

3 6 1 1 1 0

4 8 1 1 3 0

5 9 2 1 1 0

6 7 8 9 10 10 9 9 1 1 2 2 2 2 1 1 1 1 1 1 0 0 0 0

10 11 12 13 14 8 8 4 6 2 1 1 0 0 0 2 0 1 0 0 1 1 0 1 0 1 1 1 1 1

At this point the ﬁrst quartic splits oﬀ the system, and then the second quartic splits; after this, the line splits. Now the ﬁrst quartic splits again, and then the third quartic splits, and ﬁnally the line splits again. The residual system has degree ten: 10 4 4 4 1

1 2 1 1 1 0

2 2 1 1 1 0

3 2 1 1 1 0

4 2 1 1 3 0

5 3 2 1 1 0

6 5 1 2 1 0

7 5 1 2 1 0

8 3 2 1 1 0

9 10 11 12 13 14 3 1 3 1 3 0 2 1 1 0 0 0 1 2 0 1 0 0 1 1 1 0 1 0 0 1 1 1 1 1

This system is easily seen to be empty, by applying Cremona transformations. Several other applications of the Three-point Lemma and the Four-Point Lemma have been presented in [1]. In particular the system L38 (1210 ) ([12],[8]) of virtual dimension −1 can also be proved to be empty rather directly using the reductions aﬀorded by the Lemmas.

References [1] C. Bocci and R. Miranda: Topics on Interpolation Problems in Algebraic Geometry, Notes for a course in Interpolation Theory, Torino, Fall 2003, to appear in Rendiconti Seminario Mathematico di Torino. [2] A. Buckley and M. Zompatori: Linear systems of plane curves with a composite number of base points of equal multiplicity, Trans. Amer. Math. Soc., 355 (2003), no. 2, 539–549. [3] C. Ciliberto and R. Miranda: Degenerations of Planar Linear Systems, J. Reine Angew. Math. 501 (1998), 191–220. [4] C. Ciliberto and R. Miranda: Linear Systems of Plane Curves with Base Points of Equal Multiplicity, Trans. Amer. Math. Soc. 352 (2000), 4037-4050. [5] C. Ciliberto and R. Miranda: The Segre and Harbourne–Hirschowitz Conjectures, In: Applications of algebraic geometry to coding theory, physics and computation (Eilat 2001), 37–51, NATO Sci. Ser. II Math. Phys. Chem., 36, Kluwer Acad. Publ., Dordrecht, (2001).

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[6] L. Evain: La fonction de Hilbert de la r´ eunion de 4h points g´en´eriques de P2 de mˆ eme multiplicit´e, J. of Alg. Geom., 8 (1999). [7] L. Evain: Computing limit linear series with inﬁnitesimal methods, preprint; arXiv:math.AG/0407143 [8] A. Gimigliano: On linear systems of plane curves, Thesis, Queen’s University, Kingston (1987). [9] B. Harbourne: The Geometry of rational surfaces and Hilbert functions of points in the plane, Can. Math. Soc. Conf. Proc., vol. 6 (1986), 95–111. [10] B. Harbourne and J. Ro´e: Linear systems with multiple base points in P2 , Adv. Geom. 4 (2004), 41–59 [11] B. Harbourne and J. Ro´e: Extendible Estimates of Multi-Point Seshadri Constants, arXiv:math.AG/0309064. [12] A. Hirschowitz: La m´ ethode d’Horace pour l’interpolation a ` plusieurs variables, Manus. Math. 50 (1985), 337–388 [13] A. Hirschowitz: Une conjecture pour la cohomologie des diviseurs sur les surfaces rationnelles g´en´eriques, J. Reine Angew. Math., vol. 397 (1989) 208-213. [14] V. V. Kulikov: Degenerations of K3 Surfaces and Enriques Surfaces, Math. USSR Izvestija 11 (1977), 957–989. [15] T. Mignon, Syst`emes de courbes planes ` a singularit´es impos´ ees: le cas des multiplicit´es inf´erieures ou ´ egales ` a quatre, J. Pure Appl. Algebra 151 (2000), no. 2, 173–195 [16] M. Nagata: On Rational Surfaces II, Mem. Coll. Sci. Univ. Kyoto, Ser. A, Math., 33 (1960), 271–293. [17] B. Segre: Alcune questioni su insiemi ﬁniti di punti in geometria algebrica, Atti Convegno Intern. di Geom. Alg. di Torino, (1961), 15–33. [18] S. Yang: Linear series in P2 with base points of bounded multiplicity, preprint; math.AG/0406591

Ciro Ciliberto Dipartimento di Matematica Universit` a di Roma Tor Vergata Via della Ricerca Scientiﬁca 00133 Roma, Italia Email: [email protected] Rick Miranda Department of Mathematics Colorado State University Campus Delivery - 1801 Fort Collins, CO 80523-180, USA Email: [email protected]

Negative curves on very general blow-ups of P2 Tommaso de Fernex∗

Abstract. This note contains new evidence to a conjecture, related to the Nagata conjecture and the Segre-Harbourne-Gimigliano-Hirschowitz conjecture, on the cone of eﬀective curves of blow-ups of P2 at very general points. 2000 Mathematics Subject Classiﬁcation: 14H51, 14H50, 14J26

Introduction The solution to the problem of determining the dimension of every linear system of curves in P2 with assigned multiplicities at general points is predicted by the equivalent conjectures of Segre, Harbourne, Gimigliano and Hirschowitz [Se, Ha1, Gi, Hi], hereafter “SHGH conjecture”. In this paper we are interested in the following weaker form of the conjecture: Conjecture 0.1. Suppose that C is an integral curve with negative selfintersection on the blow-up Y of P2 at a set of points in very general position. Then C is a (−1)-curve of Y (that is, a smooth rational curve with self-intersection −1). If one also includes the conjecture that H 1 (Y, OY (D)) = 0 for every eﬀective nef divisor D on any such Y , then this becomes one of the formulations of the SHGH conjecture [Ha2]. There is an interesting relationship between Conjecture 0.1, which of course contains Nagata’s conjecture [Na1], and the symplectic packing problem in dimension four [MP, Bi]. In the ﬁrst section of this paper, we will review how the above conjecture characterizes the geometry of the cone of eﬀective curves of Y . It is not diﬃcult to see that the conjecture is satisﬁed by all rational curves (see Proposition 2.4 below). There are several results nowadays, giving evidence to the ∗ Research partially supported by the University of Michigan Rackham Research Grant and the MIUR of the Italian Government, National Research Project “Geometry on Algebraic Varieties” (Coﬁn 2002).

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SHGH conjecture, that are valid under suitable assumptions on the multiplicities assigned to the centers of the blow-up. These include [AH, Ha3, CM1, CM2, Mi, Ya]. In the same spirit, we prove the following result. Theorem 0.2. Conjecture 0.1 is satisﬁed by all curves whose image on P2 is a curve with a singularity of multiplicity 2 at one of the centers of the blow-up. The proof of this theorem is based on a local study of dynamic self-intersection, very much inspired to the methods in [EL] and [Xu]. The novelty here with respect to [EL, Xu] is to consider deformations with two-parameters families: it is indeed by computing the Kodaira-Spencer map from diﬀerent directions of deformation that we produce a linear system on C giving an isomorphism to P1 and showing that C 2 = −1. A similar result is proven to hold for an arbitrary smooth projective surface in place of P2 (see Theorem 2.5 below). An analogous result on linear systems on smooth projective surfaces was given in [dVL, Theorem 4.1] under the assumption that the “specialty” of the system (namely, the gap between its dimension and its expected dimension) increases by imposing a double point. We remark that this hypothesis does not translate well to the context of determining negative curves on the blow-up. To the best of our knowledge, there are no other results in which the assumptions only involve one of the multiplicities. The precise notion of “very general position” adopted in this paper is given in Deﬁnition 2.1 below. Throughout this paper we work over the complex numbers. Acknowledgements The author would like to thank L. Ein, B. Harbourne, A. Laface, R. Lazarsfeld, R. Miranda and S. Yang for useful discussions.

1. Geometry of the cone of curves In this section we discuss the implication that Conjecture 0.1 has on the geometry of the cone of eﬀective curves of the blow-up of P2 . This section is mostly of an expository nature, as the material here contained is probably well known to the specialist. We start by ﬁxing some notation. Let X be a smooth projective surface. Let N (X) := (Pic(X)/ ≡) ⊗ R and ρ := rk Pic(X). By the Hodge index theorem, we can identify N (X) with Rρ in such a way that the intersection product is given by the matrix diag(1, −1, . . . , −1). For a divisor D on X we denote by D<0 (resp. D≤0 , D≥0 , D⊥ ) the subset of N (X) deﬁned by [D] · x < 0 (resp. [D] · x ≤ 0, [D] · x ≥ 0, [D] · x = 0). Let NE(X) ⊂ N (X) be the closure of the cone spanned by the classes of eﬀective curves on X, and Nef(X) ⊂ N (X) be the closure of the cone spanned by the classes of ample divisors on X. By Kleiman’s criterion, these two cones are put in duality by the intersection product in N (X). Fix an ample class H on X, let Pos(X) := {α ∈ N (X) | α2 ≥ 0, α · H ≥ 0},

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and let Nul(X) denote the boundary of Pos(X). Then Nul(X) is supported by the quadratic equation x21 = x22 + · · · + x2ρ . Note that Nef(X) ⊆ Pos(X) ⊆ NE(X). It follows by a result of Campana and Peternell [CP] that ∂ NE(X) is supported by Nul(X) and countably many hyperplanes, so we can write ∂ NE(X) = B1 B2 , where B1 is the closure (in ∂ NE(X)) of the union of the facets of ∂ NE(X), and B2 is supported by Nul(X). The following is a more precise formulation of Conjecture 0.1. Conjecture 1.1. Let Y be the blow-up of P2 at a set of r points in very general position. Then, writing ∂ NE(Y ) = B1 B2 as above (setting X = Y ), the extremal rays of B1 that do not lie on Nul(Y ) are spanned by classes of (−1)-curves, and we have B1 = ∂ NE(Y ) ∩ KY≤0 and B2 = ∂ NE(Y ) ∩ KY>0 . In particular, NE(Y ) ∩ KY≥0 = Pos(Y ) ∩ KY≥0 . In the following, we review the equivalence of the two conjectures. We begin with a general consideration on the clustering of extremal rays. As at the beginning of the section, let X be a smooth projective surface. We consider the metric on the set of rays in N (X) given by the angular distance: for any two rays R1 and R2 in N (X), we set d(R1 , R2 ) to be the angle between them. For an extremal ray R of NE(X), we set d(R) = inf{d(R, R ) | R is an extremal ray of NE(X) diﬀerent from R}. Lemma 1.2. If R is an extremal ray of NE(X), then d(R, Nul(X)) ≤ d(R); in particular, every Cauchy sequence of extremal rays of NE(X) converges to a ray on Nul(X). Moreover, if ρ ≥ 4 and F is a facet of NE(X) such that ∂F ∩Nul(X) = ∅, then F has inﬁnitely many extremal rays forming a Cauchy sequence converging to ∂F ∩ Nul(X). Proof. We can assume that R ⊂ Nul(X). By [Ko, Lemma II.4.2], R is spanned by the class of a curve C with C 2 < 0, and every other extremal ray of NE(X) is contained in the half space C ≥0 [Laz, Exercise 1.4.33(ii)]. Thus the inequality d(R, Nul(X)) ≤ d(R) will follow once we show that the orthogonal projection (in the euclidian metric) of R onto C ⊥ is contained in Pos(X). This can be easily checked as follows. Let , denote the standard inner product in Rρ , and let ( · ) denote the intersection product deﬁned by the diagonal matrix diag(1, −1, . . . , −1). Let h = (1, 0, . . . , 0) ∈ Rρ , ﬁx a vector c ∈ Rρ such that (c · h) > 0 and (c · c) < 0, and consider the orthogonal projection π : Rρ → {x ∈ Rρ | (c · x) = 0}. It is an exercise to check that (c · c)2 (π(c) · π(c)) = −1 (c · c) c, c 2

and (π(c) · h) = (c · h) −

(c · c) · c, h . c, c

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Since (c·c)2 ≤ c, c 2 , the ﬁrst equation gives (π(c)·π(c)) ≥ 0. The second equation gives (π(c)·h) > 0. Therefore the projection of R onto C ⊥ in contained in Pos(X). To prove the second part of the lemma, we note that F is supported by the hyperplane H that is tangent to Nul(X) along the ray ∂F ∩ Nul(X); this follows by the convexity of NE(X) and the inclusion Pos(X) ⊆ NE(X). If F has only ﬁnitely many rays, then we ﬁnd another facet F containing ∂F ∩ Nul(X) that, for the same reason, is also supported by H. Since this is impossible, F must have inﬁnitely many extremal rays, and these cluster to ∂F ∩ Nul(X) by what proved in the ﬁrst part of the lemma.

Now we can go back to the question on the equivalence between the two conjectures. Recall that we are denoting by Y the blow-up of P2 at a set of r points in very general position. Clearly both conjectures are certainly true for r ≤ 2, so we can assume that r ≥ 3, hence rk Pic(Y ) ≥ 4. Since one direction is obvious, let us assume that the only integral curves C ⊂ Y with C 2 < 0 are the (−1)-curves. Write ∂ NE(Y ) = B1 B2 as described above. The extremal rays of B1 not lying on Nul(Y ), which are also extremal rays of NE(Y ), are spanned by classes of (−1)-curves. In particular, by adjunction formula and Lemma 1.2, we have B1 ⊂ KY≤0 . On the other hand, the Cone Theorem implies that B2 ⊂ KY≥0 . In fact, observing that ∂ NE(Y ) ∩ KY<0 = ∅ and that B2 is open in ∂ NE(Y ), we actually have B2 ⊂ KY>0 . Therefore we conclude that B1 = ∂ NE(Y ) ∩ KY≤0 and B2 = ∂ NE(Y ) ∩ KY>0 . The last assertion follows by continuity. Remark 1.3. As it is well known, an interest in Conjecture 1.1 comes from the search for examples of irrational Seshadri constants. Indeed, if true, the conjecture would allow us to construct such examples. This is easy to see: let Y be the blowup of P2 at r ≥ 9 points in very general position, and denote by H the pull-back on Y of the hyperplane class of P2 and by E the exceptional divisor of Y √→ P2 . Then, for a very general point √ q ∈ Y and for every rational 0 < a < 1/ r, we would have q (H − aE) = 1 − ra2 . Note that, for most (rational) values of a, this is an irrational number.

2. Negative curves on P2 blown up at very general points The necessity to assume very general position in the conjecture is clear in the case of more than 9 points. Indeed the blow-up of P2 at 9 points will generally carry inﬁnitely (−1)-curves [Na2, Theorem 4a], and certainly one cannot allow to blow up points lying on these curves. In this paper we adopt the following notion of very general position. Deﬁnition 2.1. We say that a set {x1 , . . . , xr } of r ≥ 1 distinct points on a smooth projective surface X is in very general position if the following condition

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is fulﬁlled. For every integral curve C ⊂ X, the pair (C, (x1 , . . . , xr )) belongs to an irreducible algebraic family {(Ct , (x1,t , . . . , xr,t )) | t ∈ T }, where Ct ⊂ X is an integral curve and (x1,t , . . . , xr,t ) ∈ X r for every t ∈ T , satisfying the following properties: (i) pg (Ct ) = pg (C) and multxi,t Ct = multxi C for every t ∈ T and i = 1, . . . , r; (ii) the morphism ψ : T → X r given by ψ(t) = (x1,t , . . . , xr,t ) is an isomorphism to an open subset of X r . We say that a point x ∈ X is a very general point if the set {x} is in very general position. Remark 2.2. For every family of integral curves on X, the set of r-ples (x1 , . . . , xr ) ∈ X r for which the condition of the deﬁnition is not satisﬁed by some curve of the family is contained in a proper closed subvariety of X r . We conclude that the complement of the locus in X r of points corresponding to subsets of X in very general position is a countable union of proper closed subvarieties. Any subset of a set of points in very general position is a set of points in very general position. Given a set of points in very general position, the image of any of the points of the set on the blowing-up of the surface at the residual points of the set is a very general point. The blow-up of P2 at at most 8 points in general position is a Del Pezzo surface. This surface contains ﬁnitely many (−1)-curves, and these are the only integral curves with negative self-intersection. The picture in the case of 9 points is well known too [Na2, Proposition 12]; a proof of the following statement will also be given below. Proposition 2.3. Every integral curve with negative self-intersection on the blowup of P2 at a set of 9 points in very general position is a (−1)-curve. Very little is known on the cone of eﬀective curves of the blow-up of P2 at more than 9 points. If we restrict our attention to rational curves, then it is easy to get the desired statement. The following proposition can be viewed as the ﬁrst step with respect to another formulation of the SHGH conjecture, due to Harbourne [Ha4], which says that if C is any irreducible and reduced curve on the blow-up of P2 at very general points, then C 2 ≥ pa (C) − 1, where pa (C) is the arithmetic genus of C. Proposition 2.4. Let Y be the blow-up of P2 at a set of points in very general position, and assume that C is a integral rational curve on Y with C 2 < 0. Then C is a (−1)-curve. Proof. To prove that C is a (−1)-curve, it suﬃces to show that KY · C < 0, as the conclusion then follows by adjunction. Let B ⊂ P2 be the image of C. We can assume that B is a curve, otherwise C would be exceptional, hence a (−1)-curve.

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Let p1 , . . . , pr ∈ P2 be the centers of the blow-up, and let mi = multpi B. We can assume that m1 > 0. For short, let p = (p1 , . . . , pr ), By the deﬁnition of very general position, the pair (B, p) belongs to an irreducible algebraic family {(Bt , pt ) | t ∈ T }, where pt = (p1,t , . . . , pr,t ) ∈ (P2 )r , Bt ⊂ X is an integral rational curve with multpi,t Bt = mi for every t ∈ T , and the morphism ψ : T → (P2 )r given by ψ(t) = pt is an isomorphism to an open subset of (P2 )r . Fix r points q1 , . . . , qr in general position on a smooth cubic Γ ⊂ P2 . Let Z ⊂ (P2 )r be a smooth, irreducible curve passing through p and q = (q1 , . . . , qr ), and consider the open set U = Z ∩ ψ(T ) of Z. Note that U is not empty, and that q is in the closure of U . The family {(Bt , pt ) | t ∈ ψ −1 (U )} determines an eﬀective divisor B ⊂ P2 × U , whose restriction to P2 × {pt } (for every t ∈ ψ −1 (U )) is the divisor Bt . Viewing B as a scheme, we take its ﬂat closure B inside P2 × Z, and let B0 be the restriction of B to P2 × {q}. Then B0 is an eﬀective divisor on P2 (see for instance [Har, Example III.9.8.5]). Since multpi,t Bt = mi for every pt ∈ U , we have multqi B0 ≥ mi by the semi-continuity of the multiplicity. Moreover, since Bt is a rational curve for every t ∈ ψ −1 (U ), so is every irreducible component of B0 , hence Γ is not contained in the support of B0 . Thus −KP2 · B = Γ · B0 ≥

r

mi .

(2.1)

i=1

If this inequality is strict, then we have KY ·C = KP2 ·B + mi < 0. Therefore, to conclude the proof, it is enough to show that the inequality in (2.1) is strict. Suppose this is not the case. Then multqi B0 = mi and OP2 (B0 )|Γ = OΓ ( mi qi ). Moreover, by the way we chose the qi , we can ﬁx a diﬀerent deformation in which q1 is replaced by another general point q1 of Γ while the other qi are kept the same, obtaining in this way another curve B0 . Since the previous arguments also apply to B0 , we have multq1 (B0 ) = m1 and multqi (B0 ) = mi for i ≥ 2, and moreover r r

mi qi = OP2 (B0 )|Γ = OP2 (B0 )|Γ = OΓ m1 q1 + mi q i . OΓ m1 q1 + i=2

i=2

q1 ))

This implies that OΓ (m1 (q1 − = OΓ . But Γ is an elliptic curve, and after ﬁxing q1 as zero, we know that there are ﬁnitely many m1 -torsion points on Γ, a contradiction.

Proof of Proposition 2.3. Let Y be the blow-up of P2 at a set of 9 points in very general position, and suppose that C ⊂ Y is an integral curve with C 2 < 0. Since −KY is nef, we have KY · C ≤ 0. By adjunction, we conclude that C is rational, hence it is a (−1)-curve by Proposition 2.4.

The following is the main result of this paper. Choosing S = P2 in the statement gives the theorem stated in the introduction. Theorem 2.5. Let S be a smooth projective surface, and let f : Y → S be the blowing up of S at a set Σ of points in very general position. Let C ⊂ Y be an

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integral curve with negative self-intersection, and assume that f (C) is a curve with a singularity of multiplicity 2 at one of the points of Σ. Then C is a (−1)-curve of Y . Proof. Let p ∈ Σ be the double point of f (C) whose existence we are assuming, and let X be the blow-up of S at Σ \ {p}. Then f factors through the blow-up g : Y → X of the image x ∈ X of p. Let D = g(C). Note that multx D = 2. Since x is a very general point of X, there exists an irreducible algebraic family of integral curves with marked points {(Dt , xt ) | t ∈ T } ⊂ X × X with D = Dt∗ and x = xt∗ for some t∗ ∈ T , such that the morphism ψ : T → X given by ψ(t) = xt is an isomorphism to an open subset of X and, moreover, multxt Dt = 2 for all t ∈ T . The Kodaira-Spencer map induced by any 1dimensional degeneration t → t∗ inside T deﬁnes a section of the normal bundle N := ND/X of D in X. Bearing in mind that we are dealing with a family of irreducible and reduced curves with marked singularities, these sections are non-zero whenever the degeneration t → t∗ is performed along a curve of T that is smooth at t∗ . As in the proof of [EL, Lemma 1.1], we reduce to a local computation in some open set Ω in C2 . We ﬁx local coordinates u = (u1 , u2 ) in Ω. Let f = f (u) be the holomorphic function locally deﬁning D. Let us start considering the case in which p is an ordinary node. Writing f as a power series centered at (0, 0), we can assume that the coordinates are chosen so that f (u) = u1 u2 + (higher degree terms).

(2.2)

We can reduce to the case when T is a small disk in C2 , with t∗ = (0, 0), and ﬁx coordinates t = (t1 , t2 ) in T such that ti = ui ψ. The total space of the deformation is deﬁned in Ω × T by a power series F = F (u, t). The deformation determines a Kodaira-Spencer map ρ : Tt∗ (T ) → H 0 (D, N ), which is non-trivial by our previous assumptions. This map is locally deﬁned by ) ∂ ∂ ∂F ∂F ) + λ2 (u, 0) + λ2 (u, 0) ) =: τλ . (2.3) ρ λ1 = λ1 ∂t1 ∂t2 ∂t1 ∂t2 C In view of the linearity of this map, the sections τλ ﬁll up a non-trivial linear subspace in H 0 (D, N ) as λ varies in C2 . Mimicking [EL], we consider the function Φ(u, t) := F (u + x(t), t), where x(t) = (x1 (t), x2 (t)) are the coordinate of the marked point xt of Ct . Note that Φ ∈ (u1 , u2 )2 for all t. We expand Φ as a power series in (t1 , t2 ). The coeﬃcients of the two terms of degree 1 are equal to ∂f ∂x1 ∂f ∂x2 ∂F ∂Φ = (u) · (0) + (u) · (0) + (u, 0), ∂ti ∂u1 ∂ti ∂u2 ∂ti ∂ti

i = 1, 2.

(2.4)

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These are functions of u, and both are contained in (u1 , u2 )2 . Note that ∂xj /∂ti (0) = δij . Then, by combining (2.2), (2.3) and (2.4), we see that (λ1 u2 + λ2 u1 )|C + τλ ∈ m2x ,

(2.5)

where mx is the maximal ideal of x in D. Note that (λ1 u2 + λ2 u1 )|C ∈ mx . We conclude that τλ ∈ H 0 (D, N ⊗ mx ). Then τλ gets pulled back by g|C : C → D to a section σλ of (g|C )∗ N that vanishes at the two pre-images of x. After suitably denoting these two pre-images by y1 and y2 , we actually get sections σ(1,0) ∈ H 0 (C, (g|C )∗ N ⊗ m2y1 ⊗ my2 ) and σ(0,1) ∈ H 0 (C, (g|C )∗ N ⊗ my1 ⊗ m2y2 ) when λ ∈ {(1, 0), (0, 1)}. This implies that deg(Div(σλ )) ≥ 3 for every λ = 0. Since D2 < 4, this yields deg(Div(σλ )) = 3, that is D2 = 3, and thus C 2 = −1. In fact, we observe that the linear system | Div(σλ )| contains a pencil parameterized by λ. This pencil has base points at y1 and y2 and movable part of degree 1, which deﬁnes an isomorphism to P1 . It remains to discuss the case when x is not an ordinary node of D. We now explain why this case cannot occur. Using analogous notation as in the previous discussion, we get the following local equation of D: f = u21 + (higher degree terms). Arguing as before, we see this time that τ(0,1) ∈ H 0 (C, N ⊗ m2x ). This implies that

deg(Div(σ(0,1) )) ≥ 4, which is impossible.

References [AH] J. Alexander and A. Hirschowitz, Polynomial interpolation in several variables, J. Algebraic Geom. 4 (1995), 201–222. [Bi] P. Biran, Symplecting packing in dimension 4, Geom. Funct. Anal. 7 (1997), 420–437. [CP] F. Campana and T. Peternell, Algebraicity of the ample cone of projective varieties, J. Reine Angew. Math. 407 (1990), 160–166. [CM1] C. Ciliberto and R. Miranda, Degenerations of planar linear systems, J. Reine Angew. Math. 501 (1998), 191–220. [CM2] C. Ciliberto and R. Miranda, Linear systems of plane curves with base points of equal multiplicity, Trans. Amer. Math. Soc. 352 (2000), 4037–4050. [dVL] C. de Volder and A. Laface, Linear systems on generic K3 surfaces, to appear in Bull. Belg. Math. Soc., math.AG/0309073. [EL] L. Ein and R. Lazarsfeld, Seshadri constants on smooth surfaces, Ast´erisque (1993), no. 218, 177–186, Journ´ees de G´eom´etrie Alg´ebrique d’Orsay (Orsay, 1992). [Gi] A. Gimigliano, On linear systems of plane curves, Ph D thesis, Queen’s University (1987).

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[Ha1] B. Harbourne, The geometry of rational surfaces and Hilbert functions of points in the plane, Can. Math. Soc. Conf. Proc. 6 (1986), 95–111. [Ha2] B. Harbourne, Points in good position in P2 , in Zero-Dimensional Schemes, Proceedings of the International Conference, Ravello, 1992 (F. Orecchia and L. Chiantini Eds.), de Gruyter, pp. 213–229, 1994. [Ha3] B. Harbourne, Rational surfaces with K 2 > 0, Proc. Amer. Math. Soc. 124 (1996), 727–733. [Ha4] B. Harbourne, The (unexpected) importance of knowing α, preprint. [Har] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977. [Hi] A. Hirschowitz, Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles g´en´eriques, J. Reine Angew. Math. 397 (1989), 208–213. [Ko] J. Koll´ ar, Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer-Verlag, Berlin, 1996. [Laz] R. Lazarsfeld, Positivity in Algebraic Geometry, I, Ergeb. Math. Grenzgeb. (3) 48, Springer-Verlag, Berlin, 2004. [MP] D. McDuﬀ and L. Polterovich, Symplecting packings and algebraic geometry, With an appendix by Y. Karshon, Invent. Math. 115 (1994), 405–434. [Mi] R. Migon, Syst`eme de courbes planes ` a singularit´es impos´ees: le cas des multiplicit´es inf´erieures ou ´egales ` a quatre, J. Pure Appl. Algebra 151 (2000), 173–195. [Na1] M. Nagata, On the 14-th problem of Hilbert, Amer. J. Math. 81 (1959), 766–772. [Na2] M. Nagata, On rational surfaces, II, Mem. Coll. Sci. Univ. Kyoto, Ser. A, Vol. XXXIII, Math. No. 2 (1960), 271–293. [Se] B. Segre, Alcune questioni su insiemi ﬁniti di punti in Geometria Algebrica, Atti del Convegno Internazionale di Geometria Algebrica, Torino (1961). [Ya] S. Yang, Linear series in P2 with base points of bounded multiplicity, preprint, math.AG/0406591. [Xu] G. Xu, Curves in P2 and symplectic packings, Math. Ann. (1994), 609–613. Tommaso de Fernex Department of Mathematics University of Michigan East Hall, 530 Church Street Ann Arbor, MI 48109-1109, USA Email: [email protected]

Linear congruences and hyperbolic systems of conservation laws Pietro De Poi and Emilia Mezzetti∗

Abstract. S. I. Agafonov and E. V. Ferapontov have introduced a construction that allows naturally associating to a system of partial diﬀerential equations of conservation laws a congruence of lines in an appropriate projective space. In particular hyperbolic systems of Temple class correspond to congruences of lines that place in planar pencils of lines. The language of Algebraic Geometry turns out to be very natural in the study of these systems. In this article, after recalling the deﬁnition and the basic facts on congruences of lines, Agafonov-Ferapontov’s construction is illustrated and some results of classiﬁcation for Temple systems are presented. In particular, we obtain the classiﬁcation of linear congruences in P5 , which correspond to some classes of T -systems in 4 variables. 2000 Mathematics Subject Classiﬁcation: Primary 14M15, 35L65 Secondary 53A25, 53B50

Introduction Linear congruences of lines in Pn are the (irreducible) subvarieties of dimension n+1 n − 1 of the Grassmannian G(1, n), embedded in P( 2 )−1 by the Pl¨ embeducker ding, obtained by the intersection with a linear space of dimension n2 . Such a congruence of lines B has, in general, order one, i.e. through a general point in Pn there passes only one line of B. Moreover the family of lines parameterized by B can be characterized as the set of the (n − 1)-secant lines of the fundamental locus Φ ⊂ Pn , where Φ is deﬁned by the property that through a point in it there pass inﬁnitely many lines of B. This article deals with a recent discovered application of congruences of lines to mathematical physics (precisely to hyperbolic systems of conservation laws) due to S. I. Agafonov and E. V. Ferapontov: see [AF96] and [AF99]. More prei ∂f i (u) = 0, cisely, to a system of conservation laws, which has the form ∂u ∂t + ∂x with i = 1, . . . , n − 1, they associate an (n − 1)-parameter family B of lines in Pn , ∗ This research was partially supported by the DFG Forschungsschwerpunkt “Globale Methoden in der Komplexen Geometrie” for the ﬁrst author, by MIUR, project “Geometria sulle variet` a algebriche” for the second author and by funds of the University of Trieste (fondi 60%) for both.

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deﬁned by the parametric equations yi = ui λ−f i (u)μ, i = 1, . . . , n−1 and y0 = λ, yn = μ, where (λ : μ) ∈ P1 are the parameters of a line of B, u = (u1 , . . . , un−1 ) are the (local) parameters of B and (y0 : · · · : yn ) are the homogeneous coordinates on Pn . It turns out that with this correspondence the basic concepts of the theory of the systems of conservation laws acquire a clear and simple projective interpretation. For instance, for a particular class of systems of conservation laws, the so called T -systems (see [AF99] and [AF01]), their corresponding family of lines B is characterized by the fact that the lines of B passing through a point of its focal locus form a planar pencil of lines; moreover, a reciprocal transformation (see [AF01] for a deﬁnition) of one of these systems of conservation laws corresponds to a projectivity in Pn , and vice versa. Therefore, the classiﬁcation of the T -systems is equivalent to the study of these families of lines B. Classically, the study of congruences of lines in P3 was started by E. Kummer in [Kum66], in which he gave a classiﬁcation of those of order one. More recently Z. Ran in [Ran86] studied the surfaces of order one in a general Grassmannian G(r, n), i.e. families of r-planes in Pn for which the general (n − r − 2)-plane meets only one element of the family. He gave a classiﬁcation of such surfaces, obtaining in particular, in the case of P3 , a modern and more correct proof of Kummer’s classiﬁcation. The congruences of lines in P4 , with special regard to those of order one, have been considered by G. Marletta in [Mar09a] and [Mar09b]. The classiﬁcation of the linear ones has been given by G. Castelnuovo in [C91], where a detailed description of these particular subvarieties of G(1, 4) is obtained. These classical results in P4 have been analysed and extended by P. De Poi in [DP01], [DPi], [DPii], and [DPiii]. A general fact about linear congruences in Pn is that the lines of the family passing through a general focus form a linear pencil and the plane of this pencil cuts the focal locus residually along a plane curve of degree n − 2. So linear congruences always deﬁne Temple systems. Conversely for n ≤ 4 in [AF01] it has been proved that all families of T -systems are even algebraic, and more precisely linear congruences. In higher dimensions nothing is known, in particular also a complete classiﬁcation of the linear congruences of lines is still missing. So in this paper we have initiated a systematic study of the more simple unknown case, that of linear congruences in P5 . For general linear congruences in P5 , the focal locus is a smooth Palatini threefold, which is a scroll over a cubic surface S in P3 (see [O92], [FM02]). This surface S can be realized as follows: let the congruence B be deﬁned as G(1, 5) ∩ Δ, for a 10-dimensional linear space Δ. The dual of the Grassmannian is a cubic hypersurface (the “Pfaﬃan”, see Section 4), then S is naturally identiﬁed with the ˇ 5) with the dual of Δ. Classifying linear congruences in P5 intersection of G(1, ˇ with respect to G(1, ˇ 5) amounts to describe all special positions of the 3-space Δ ˇ 5), the focal locus and to its singular locus. For instance, when S meets Sing G(1, acquires some linear irreducible components. Particularly interesting are the cases when S splits: the description of these congruences relies on a recent classiﬁcation

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of the linear systems of 6 × 6 skew-symmetric matrices of constant rank 4 up to the natural action of the projective linear group PGL6 ([MM04]). This article is structured as follows: in Section 1, the basic deﬁnitions connected to congruences of lines in the algebraic setting are given. In Section 2 we recall some deﬁnitions and results about systems of conservation laws, reciprocal transformations and systems of Temple class. In Section 3 the correspondence between systems of conservation laws and families of lines is illustrated, with special regard to systems of conservation laws of Temple class. In Section 4 we collect some general facts about linear congruences. Finally, in Section 5 we study the linear congruences in P5 . We ﬁrst consider the case in which the cubic surface ˇ 5), these points S has one or more singular points on the singular locus of G(1, correspond to some 3-dimensional linear spaces that enter in the focal locus. We then study the congruences such that the surface S is reducible, getting four types of congruences. In some cases the focal locus has a parasitic component, i.e. an irreducible, maybe embedded component, which is not met by a general line of B. In a forthcoming paper we plan to apply these results to the classiﬁcation of Temple systems in 4 variables. We have to point out that the classiﬁcation considered here holds over an algebraically closed ﬁeld, so it will be necessary to reﬁne it over the real ﬁeld. Acknowledgements. We wish to thank Jenya Ferapontov for introducing us to this beautiful connection between algebraic geometry and partial diﬀerential equations. We also thank Dario Portelli and Giorgio Tondo for interesting discussions.

1. Notation, deﬁnitions and preliminary results In the realm of algebraic geometry, we will work with schemes and varieties over C, with standard notation and deﬁnitions as in [Har77]. A variety will always be projective. We refer to [DP01] and [DP03] for general results and references about families of lines, focal diagrams and congruences, and to [GH78] for notations about Schubert cycles. In particular we denote by σa0 ,a1 the Schubert cycle of the lines in Pn contained in a ﬁxed (n − a1 )-dimensional subspace H ⊂ Pn and which meet a ﬁxed (n − 1 − a0 )-dimensional subspace Π ⊂ H. Deﬁnition 1.1. A congruence of lines B in Pn is a ﬂat family of lines of dimension n − 1, and we can think of it as an (n − 1)-dimensional subvariety of the Grassmannian G(1, n). Its order a0 is the number of lines passing through a general point in Pn . Throughout this article, we will denote by Λ := {(b, P ) | P ∈ Λ(b)} the incidence correspondence associated to the family B, with its natural projections f and p to Pn and to B respectively. If b ∈ B, then Λ(b) := f (Λb ) denotes the line parametrised by b and Λb := p−1 (b) the corresponding subset of Λ. We can

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summarize all in the diagram: f

Λb ⊂ Λ ⊂ B × Pn −−−−→ Pn ⊃ Λ(b) ⏐ ⏐ ⏐ ⏐p + +

(1.1)

b ∈ B. Note that f is surjective if and only if a0 > 0. In this case f is a map of degree a0 . The basic objects when one deals with congruences are the focal and fundamental loci. Deﬁnition 1.2. Let B be a congruence of lines; then the focal divisor of the family B is the ramiﬁcation divisor R ⊂ Λ of f : Λ → Pn ;

(1.2)

the schematic image F of R under f is called the focal locus: F = f (R) ⊂ Pn . The foci of B are the branch points of f . The fundamental locus Φ is the set of points y contained in more lines of the family than expected: dim f −1 (y) > n − dim f (Λ). Example 1.3. The family of the tangent lines to a curve C (which is not a line) in P2 is the most simple example of a congruence. Other examples are the family of the secant lines to a curve in P3 or the one of the tangent lines to a surface in P4 . Remark 1.4. The fundamental locus Φ is in general properly contained in the focal locus F , but if B is a congruence of order one, then Φ = F and the codimension of F in Pn is ≥ 2. The following recent result gives to converse statement: Theorem 1.5. (F. Catanese, P. De Poi, 2004, [DP04]) Let B be a congruence such that the fundamental locus Φ coincides (set-theoretically) with the focal locus F ; then the order of B is zero or one. Remark 1.6. It is important to note that the focal locus F often has some unexpected components. For example, let B be the congruence of the secant lines to a curve C in P3 . If L is a line meeting C at two points x, y such that the tangent lines to C at x and y are incident, then the tangent plane to B at the point corresponding to L is contained in the Grassmannian and L ⊂ F . Since a curve in P3 has in general a one-dimensional family of secant lines of this type (called stationary secants), the congruence B will have a focal surface. The only curve in P3 without stationary secant lines is the twisted cubic. For more details, see [ABT].

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The focal locus of a congruence may also have a component which is not met by a general line of B. Such a component is called a parasitic component. Some explicit examples will be described in Sections 4 and 5. From now on, we assume that f is surjective. In this case, the fundamental locus is the set of points P ∈ Pn for which the ﬁbre of the map f : Λ → Pn has positive dimension. Theorem 1.7. (C. Segre, 1888, [Seg88], C. Ciliberto - E. Sernesi, 1992, [CS92]) With notations as above, given a congruence B, for the general b ∈ B, the corresponding line Λ(b) ⊂ Pn contains exactly n − 1 foci (counting multiplicities) which are foci for the line Λ(b). Otherwise Λ(b) ⊂ F . Moreover, if dim(F ) = n − 1, then Λ(b) is tangent to F at its (smooth) focal non-fundamental points. Let us now give some interesting and useful examples of ﬁrst order congruences. • Congruences of order one in P3 (E. Kummer, 1866, [Kum66], Z. Ran, 1986, [Ran86]): these are all classiﬁed and can be divided in three cases: 1. F is a skew cubic, B is the family of its secant lines. In the Pl¨ ucker embedding, B is a Veronese surface; 2. F = C ∪ L, C is a rational curve of degree d, L is a (d − 1)-secant line of C, B is the family of lines meeting both C and L; 3. F is a double structure on a line L, B is a union of pencils of lines with centres at the points of L. • Linear congruences in P4 (G. Castelnuovo, 1891, [C91]): if B = G(1, 4) ∩ P6 , then F is a projected Veronese surface or a degeneration of it, B is the family of the trisecant lines of F . The order of B is 1. The lines in B through a general point of F form a planar pencil and cut a conic on F . • (n − 1)-secant lines of smooth codimension two subvarieties in Pn (P. De Poi, 2003, [DP03]): the congruences of order one formed by these lines are completely described, they are all contained in Pn , with n ≤ 5.

2. Systems of conservation laws In the realm of mathematical physics, we will work over R. All the functions are—at least—C 1 . Deﬁnition 2.1. A system of conservation laws is a quasi-linear system of ﬁrst order partial diﬀerential equations of the form ∂ui ∂f i (u) + = 0, ∂t ∂x

i = 1, . . . , n − 1

(2.1)

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where u(x, t) = (u1 (x, t), . . . , un−1 (x, t)) are the unknown functions and the f i (u)’s are functions deﬁned over a domain Ω ⊂ Rn−1 . The system can be written: uit + Σj

∂f i (u) j u = uit + Jf (u) · ux = 0, ∂uj x

i = 1, . . . , n − 1

(2.2)

where Jf denotes the Jacobian matrix of f . The system is called hyperbolic (resp. strictly hyperbolic) if all the eigenvalues of Jf are real (resp. real and distinct). Deﬁnition 2.2. If the system (2.1) is strictly hyperbolic, the eigenvalues λ1 (u) < λ2 (u) < · · · < λn−1 (u) are called characteristic velocities. The integral trajectories γi of the ﬁelds of eigenvectors vi are called the rarefaction curves: γ˙ i (t) = vi (γi (t)). Remark 2.3. Given a strictly hyperbolic system of conservation laws as above, through a point u ∈ Ω ⊂ Rn−1 there pass n − 1 rarefaction curves. Indeed each eigenvector of the eigenvalue λi (u), for i = 1, . . . , n − 1, has the direction of the tangent line to the curve γi .

2.1. Temple systems. These systems, which are strictly hyperbolic, were introduced by B. Temple in [Tem83]. They naturally arise in the theory of equations of associativity of 2D topological ﬁeld theory (see [Dub96]). Deﬁnition 2.4. A strictly hyperbolic system (2.1) is said to be linearly degenerate if Li (λi )(u) = 0

∀i

where Li is the Lie derivative in the direction of vi . This means that the eigenvalue λi is constant along the rarefaction curve γi . Deﬁnition 2.5. A strictly hyperbolic system (2.1) is said to be a Temple system or a T -system if it is linearly degenerate and the rarefaction curves are lines in the coordinates (u1 , . . . , un−1 ) (see [Tem83]).

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2.2. Reciprocal transformations. Let us deﬁne two new independent variables (X, T ) by

αi (ui dx + f i (u)dt) + μdx + νdt, dX :=

(2.3)

i

dT :=

α ˜ i (ui dx + f i (u)dt) + μ ˜dx + ν˜dt.

(2.4)

i

By (2.1) the two 1-forms on the right are closed, so (2.1) takes the new form ∂F i (U) ∂U i + = 0, ∂T ∂X

i = 1, . . . , n − 1

(2.5)

Deﬁnition 2.6. Transformations of the form (2.3), (2.4) are called reciprocal. Reciprocal transformations are known to preserve the class of T -systems (see [AF96]).

3. The correspondence Agafonov and Ferapontov ([AF96] and [AF99]) associate to a system of conservation laws (2.1) an (n − 1)-parameter family B of lines in Pn , i.e. a congruence, deﬁned by the parametric equations ⎧ ⎪ ⎨y0 = λ, yi = ui λ − f i (u)μ, i = 1, . . . , n − 1, ⎪ ⎩ yn = μ where • (λ : μ) ∈ P1 are homogeneous coordinates on a line of B; • u = (u1 , . . . , un−1 ) are the (local) parameters of B; • (y0 : · · · : yn ) are homogeneous coordinates on Pn . So for every u in the domain Ω, the above equations deﬁne a line Λ(u). A dictionary can be written translating properties of the system (2.1) to properties of the family of lines, and conversely. For example, reciprocal transformations of the system correspond to projectivities in Pn . The eigenvalues λ1 (u), . . . , λn−1 (u) are in natural bijection with the foci of B on the line Λ(u). In particular, the system is strictly hyperbolic if and only if on a general line of B there are n − 1 distinct foci of the congruence. The rarefaction curves correspond to developable ruled surfaces in Pn .

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3.1. Temple systems again. In general for a hyperbolic system of conservation laws the corresponding focal locus of the congruence B is a hypersurface F and the lines of the family are tangent to F at the n − 1 foci. For a Temple system the situation is diﬀerent. Theorem 3.1. ( [AF96]) The congruences of lines B associated to Temple systems are characterized by the properties that every focus is a fundamental point and that the rarefaction curves correspond to planar pencils of lines of B. By Theorem 1.5 it follows that dim F = n − 2 and the order of B is one. The classiﬁcation of the T -systems up to reciprocal transformations is equivalent to the classiﬁcation of non-necessarily algebraic congruences of lines of order one with planar pencils. Example 3.2. The wave equation. The equation ftt = fxx can be rewritten as a system of two conservation laws ( u1t = u2x u2t = u1x . The associated congruence B in P3 is formed by the lines meeting two ﬁxed skew lines L and L . On each line Λ of B there are two distinct foci, its intersections with L and L . This is also an example of a Temple system. The associated congruence of lines is linear. It is rather easy to prove (see Proposition 4.2) that all linear congruences, in any projective space, are associated to some Temple system. Conversely: Theorem 3.3. ( [AF96], [AF01]) All Temple systems in 2 and 3 variables give rise to linear congruences. In particular they are all algebraic. Agafonov and Ferapontov have conjectured that congruences of lines whose developable surfaces are planar pencils of lines have always algebraic focal varieties (possibly reducible and singular).

4. Linear congruences of lines As we said in the introduction, a linear congruence of lines B in Pn has the form B = G(1, n) ∩ Δ, where Δ is a linear subspace of dimension n2 of P(∧2 V ), with V := H 0(OPn(1))∗ , the space of the Pl¨ ucker embedding of the Grassmannian. The n+1 coordinates of a line ∈ G(1, n) can be interpreted as entries of a 2 skew-symmetric (n+1)×(n+1) matrix (pij )i,j=0,...,n of rank 2. A hyperplane H in P(∧2 V ) has an equation of the form Σni,j=0 aij pij = 0, and the aij ’s are coordinates of H as a point in the dual space P(∧2 V ∗ ). Clearly, also the dual coordinates aij

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can be interpreted as entries of a skew-symmetric matrix A. Δ is the intersection of the n − 1 hyperplanes H1 , . . . , Hn−1 with equations: n

n

···

a1ij pij = 0,

i,j=0

an−1 ij pij = 0

(4.1)

i,j=0

associated to matrices A1 , . . . , An−1 . In the dual space H1 , . . . , Hn−1 generate the ˇ dual (n − 2)-space Δ. Some general results about fundamental varieties of linear congruences are given in [BM01]; in particular, it is proved that the focal locus of a linear congruence B is the degeneracy locus F of a morphism of sheaves of the form ⊕(n−1)

→ ΩPn (2).

(4.2)

H 0 (ΩPn (2)) ∼ = (∧2 V )∗ ,

(4.3)

φ : OPn

Explicitly, there is an isomorphism:

and so a global section of ΩPn (2) is a skew-symmetric matrix of type (n+1)×(n+1) with entries in the base ﬁeld. Then, the morphism φ in (4.2) is deﬁned by the n− 1 skew-symmetric matrices A1 , . . . , An−1 . The corresponding degeneracy locus F in Pn is deﬁned by the equations n−1

λi Ai [X] = 0

(4.4)

i=1

for some [λ] = (λ1 , . . . , λn−1 ) = (0, . . . , 0), where [X] denotes the column matrix of the coordinates. Since the matrices A1 , . . . , An−1 are skew-symmetric, the situation changes whether n is even or odd. Proposition 4.1. ( [DP03]) If F is the focal locus of a linear congruence B in Pn , then 1. for B general, F is smooth if dim(F ) ≤ 3; 2. if n is even, for each [λ] ∈ Pn−2 equation (4.4) has at least one solution, and F is rational; n−1 3. if n is odd, the vanishing of the Pfaﬃan of the matrix i=1 λi Ai deﬁnes a hypersurface Z of degree (n + 1)/2 in Pn−2 (in which λ1 , . . . , λn−1 are the coordinates). Furthermore, if φ is general, for a ﬁxed point [λ] ∈ Z, equation (4.4) has a line contained in F as solution, and F results to be a scroll over (an open set of ) Z. Besides, in both cases deg(F ) =

n2 − 3n + 4 . 2

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Another known result, which can be deduced by the above description of the focal locus of a linear congruence and which explains the fact that linear congruences give Temple systems is the following: Proposition 4.2. Let B be a linear congruence in Pn , with focal locus F ⊂ Pn ; then B has order one and is the closure of the family of the (n − 1)-secant lines to F . Moreover, if P is a general point in F , the family of the lines of B through P is a pencil, whose plane intersects F , out of P , in a curve of degree n − 2. In particular, a linear congruence corresponds to a Temple system. Proof. The ﬁrst assertion follows from standard Schubert calculus and the second one can be deduced from Theorems 1.5 and 1.7. We give here a diﬀerent direct proof of both facts which relies on the previous observations. From equation (4.4) (or equations (4.1)) we infer that through the general point P ∈ Pn with coordinates [X] = [x0 , . . . , xn ] there passes only the line of B whose coordinates are the maximal minors of 1 ⎞ ⎛ 1 ... i a0i xi i ani xi ⎟ ⎜ .. A := ⎝ (4.5) ⎠. . n−1 n−1 i a0i xi i ani xi Moreover, if we think of A as a matrix with linear entries in Pn , the degeneracy locus of A is F . Now, we can ﬁx one line ∈ B and without loss of generality we can suppose that it does not intersect the (n − 2)-dimensional space deﬁned by x0 = x1 = 0; then F ∩ is deﬁned by the determinant of the submatrix of A formed by the last (n − 1) columns, and therefore – if is general – F ∩ is a zero dimensional scheme of length (n − 1). Now, if P is a general focal point, by the linearity the lines of B through it form a pencil P : in fact in equation (4.4) we can suppose that A1 [X] = 0 and A2 [X] = 0, . . . , An−1 [X] = 0, i.e. P is the centre of the linear complex A1 . Then ucker coordinates are given by the lines of P are contained in the plane whose Pl¨ the (n − 2) × (n − 2)-minors of ⎞ 2 ⎛ 2 ... i a0i xi i ani xi ⎟ ⎜ .. A := ⎝ ⎠. . n−1 n−1 i a0i xi i ani xi The plane πP of the pencil intersects F in P and in a curve of degree (n − 2): in fact πP intersects the hypersurface V deﬁned by a (ﬁxed) minor of (4.5) in a curve of degree (n − 1) which splits in the line of the congruence through P and contained in V and residually in a curve which must be contained in F .

ˇ n) parametrises the tangent hyThe dual variety of the Grassmannian G(1, perplanes to G(1, n). It is deﬁned by the maximal Pfaﬃans of the matrix A, ˇ n of degree n+1 , deﬁned by the therefore if n is odd, it is the hypersurface in P 2 Pfaﬃan, while if n is even, it has codimension 3. In the case n odd, using co-

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ˇ ˇ ˇ ordinates (λ1 , . . . , λn−1 ) in Δ, the intersection S := G(1, n) ∩ Δ is deﬁned by Pfaﬀ( i λi Ai ) = 0, hence it coincides with the hypersurface Z of Proposition 4.1. 4.0.1. Linear congruences in P3 and P4 . We end this section by brieﬂy recalling the classiﬁcation of the linear congruences in low dimensional projective spaces. In P3 , the situation is very simple: G(1, 3) ⊂ P5 is the Klein quadric, its dual is ˇ is a line. Correspondingly, we have the following cases: if Δ ˇ again a quadric, and Δ ˇ is general, it intersects G(1, 3) at two distinct points and the congruence represents ˇ can be tangent and therefore the join of the two corresponding lines. The line Δ ˇ intersects G(1, 3) in a double point. Correspondingly, we have a congruence which has as focal locus a double line (and the congruence is a subset of the set of lines ˇ 3): in this case, we do ˇ ⊂ G(1, meeting its support). Finally, it can happen that Δ not have a congruence, since the corresponding family of lines in P3 has dimension greater than three. In P4 , the situation is more complicated: G(1, 4) ⊂ P9 has dimension six and ˇ G(1, ˇ 4) = ∅: then the congruence is ˇ ∼ degree ﬁve. Δ = P2 and in the general case Δ∩ ˇ G(1, ˇ 4) is non-empty given by the trisecants to a projected Veronese surface. If Δ∩ and ﬁnite, then its length is at most 3. If it is a single point, this point gives us a focal plane: the congruence is given by the secant lines to a cubic scroll which ˇ 4) is two points, then we have the lines meeting ˇ ∩ G(1, meet this plane also. If Δ ˇ 4) is given by three ˇ ∩ G(1, two ﬁxed planes and which meet a quadric also. If Δ points, we get the lines meeting three planes; the focal locus has also a fourth component, which is parasitic, the plane spanned by the 3 points of intersection of the three planes two by two. Of course also limit cases of these are possible, ˇ ∩ G(1, ˇ 4) is a double point, etc. Finally, Δ ˇ ∩ G(1, ˇ 4) can be a curve or e.g. if Δ ˇ ˇ Δ ⊂ G(1, 4): several cases are possible but they don’t deﬁne a congruence, since the corresponding family of lines in P4 has dimension greater than four (see the original paper of Castelnuovo [C91]). The article [AF01] contains the interpretation of this classiﬁcation in terms of that of Temple systems in three variables up to reciprocal transformations.

5. Linear congruences in P5 A linear congruence of lines B in P5 is of the form B = G(1, 5) ∩ Δ, where Δ is a linear space of dimension 10, intersection of 4 hyperplanes with equations: 5

i,j=0

aij pij = 0,

5

i,j=0

bij pij = 0,

5

i,j=0

cij pij = 0,

5

dij pij = 0

(5.1)

i,j=0

associated to skew-symmetric 6 × 6 matrices A, B, C, D. ˇ 5) in P ˇ 14 The dual variety of the Grassmannian is the cubic hypersurface G(1, ˇ deﬁned by the Pfaﬃan. One can think of G(1, 5) as the locus of skew-symmetric

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matrices (in the (aij )’s coordinates) of rank at most four. Using coordinates ˇ 5) ∩ Δ ˇ the intersection S := G(1, ˇ is deﬁned by Pfaﬀ(aA + bB + (a, b, c, d) in Δ, cC + dD) = 0, hence, in general, it is a cubic surface. ˇ 5) G(1, 5) associates to a tangent hyThe rational Gauss map γ : G(1, ˇ 5)), which is perplane its unique tangency point. It is regular outside Sing(G(1, naturally isomorphic to G(3, 5) and corresponds to the matrices of rank two (which indeed is dually isomorphic to G(1, 5) also). If A is such a matrix, the corresponding 3-space πA is the projectivised kernel of A. As a hyperplane section of G(1, 5), A represents the lines in P5 meeting πA . In what follows we will always identify ˇ 5) with G(3, 5). Sing G(1, For general Δ, the image γ(S) is a 2-dimensional family of lines, whose union is a smooth 3-fold F of degree 7 and sectional genus 4, called Palatini scroll. The lines of the congruence B are the 4-secant lines of F and F is the focal locus of B (see also [O92], [BM01], [FM02] and [DP03]). More in general: Proposition 5.1. Let Δ be a linear space of dimension 10 in P14 such that S := ˇ ∩ G(1, ˇ 5) is a reduced cubic surface and S ∩ G(3, 5) is empty. The lines b ∈ γ(S) Δ span (set-theoretically) the focal divisor F of the linear congruence B = G(1, 5)∩Δ, i.e. the map f of (1.2) drops rank exactly at the pairs (b, P ) such that b ∈ γ(S) and P ∈ Λ(b). The lines of B are the 4-secants of F . Proof. Since γ(S) ⊂ B is the set of points whose tangent hyperplanes (to G(1, 5)) contain Δ(= B ), we have one inclusion recalling that for these points the global characteristic map—see for example Deﬁnition 1 of [DP01]—of the congruence drops rank. For the other inclusion, given a point P ∈ F ∩ Λ(b), by deﬁnition, through a smooth b ∈ B there passes at least one tangent direction to G(1, 5) \ B contained in Δ: we can ﬁnd—by dimensional reasons—a tangent hyperplane of P

containing Δ also. The last assertion follows from Theorem 1.7. Classifying linear congruences in P5 amounts to describing all special positions ˇ 5) and to its singular locus. As we will see ˇ with respect to G(1, of the 3-space Δ the situation is rather complicated and several diﬀerent cases are possible. Remark 5.2. Actually, we are mainly interested in the case of “true” linear congruences, i.e. if the intersection B = Δ ∩ G(1, 5) is proper and the focal locus has dimension three. Therefore, in what follows we will not give many details on the cases such that B has dimension > 4 or F has dimension > 3. ˇ 5), then Δ ˇ is contained in G(1, ˇ Proposition 5.3. With the above notation, if Δ ˇ ˇ meets G(3, 5) and the focal locus F of B has dimension > 3. If Δ ∩ G(1, 5) is a cubic surface S, then dim F > 3 if and only if S intersects G(3, 5) at least along a curve. Proof. The ﬁrst claim follows from Corollary 11 of [MM04]. The second and the third one are Theorem 4.3 and Theorem 4.9 of [FM02].

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From now on, we will always assume that the congruence B is obtained from a ˇ 5) is proper i.e. a cubic surface S. ˇ ∩ G(1, 10-space Δ such that the intersection Δ From [FM02], Remark 4.4, it follows that every cubic surface in P3 can be realized in this way. If S is smooth, then F is always a Palatini scroll. For any S, possibly singular and/or reducible, such that dim F = 3, the Hilbert polynomial of F is PF (t) = 7/6t3 + 2t2 + 11/6t + 1. The equations of F can be written explicitly as maximal minors of the following 4 × 6 matrix (see [FM02]): ⎞ ⎛ i a0i xi . . . . i a5i xi ⎟ ⎜ i b0i xi . . . . i b5i xi ⎟ (5.2) M =⎜ ⎝ ⎠ i c0i xi . . . . i c5i xi . . . . i d0i xi i d5i xi that we will also write in the form ⎛ L10 L11 ⎜L20 L21 ⎜ ⎝L30 L31 L40 L41

. . . .

. . . .

. . . .

⎞ L15 L25 ⎟ ⎟. L35 ⎠ L45

(5.3)

The lines of B through a general point P in F form a linear pencil contained in a plane cutting F , out of P , along a plane cubic curve. The coeﬃcients of the equations of this plane are the lines of the matrix (5.2) computed at P .

It is well known that a cubic surface with isolated singularities can have at most 4 double points, or one triple point if it is a cone, and that an irreducible cubic with a singular curve is necessarily ruled with a double line. Observe also that if A is ˇ is tangent to G(1, ˇ 5) at A. We a singular point of S, then either A ∈ G(3, 5) or Δ will now consider the various possibilities for S, studying the linear congruences B and their focal loci. We will also give some explicit examples, constructed using CoCoA(see [CoCoA]). We begin by studying two special situations for the singularities of S, i.e. the cases when one or more singular points belong to G(3, 5) and when S is reducible not meeting G(3, 5).

5.1. S with singular points on G(3, 5). 5.1.1. Only one singular point. Let A be a singular point of S belonging to G(3, 5). In this case Δ is contained in the linear complex of the lines meeting the 3-space πA , hence each line of the congruence B intersects πA . We get that F splits as F = πA ∪ Y , where Y is an arithmetically Cohen-Macaulay (aCM for short) threefold of degree 6 and sectional genus 3, deﬁned by the maximal minors of a 3 × 4 matrix of linear forms. Indeed, if we choose a system of coordinates such that πA has equations x0 = x1 = 0, then the only non-zero coordinate of A is a01

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and the matrix (5.3) becomes ⎛ x1 ⎜L20 ⎜ ⎝L30 L40

−x0 L21 L31 L41

0 0 . . . . . .

⎞ 0 0 . L25 ⎟ ⎟. . L35 ⎠ . L45

(5.4)

One checks that πA is a component of F and the residual is deﬁned by the 3 × 3 minors of the matrix formed by the last 3 rows and 4 columns of (5.4). ˇ are general, then S is smooth outside A. Y results If the other generators of Δ to be a singular Bordiga scroll, with 6 singular points all belonging to Y ∩ πA , which is a smooth quadric surface. The foci on a general line of the congruence are contained one in πA and the other three in Y , so the lines of B are the trisecants of Y meeting also πA . The lines of B through a point of πA cut Y at the points of a plane cubic, whereas those through a point P in Y cut Y residually along a conic C and πA along a line L. 5.1.2. Two singular points. Assume now that S has also a second singular point B on G(3, 5). Then F is of the form F = πA ∪ πB ∪ Z, where Z is an aCM threefold of degree 5 and sectional genus 2. To see this, let us note that the 3-spaces πA and πB are in general position (otherwise the line AB would be contained in G(3, 5)), so we can assume that πA is deﬁned by x0 = x1 = 0 and πB by x2 = x3 = 0. So the matrix M becomes ⎞ ⎛ 0 0 0 x1 −x0 0 ⎜ 0 0 ⎟ 0 x3 −x2 0 ⎟. ⎜ (5.5) ⎝L30 L31 . . . L35 ⎠ L40 L41 . . . L45 By developing its 4 × 4 minors, we get that the two 3-spaces πA and πB are irreducible components of F and the residual is deﬁned by the 2 × 2 minors of x0 L30 + x1 L31 L34 L35 , x0 L40 + x1 L41 L44 L45 or, equivalently, of

x2 L32 + x3 L33 x2 L42 + x3 L43

L34 L44

L35 . L45

ˇ are general, Z is therefore a Castelnuovo threefold. If the other generators of Δ then S is smooth outside A and B. Z results to be singular at 8 points, 4 of them belonging to QA := Z ∩πA , and the other 4 to QB := Z ∩πB . QA and QB are both quadric surfaces, linear sections of the unique quadric containing Z, of equation L34 L45 − L35 L44 = 0. They intersect along the line πA ∩ πB . Clearly the lines of B are the secants of Z meeting also πA and πB . The lines of B through a point in πA or πB cut Z along the points of a conic, whereas those through a point P in Z cut Z, πA and πB residually each along a line.

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5.1.3. Three singular points. We assume now that S has also a third singular point C on G(3, 5). Then F is of the form F = πA ∪ πB ∪ πC ∪ V , where V is a complete intersection of two quadrics. Indeed, as in the previous case the 3-spaces πA , πB and πC are two by two in general position, so we can assume that πA is deﬁned by x0 = x1 = 0, πB by x2 = x3 = 0 and πC by x4 = x5 = 0. The matrix M takes the form ⎛ ⎞ x1 −x0 0 0 0 0 ⎜0 0 ⎟ 0 x3 −x2 0 ⎜ ⎟. ⎝0 0 0 0 x5 −x4 ⎠ L0 . . . . L5 It is easy to see that the residual of the 3 spaces πA , πB and πC is deﬁned by the equations x0 L0 + x1 L1 = x4 L4 + x5 L5 = 0, or equivalently by x2 L2 + x3 L3 = x5 L5 + x4 L4 = 0. V is therefore a Del Pezzo threefold. Again, if the other ˇ are general, then S is smooth outside A, B and C. Also V results generators of Δ to be smooth. The intersections of V with the spaces πA , πB and πC are quadric surfaces intersecting two by two along lines. The lines of B are the lines meeting simultaneously πA , πB , πC and V .

5.1.4. Four singular points. We consider ﬁnally the case in which S has also a fourth singular point D on G(3, 5). Then F is of the form F = πA ∪ πB ∪ πC ∪ πD ∪ W , where W is a rational normal cubic scroll. It is clear that the four spaces πA , πB , πC and πD are all components of the focal locus F , and that the lines of B are characterized by the property of meeting simultaneously all of them. The component W , that must exist by degree reasons, arises in the following way. Let H be a hyperplane containing πA , it intersects the other three spaces along planes, call them αB , αC and αD , and BH , the restriction of B to H, is formed by the lines meeting them. There is a uniquely determined fourth plane meeting αB , αC and αD along lines, and it is necessarily contained in the focal locus of BH (it is a “parasitic plane”, see [DPi], because its lines intersect all the three planes αB , αC and αD but a general line of BH does not meet it). If we let H vary, we get a 1-dimensional family of planes, whose union is W .

5.1.5. Special degenerate cases. These 4 examples don’t exhaust the list of ˇ could possibilities for the surfaces S meeting G(3, 5). For instance the space Δ be contained in the tangent space to G(3, 5) at one of the points of intersection, or intersect it along a plane or a line. In this situation four, three or two of the ˇ with G(3, 5) get identiﬁed. Hence some components points of intersection of Δ of the focal locus appear with multiplicity greater than one and the 4 foci on a general line of B are not distinct. A particular case is when S is a cone of vertex a point A of G(3, 5), then the 3-space corresponding to the vertex counts twice as component of F .

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5.2. S with a double line not meeting G(3, 5). Assume that S is reducible and disjoint from G(3, 5), therefore of the form S = π ∪ Q where π is a plane and Q a (possibly reducible) quadric. ˇ 5) and not meeting G(3, 5) are In [MM04] the linear spaces contained in G(1, classiﬁed up to the natural action of PGL6 . They can be interpreted also as linear spaces of skew-symmetric matrices of constant rank 4. It results that there are two orbits of lines, an open irreducible orbit of dimension 22, and a codimension one closed orbit. Note that the Gauss map γ, if restricted to such a line , is regular and deﬁned by the derivatives of the cubic ˇ 5)), which have degree 2. Therefore γ() Pfaﬃan polynomial (the equation of G(1, 14 is embedded in P as a conic. The lines parametrised by this conic represent a ruling of a smooth quadric or the lines of a quadric cone, respectively, for the two orbits. The planes are distributed in 4 orbits, all of dimension 26. The Gauss image of such a plane π is embedded in P14 as a Veronese surface V . The four orbits correspond precisely to the possible double Veronese embeddings of the plane in G(1, 5). The lines represented by V have the following geometrical interpretation, in the four cases (see [SU04]): 1. the secant lines of a skew cubic curve C embedded in a 3-space L; 2. the lines contained in a quadric 3-fold Γ and meeting a ﬁxed line r contained in Γ; 3. the lines joining the corresponding points in a ﬁxed isomorphism between two disjoint planes; 4. the lines of a cone C(V) over a projected Veronese surface. ˇ is generated by one of these four types of planes plus one general point of If Δ 14 ˇ P , then the residual component of π in S is a quadric surface of maximal rank. We describe now the corresponding congruences and their focal loci. 5.2.1. Case 1. F = L ∪ X, where L is a scheme whose support is L, having C as embedded component, and X is a singular threefold of degree 6 with Hilbert polynomial PX (t) = t3 + 3t2 + 2; X meets L along a quartic surface with C as singular locus. Through a general point of L there passes a 1-dimensional family of lines of B, all contained in L, and through a general point of C a 2-dimensional family. L and C are components of the focal locus, but a general line of B does not meet L nor C, they are therefore both parasitic components. On a general line of B the 4 foci lie all on X. It follows that X is an example of (singular) threefold containing a 3-dimensional family of plane cubic curves (see [MP96]).

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An explicit example of this type of congruence is given by the following skewˇ symmetric matrix, which represents Δ: ⎞ ⎛ 0 −d a b c 0 ⎜. 0 0 a b c ⎟ ⎜ ⎟ ⎜. . 0 d 0 d⎟ ⎜ ⎟. ⎜. . . 0 0 0 ⎟ ⎟ ⎜ ⎝. . . . 0 −d⎠ . . . . . 0 5.2.2. Case 2. F = Γ ∪ Y , where Γ is a scheme whose support is Γ, having r as embedded component, and Y is a quintic threefold with Hilbert polynomial PY (t) = 5/6t3 + 5/2t2 + 5/3t + 1. Through a general point in Γ there passes a 1-dimensional family of lines, through a general point in r a 2-dimensional family. Hence r is a parasitic scheme. Y meets Γ in a quartic surface which is singular along r. An explicit example is provided by the matrix: ⎛ ⎞ 0 a b c d 0 ⎜ . 0 0 d c b⎟ ⎜ ⎟ ⎜ . . 0 0 d 0⎟ ⎜ ⎟ ⎜ . . . 0 0 d⎟ . ⎜ ⎟ ⎝ . . . . 0 a⎠ . . . . . 0 5.2.3. Case 3. F = Z1 ∪ Z2 , where Z1 is a rational cubic scroll, i.e. P1 × P2 , ˇ and Z2 is a singular Del Pezzo which is the union of the lines parametrised by Δ, threefold, complete intersection of two quadrics. Z1 ∩ Z2 is a quartic surface, rational normal scroll of type (2, 2), which cuts a conic on each plane of Z1 . The singular locus of Z2 is the union of two conics. The foci of a general line lie two on each component of F , so B is formed by the lines which are secant both Z1 and Z2 . An example of this congruence is the following: ⎛ ⎞ 0 a b d 0 0 ⎜ . 0 c 0 d 0⎟ ⎜ ⎟ ⎜ . . 0 0 0 d⎟ ⎜ ⎟ ⎜ . . . 0 a b⎟ . ⎜ ⎟ ⎝ . . . . 0 c⎠ . . . . . 0 5.2.4. Case 4. F = C(V) ∪ T , where T is a cubic threefold contained in a hyperplane of P5 passing through the vertex of the cone. The lines of B are the trisecants of F = C(V), meeting also T .

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An example of this congruence is ⎛ 0 a ⎜. 0 ⎜ ⎜. . ⎜ ⎜. . ⎜ ⎝. . . .

the following: ⎞ b c d 0 c d b 0⎟ ⎟ 0 0 0 d⎟ ⎟. . 0 a 0⎟ ⎟ . . 0 0⎠ . . . 0

5.2.5. Special degenerate cases . In all the four cases, for special choices of ˇ the rank of the quadric Q can decrease, giving raise to the 4th generator of Δ, various degenerations of the focal loci of the just considered congruences.

5.3. Classiﬁcation of linear congruences in P5 . Table 5.3 summarizes the sketch of classiﬁcation of linear congruences of lines in P5 with 3-dimensional focal locus. Note that the smooth Palatini threefolds, which are scroll over cubic surfaces, have a non-trivial moduli space. Indeed the moduli space of cubic surfaces is rational of dimension 4, and the moduli space of rank 2 vector bundles E on a cubic surface S, such that P(E) is a Palatini scroll, has dimension 5 (see [FM02]). We ﬁnish this article with a few words about the singular Palatini scrolls and the corresponding congruences. Actually, the singular Palatini scrolls which come out from a surface S with (isolated) double points Pi (such that in fact Δ is tangent to ˇ(G(1, 5) \ G(3, 5)) at Pi ) have only isolated singularities and can be described as the smooth ones. Therefore also the congruences associated to these can be described as in the case of the smooth Palatini scroll. Let us pass now to the two more problematic Palatini scrolls, i.e. the ones which come from an S that is either a cubic cone or a ruled surface, singular along a line. In both cases, S has a 1-dimensional family of lines, and these lines correspond to quadric surfaces in P5 . ˇ 5) is a cubic cone of vertex A must be contained ˇ such that Δ∩ ˇ G(1, A 3-space Δ ˇ 5) at A, and intersect the in the quadric IIA , the second fundamental form of G(1, vertex of IIA only at A. An example is the following: ⎛ ⎞ 0 c a d−c b 0 ⎜. 0 d a 0 b⎟ ⎟ ⎜ ⎜. . 0 c d 0⎟ ⎜ ⎟. ⎜. . . 0 0 c⎟ ⎜ ⎟ ⎝. . . . 0 0⎠ . . . . . 0 The Palatini threefolds obtained in this way result to be singular along the line corresponding to A. This Palatini scroll can indeed be constructed as the smooth one, and also the congruence does not give problems.

Linear congruences and hyperbolic systems of conservation laws

Surface S

Sing S

Focal locus F smooth Palatini scroll

smooth irreducible with isolated singularities P1 , . . . , Pk , k≤4

Pi ∈ / G(3, 5)

singular scroll

Palatini

Pi ∈ G(3, 5)

the 3-space πPi is an irreducible component of F

irreducible ruled with a double line l

l ∩ G(3, 5) = ∅

singular scroll

reducible π ∪ Q, π a plane, Q a quadric

conic in π, disjoint from G(3, 5)

L ∪ X, deg X = 6

Palatini

Γ ∪ Y , deg Y = 5 Z1 ∪Z2 , deg Z2 = 4, Z1 = P1 × P2 C(V)∪T , deg T = 3

227

Remarks F has moduli F can be singular at a point or along a line the residual intersects πPi along a smooth quadric F is singular along a quadric ˇ is tancone, Δ ˇ 5) gent to G(1, along l L is a parasitic component Γ has a parasitic line as embedded component the lines of B are secant both Z1 and Z2 the lines of B are the trisecants C(V) meeting T

successive degenerations of the previous cases Table 1 ˇ must be To obtain a ruled cubic S with double line , disjoint from G(3, 5), Δ ˇ 5) at P , for all P ∈ l. contained in the intersection of the tangent spaces to G(1, One checks that such a 3-space exists only if is a line of the closed orbit (see Subsection 5.2), whereas in the other case no 3-space is contained in the required intersection. The Palatini scroll constructed in such a way is singular along the quadric cone Q which correspond to . Now we consider the secant lines to F which meet also Q. They form – by dimensional reasons – a congruence. Therefore this is the congruence which we are looking for, since on a general line of it we have two distinct foci plus a double focus.

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Note that this is only a sketch of classiﬁcation. The task of writing a complete classiﬁcation of linear congruences up to isomorphism seems to us out of reach, because of the multitude of special cases, due to the moduli of cubic surfaces (see ˇ 5). for instance [BL98]) and to their several diﬀerent embeddings in G(1, In a forthcoming paper we plan to apply these results to the classiﬁcation of Temple systems in 4 variables. We have to point out that this classiﬁcation holds on an algebraically closed ﬁeld, so it will be necessary to reﬁne it over the real ﬁeld.

References [AF96] S. I. Agafonov and E. V. Ferapontov, Systems of conservation laws from the point of view of the projective theory of congruences, Izv. Ross. Akad. Nauk. Ser. Mat. 60 (1996), no. 6, 3–30. [AF99]

, Theory of congruences and systems of conservation laws, J. Math. Sci. (New York) 94 (1999), no. 5, 1748–1794, Geometry, 4.

[AF01]

, Systems of conservation laws of Temple class, equations of associativity and linear congruences in P4 , Manuscripta Math. 106 (2001), no. 4, 461–488.

[ABT] E. Arrondo, M. Bertolini and C. Turrini, A focus on focal surfaces, Asian J. Math. 5 (2001), no. 3, 535–560. [BM01] D. Bazan and E. Mezzetti, On the construction of some Buchsbaum varieties and the Hilbert scheme of elliptic scrolls in P5 , Geom. Dedicata, 86 (2001), no. 1–3, 191–204. [BL98] M. Brundu and A. Logar, Parametrization of the orbits of cubic surfaces, Transformation Groups 3 (1998), 209–239. [C91] G. Castelnuovo, Ricerche di geometria della retta nello spazio a quattro dimensioni, Atti R. Ist. Veneto Sc., ser.VII, 2 (1891), 855–901. [CS92] C. Ciliberto ed E. Sernesi, Singularities of the theta divisor and congruences of planes, J. Algebr. Geom. 1 (1992), no. 2, 231–250. [CoCoA] CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra, Available at http://cocoa.dima.unige.it. [DP01] P. De Poi, On ﬁrst order congruences of lines of P4 with a fundamental curve, Manuscripta Math. 106 (2001), 101–116. [DP03]

, Threefolds in P5 with one apparent quadruple point, Communications in Algebra 31 (2003), no. 4, 1927–1947.

[DP04]

, Congruences of lines with one-dimensional focal locus, Port. Math. (N.S.) 61 (2004), no. 3, 329–338.

[DPi]

, On ﬁrst order congruences of lines in P4 with irreducible fundamental surface, Mathematische Nachrichten 278 (2005), no. 4, 363–378

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[DPii]

, On ﬁrst order congruences of lines in P4 with non-reduced fundamental surface, Preprint ArXiv:math.AG/0407341, submitted.

[DPiii]

, On ﬁrst order congruences of lines in P4 with reducible fundamental surface, in preparation.

[Dub96] B. Dubrovin, Geometry of 2D topological ﬁeld theories, in Lect. Notes Math. Vol. 1620 (1996), Springer Verlag 120–348. [FM02] M. L. Fania and E. Mezzetti, On the Hilbert scheme of Palatini threefolds, Adv. Geom 2 (2002), 371–389. [GH78] P. Griﬃths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, 1978. [Har77] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, no. 52, Springer-Verlag, 1977. ¨ [Kum66] E.E. Kummer, Uber die algebraischen Strahlensysteme, insbesondere u ¨ber die der ersten und zweiten Ordnung, Abh. K. Preuss. Akad. Wiss. Berlin (1866), 1–120. [MM04] L. Manivel and E. Mezzetti, On linear spaces of skew-symmetric matrices of constant rank, Manuscripta Math. 117 (2005), no. 3, 319–331 [Mar09a] G. Marletta, Sopra i complessi d’ordine uno dell’S4 , Atti Accad. Gioenia, Serie V, Catania III (1909), 1–15, Memoria II. [Mar09b]

, Sui complessi di rette del primo ordine dello spazio a quattro dimensioni, Rend. Circ. Mat. Palermo XXVIII (1909), 353–399.

[MP96] E. Mezzetti and D. Portelli, Threefolds in P5 with a 3-dimensional family of plane curves, Manuscripta Math. 90 (1996), 365-381. [O92] G. Ottaviani, On 3-folds in P5 which are scrolls, Ann. Sc. Norm. Sup. Pisa 19(4) (1992), 451–471. [Ran86] Z. Ran, Surfaces of order 1 in Grassmannians, J. Reine Angew. Math. 368 (1986), 119–126. [Seg88] C. Segre, Un’osservazione sui sistemi di rette degli spazˆı superiori, Rend. Circ. Mat. Palermo II (1888), 148–149. [SU04] J.C. Sierra and L. Ugaglia, On double Veronese embeddings in the Grassmannian G(1, N ), Preprint ArXiv:math.AG/0410600, (2004). [Tem83] B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc. 280 (1983), no. 2, 781–795.

Pietro De Poi Mathematisches Institut, Universit¨ at Bayreuth Lehrstuhl VIII, Universit¨ atsstraße 30 D-95447 Bayreuth, Germany Email: [email protected]

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Emilia Mezzetti Dipartimento di Matematica e Informatica Universit` a degli Studˆı di Trieste Via Valerio, 12/b I-34127 Trieste, Italy Email: [email protected]

A note on the very ampleness of complete linear systems on blowings-up of P3 Cindy De Volder∗and Antonio Laface‡

Abstract. In this note we consider the blowing-up X of P3 along r general points of the anticanonical divisor of a smooth quadric in P3 . Given a complete linear system L = |dH − m1 E1 − · · · − mr Er | on X, with H the pull-back of a plane in P3 and Ei the exceptional divisor corresponding to Pi , we give necessary and suﬃcient conditions for the very ampleness (resp. base point freeness and non-speciality) of L. As a corollary we obtain a suﬃcient condition for the very ampleness of such a complete linear system on the blowing-up of P3 along r general points. 2000 Mathematics Subject Classiﬁcation: 14C20

1. Introduction In this note we work over an algebraically closed ﬁeld of characteristic 0. Let P1 , . . . , Pr be general points of the anticanonical divisor of a smooth quadric in P3 and choose some integers m1 ≥ . . . ≥ mr ≥ 0. Consider the linear system L of surfaces of degree d in P3 having multiplicities at least mi at Pi , for all i = 1, . . . , r. Let X denote the blowing-up of P3 along P1 , . . . , Pr , and let L denote the complete linear system on X corresponding to L . Let Z be a zero-dimensional subscheme of length 2 of X, then L separates Z if there exists a divisor D ∈ L such that Z ∩ D = ∅ but Z ⊂ D. The system L on X is called very ample if it separates all such Z. In Theorem 5.1 (resp. Theorem 4.1) we prove that such a system L is very ample (resp. base point free) on X if and only if mr > 0, d ≥ m1 + m2 + 1 and 4d ≥ m1 + · · · + mr + 3 (resp. d ≥ m1 + m2 and 4d ≥ m1 + · · · + mr + 2). A fundamental tool for proving these results is Theorem 3.1 which states that a system L with 2d ≥ m1 + · · · + m4 is non-special if d ≥ m1 + m2 − 1 and 4d ≥ m1 + · · · + mr + 1. ∗ The ﬁrst author is a Postdoctoral Fellow of the Fund for Scientiﬁc Research-Flanders (Belgium) (F.W.O.-Vlaanderen) ‡ The second author would like to acknowledge the support of the MIUR of the Italian Government in the framework of the National Research Project “Geometry in Algebraic Varieties” (Coﬁn 2002)

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If r ≤ 8, the points Pi are in general position on P3 and the dimension and base locus of L on X can be determined using the results from [1, 2]. The techniques used in this note are a generalization of the ones in [1, 2] and make use of the results about complete linear systems on rational surfaces with irreducible anticanonical divisor (see [3, 4]).

2. Preliminaries and notation Let L3 (d) denote the complete linear system of surfaces of degree d in P3 . Consider ¯ ¯ ∈ L3 (2) in P3 and let KQ¯ denote the canonical class on Q. a general quadric Q ¯ Then we know that −KQ¯ is just the linear system on Q induced by L3 (2), so we can consider DQ¯ ∈ −KQ¯ which is smooth and irreducible. Let P1 , . . . , Pr be general points of DQ¯ and choose integers m1 ≥ · · · ≥ mr ≥ 0. By Xr we denote the blowing-up of P3 along the points P1 , . . . , Pr , E0 denotes the pullback of a plane in P3 , by Ei (i = 1, . . . , r) we mean the exceptional divisor on Xr corresponding to Pi and π : Xr → P3 denotes the projection map. On P3 , we let L3 (d; m1 , . . . , mr ) denote the linear system of surfaces of degree d with multiplicities at least mi at Pi for all i = 1, . . . , r as well as the corresponding sheaf. By abuse of notation, on Xr , L3 (d; m1 , . . . , mr ) also denotes the invertible sheaf π ∗ (OP3 (d)) ⊗ OXr (−m1 E1 − · · · − mr Er ) and the corresponding complete linear system |dE0 − m1 E1 − · · · mr Er |. Analoguously, on P3 , L3 (d; mn1 1 , . . . , mnt t ) denotes the linear system of surfaces of degree d with multiplicities at least mi at ni of the points on DQ as well as the corresponding sheaf. Again, the same notation is used to denote the associated complete linear system and invertible sheaf on Xr . The virtual dimension of the linear system L = L3 (d; m1 , . . . , mr ) on P3 as well as on Xr is deﬁned as r d+3 mi + 2 vdim (L) := − − 1. 3 3 i=1 The expected dimension of L is then given by edim (L) := max{−1, vdim (L)}. It is then clear that dim(L) ≥ edim (L) ≥ vdim (L), and the system L is called special if dim(L) > edim (L). The system L is associated to the sections of the sheaf OPn (d)⊗IZ , where Z = mi pi is the zero-dimensional scheme of fat points. From the cohomology exact sequence associated to 0

/L

/ OPn (d)

/ OZ

/ 0,

we obtain that hi (L) = 0 for i = 2, 3. Therefore v(L) = h0 (L) − h1 (L) − 1, so that a non-empty system is special if and only if h1 (L) > 0.

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¯ on Xr is a divisor of L3 (2; 1r ), and Note that the strict transform Qr of Q ¯ along P1 , . . . , Pr . So Pic (Qr ) = f1 , f2 , e1 , . . . , er , Qr is just the blowing-up of Q ¯ and e1 , . . . , er the exceptional with f1 and f2 the pullbacks of the two rulings on Q curves. By LQr (a, b; m1 , . . . , mr ) we denote the complete linear system |af1 +bf2 − m1 e1 − . . . − mr er |, and, as before, if some of the multiplicities are the same, we also use the notation LQr (a, b; mn1 1 , . . . , mnt t ). Let Bs be the blowing-up of P2 along s general points of a smooth irreducible cubic, then Pic Bs = h, e1 , . . . , es , with h the pullback of a line and el the exceptional curves. By L2 (d; m1 , . . . , ms ) we denote the complete linear system |dh − m1 e1 − . . . − ms es |. And again, as before, if some of the multiplicities are the same, we also use the notation L2 (d; mn1 t , . . . , mnr t ). Note that −KBs = L2 (3; 1s ) contains a smooth irreducible divisor which we will denote by DBs . On Bs , a system L2 (d; m1 , . . . , ms ) is said to be in standard form if d ≥ m1 + m2 + m3 and m1 ≥ m2 ≥ · · · ≥ ms ≥ 0; and it is called standard if there exists a ˜ m ˜ e˜1 , . . . , e˜s of Pic Bs such that L2 (d; m1 , . . . , ms ) = |d˜h− ˜ 1 e˜1 −. . .− m ˜ s e˜s | base h, is in standard form. As explained in [1, §6], the blowing-up Q of the quadric along 1 general point can also be seen as a blowing-up of the projective plane along 2 general points, and LQ (a, b; m) = L2 (a + b − m; a − m, b − m). So, in particular −KQ = LQ (2, 2; 1) = L2 (3; 12 ) = −KB2 . Obviously, this means that our blowing-up Qr can also be seen as a Br+1 and LQr (a, b; m1 , m2 , . . . , mr ) = L2 (a + b − m1 ; a − m1 , b − m1 , m2 , . . . , mr ). This implies in particular that we can apply the results from [3] and [4].

3. Non-speciality Theorem 3.1. Consider L = L3 (d; m1 , . . . , mr ) on Xr with 2d ≥ m1 +m2 +m3 + m4 and m1 ≥ m2 ≥ · · · ≥ mr ≥ 0. Then h1 (L) = 0 if (1) d ≥ m1 + m2 − 1 and (2) 4d ≥ m1 + · · · + mr + 1 if r ≥ 9. Proof. We will assume that mr > 0, since otherwise we can work on Xr with r := max{i : mi > 0}. If r ≤ 8 then the points P1 , . . . , Pr are general points of P3 and the statement follows from [1, Theorem 5.3]. So we assume that r ≥ 9 and consider the following exact sequence 0

/ L3 (d − 2; m1 − 1, . . . , mr − 1)

/L

/ L ⊗ OQr

/ 0.

Then L ⊗ OQr = LQr (d, d; m1 , . . . , mr ) = L2 (2d − m1 ; (d − m1 )2 , m2 , . . . , mr ). Proceeding as in the proof of [1, Lemma 5.2] it is easily seen that this is a standard

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class. Moreover L2 (2d − m1 ; (d − m1 )2 , m2 , . . . , mr ).KBr+1 = −4d + m1 + · · · + mr ≤ −1, so we can apply [3, Theorem 1.1] to obtain h1 (L ⊗ OQr ) = 0. On the other hand, one can easily check that L := L3 (d − 2; m1 − 1, . . . , mr − 1) still satisﬁes the conditions of the theorem. Continue like this until the residue class L = L3 (d ; m1 , . . . , mr ) is such that r ≤ 8. For this class we then know that

h1 (L ) = 0 which gives us that h1 (L) = 0.

4. Base point freeness Theorem 4.1. Consider L = L3 (d; m1 , . . . , mr ) on Xr with m1 ≥ m2 ≥ · · · ≥ mr . Then L is base point free on Xr if and only if the following conditions are satisﬁed: (1) mr ≥ 0, (2) d ≥ m1 + m2 and (3) 4d ≥ m1 + · · · mr + 2 if r ≥ 8. Proof. First of all let us prove that the conditions are necessary. Obviously, if mr < 0 then mr Er ⊂ Bs(L); and if d < m1 + m2 then the strict transform of the line through P1 and P2 is contained in Bs(L). Now assume that (1) and (2) are satisﬁed, but 4d ≤ m1 + · · · + mr + 1 and consider 0

/ L3 (d − 2; m1 − 1, . . . , mr − 1)

/L

/ L ⊗ OQr

/ 0.

(4.1)

Proceeding as before, one can check that L ⊗ OQr = L2 (2d − m1 ; (d − m1 )2 , m2 , . . . , mr ) is standard and L2 (2d − m1 ; (d − m1 )2 , m2 , . . . , mr ).(−KBr+1 ) = 4d − m1 − · · · − mr ≤ 1. Using the results from [3], we obtain that L ⊗ OQr has base points, which will also be base points of L. Now assume that all three conditions are satisﬁed (as before, we may even assume mr > 0). If r ≤ 8, the result follows from [2, Theorem 6.2], so we assume that r ≥ 9 and that the result holds for r < r. Consider the exact sequence (4.1). As before, one can see that L ⊗ OQr = L2 (2d − m1 ; (d − m1 )2 , m2 , . . . , mr ) is standard and since L2 (2d − m1 ; (d − m1 )2 , m2 , . . . , mr ).KBr+1 ≤ −2 we know that L ⊗ OQr is base point free (see [3, Lemma 3.3(2)]). Also, because of Theorem 3.1, we know that h1 (L3 (d − 2; m1 − 1, . . . , mr − 1)) = 0, so L induces the complete linear system L ⊗ OQr on Qr , which means in particular that L has no base points on Qr . On the other hand it is easily checked that L3 (d−2; m1 −1, . . . , mr −1) still satisﬁes the conditions of the theorem. Continue like this until you have r < r for the residue class (i.e. repeat this reasoning mr times). Denote L3 (d − 2mr ; m1 − mr , . . . , mr−1 − mr ) by L . We then know that L + mQr ⊂ L. Since L has no

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base points on Qr , and since L is base point free by our induction hypothesis, we obtain that Bs(L) = ∅.

5. Very ampleness Theorem 5.1. Consider L = L3 (d; m1 , . . . , mr ) on Xr with m1 ≥ m2 ≥ · · · ≥ mr . Then L is very ample on Xr if and only if the following conditions are satisﬁed: (1) mr > 0, (2) d ≥ m1 + m2 + 1 (d ≥ m1 + 1 if r = 1; d ≥ 1 if r = 0) and (3) 4d ≥ m1 + · · · + mr + 3 if r ≥ 9. Proof. First of all let us note that the conditions are necessary. Obviously, if mr ≤ 0 then L cannot separate on Er ; and if d ≤ m1 + m2 then L cannot separate a zero-dimensional subscheme Z of length 2 of the strict transform of the line through P1 and P2 . In case 4d ≤ m1 + · · · + mr + 2, one can see that L cannot separate Z if it is contained in DQr . Now assume that all three conditions are satisﬁed. First of all consider the exact sequence 0

/ L3 (d; m1 + 1, m2 , . . . , mr )

/L

/ L ⊗ OE

1

/ 0.

Because of Theorem 3.1 we know that h1 (L3 (d; m1 + 1, m2 , . . . , mr )) = 0, so L induces the complete linear system L ⊗ OE1 on E1 . Since L ⊗ OE1 = L2 (m1 ), we see that L separates on E1 . Naturally, a similar reasoning can be done for any Ei , so L separates on every Ei . Moreover, one can easily check that L(Ei ) := L3 (d; m1 , . . . , mi−1 , mi + 1, mi+1 , . . . , mr ) satisﬁes all the conditions of Theorem 4.1, so that L(Ei ) is base point free on Xr . This means that L can separate Z if ∃i : Z ∩ Ei = 0 but Z ⊂ Ei . Combining the previous two results, we see that we now only need to show that L separates Z with Z ∩ Ei = ∅ for all i = 1, . . . , r. In case r = 0, 1 or 2, this is trivial. Now let us assume that r ≥ 3 and that the statement holds for r < r. First look at the case where mr = 1 and consider the exact sequence 0

/ L3 (d − 2; m1 − 1, . . . , mr−1 − 1, 0)

/L

/ L ⊗ OQr

/ 0.

Proceeding as before, one can easily see that L ⊗ OQr is standard. Moreover, 4d − m1 − · · · − mr ≥ 3 (if r ≥ 9 this is condition (3) and if 3 ≤ r ≤ 8 this follows from (2)), so [4, Theorem 2.1] implies that L ⊗ OQr is very ample. Since h1 (L3 (d − 2; m1 − 1, . . . , mr−1 − 1)) = 0 (because of Theorem 3.1) we then obtain that L separates on Qr . Using Theorem 4.1 we also obtain that L := L3 (d − 2; m1 − 1, . . . , mr−1 − 1, 0) is base point free, which implies that L separates Z if Z ∩ Qr = ∅. Let r = max{i : mi > 1} (or r = 0 if all mi = 1), then one can

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easily check that L satisﬁes all the conditions of the theorem on Xr . So, using the induction hypothesis, we have that L is very ample on Xr . But since Z on Xr is disjoint with all Ei , Z corresponds with a zero-dimensional subscheme on Xr (also disjoint with all Ei ). So we may conclude that L separates any Z. Now we assume mr > 1 and we assume that the statement holds for mr < mr . Consider the exact sequence 0

/ L3 (d − 2; m1 − 1, . . . , mr − 1)

/L

/ L ⊗ OQr

/ 0.

Proceeding similarly as for the case mr = 1 one can easily see that L separates all Z.

Remark 5.2. Let A1 , . . . , Ar be general points on P3 , let Yr be the blowing-up of P3 along those r general points and let L3 (d; m1 , . . . , mr ) (m1 ≥ m2 ≥ · · · ≥ mr ) denote the complete linear system |dE0 − m1 E1 − · · · − mr Er | on Yr . Since the very ampleness is an open property, Theorem 5.1 implies that L3 (d; m1 , . . . , mr ) is very ample on Yr if mr > 0, d ≥ m1 + m2 + 1 (d ≥ m1 + 1 if r = 1; d ≥ 1 if r = 0) and 4d ≥ m1 + · · · + mr + 3 if r ≥ 9. Of course the third condition will now no longer be a necessary condition.

References [1] Cindy De Volder and Antonio Laface. On linear systems of P3 through multiple points. Preprint, math.AG/0311447, 2003. [2] Cindy De Volder and Antonio Laface. Base locus of linear systems on the blowing-up of P3 along at most 8 general points. Preprint, math.AG/0401244, 2004. [3] Brian Harbourne. Complete linear systems on rational surfaces. Trans. Amer. Math. Soc., 289(1):213–226, 1985. [4] Brian Harbourne. Very ample divisors on rational surfaces. 272(1):139–153, 1985.

Math. Ann.,

[5] Antonio Laface and Luca Ugaglia. On a class of special linear systems of P3 . Preprint, math.AG/0311445, 2003. Cindy De Volder Department of Pure Mathematics and Computeralgebra Galglaan 2, B-9000 Ghent, Belgium Email: [email protected] Antonio Laface Dipartimento di Matematica Universit` a degli Studi di Milano Via Saldini 50, 20100 Milano, Italy Email: [email protected]

Phylogenetic algebraic geometry Nicholas Eriksson, Kristian Ranestad, Bernd Sturmfels, Seth Sullivant ∗

Abstract. Phylogenetic algebraic geometry is concerned with certain complex projective algebraic varieties derived from ﬁnite trees. Real positive points on these varieties represent probabilistic models of evolution. For small trees, we recover classical geometric objects, such as toric and determinantal varieties and their secant varieties, but larger trees lead to new and largely unexplored territory. This paper gives a self-contained introduction to this subject and oﬀers numerous open problems for algebraic geometers. 2000 Mathematics Subject Classiﬁcation: 14Q99, 92B10

1. Introduction Our title is meant as a reference to the existing branch of mathematical biology which is known as phylogenetic combinatorics. By “phylogenetic algebraic geometry” we mean the study of algebraic varieties which represent statistical models of evolution. For general background reading on phylogenetics we recommend the books by Felsenstein [11] and Semple-Steel [21]. They provide an excellent introduction to evolutionary trees, from the perspectives of biology, computer science, statistics and mathematics. They also oﬀer numerous references to relevant papers, in addition to the more recent ones listed below. Phylogenetic algebraic geometry furnishes a rich source of interesting varieties, including familiar ones such as toric varieties, secant varieties and determinantal varieties. But these are very special cases, and one quickly encounters a cornucopia of new varieties. The objective of this paper is to give an introduction to this subject area, aimed at students and researchers in algebraic geometry, and to suggest some concrete research problems. The basic object in a phylogenetic model is a tree T which is rooted and has n labeled leaves. Each node of the tree T is a random variable with k possible states (usually k is taken to be 2, for the binary states {0, 1}, or 4, for the nucleotides {A,C,G,T}). At the root, the distribution of the states is given by π = (π1 , . . . , πk ). ∗ N. Eriksson was supported by a NDSEG fellowship, K. Ranestad was supported by the Norwegian Research Council and MSRI, B. Sturmfels and S. Sullivant were partially supported by the National Science Foundation (DMS-0200729).

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On each edge e of the tree there is a k × k transition matrix Me whose entries are indeterminates representing the probabilities of transition (away from the root) between the states. The random variables at the leaves are observed. The random variables at the interior nodes are hidden. Let N be the total number of entries of the matrices Me and the vector π. These entries are called model parameters. For instance, if T is a binary tree with n leaves, then T has 2n − 2 edges, and hence N = (2n − 2)k 2 + k. In practice, there will be many constraints on these parameters, usually expressible in terms of linear equations and inequalities, so the set of statistically meaningful parameters is a polyhedron P in RN . Sometimes, these constraints are given by non-linear polynomials, in which case P would be a semi-algebraic subset of RN . Specifying this subset P means choosing a model of evolution. Several biologically meaningful choices of such models will be discussed in Section 3. Fix a tree T with n leaves. At each leaf we can observe k possible states, so there are k n possible joint observations we can make at the leaves. The probability φσ of making a particular observation σ is a polynomial in the model parameters. Hence we get a polynomial map whose coordinates are the polynomials φσ . This map is denoted n

φ : RN → Rk . The map φ depends only on the tree T and the number k. What we are interested in is the image φ(P ) of this map. In real-world applications, the coordinates φσ represent probabilities, so they should be non-negative and sum to 1. In other words, the rules of probability require that φ(P ) lie in the standard (k n − 1)n simplex in Rk . In phylogenetic algebraic geometry we temporarily abandon this requirement. We keep things simpler and closer to the familiar setting of complex n algebraic geometry, by replacing φ by its complexiﬁcation φ : CN → Ck , and by n replacing P and φ(P ) by their Zariski closures in CN and Ck respectively. As we shall see, the polynomials φσ are often homogeneous and φ(P ) is best regarded as a subvariety of a projective space. In Section 2 we give a basic example of an evolutionary model and put it squarely in an algebraic geometric setting. This relation is then developed further in Section 3, where we describe the main families of models and show how in special cases they lead to familiar objects like Veronese and Segre varieties and their secant varieties. Section 4 is concerned with the widely used Jukes-Cantor model, which is a toric variety in a suitable coordinate system. In the last section we formulate a number of general problems in phylogenetic algebraic geometry that we ﬁnd particularly important, and a list of more speciﬁc computationally oriented problems that may shed light on the more general ones.

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2. Polynomial maps derived from a tree In this section we explain the polynomial map φ associated to a tree T and an integer k ≥ 1. To make things as concrete as possible, let k = 2 and T be the tree on n = 3 leaves pictured below.

π b a c 1

2

d 3

The probability distribution at the root is an unknown vector (π0 , π1 ). For each of the four edges of the tree, we have a 2 × 2-transition matrix: a00 a01 b00 b01 Ma = Mb = a10 a11 b10 b11 c00 c01 d00 d01 Mc = Md = . c10 c11 d10 d11 Altogether, we have introduced N = 18 parameters, each of which represents a probability. But we regard them as unknown complex numbers. The unknown π0 represents the probability of observing letter 0 at the root, and the unknown b01 represents the probability that the letter 0 gets changed to the letter 1 along the edge b. All transitions are assumed to be independent events, so the monomial πu · aui · buv · cvj · dvk represents the probability of observing the letter u at the root, the letter v at the interior node, the letter i at the leaf 1, the letter j at the leaf 2, and the letter k at the leaf 3. Now, the probabilities at the root and the interior node are hidden random variables, while the probabilities at the three leaves are observed. This leads us to consider the polynomial φijk = π0 a0i b00 c0j d0k + π0 a0i b01 c1j d1k + π1 a1i b10 c0j d0k + π1 a1i b11 c1j d1k . This polynomial represents the probability of observing the letter i at the leaf 1, the letter j at the leaf 2, and the letter k at the leaf 3. The eight polynomials φijk specify our map φ : C18 → C8 . In applications, where the parameters are actual probabilities, one immediately replaces C18 by a subset P , for instance, the nine-dimensional cube in R18 deﬁned

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by the constraints a00 + a01

π0 + π1 = 1, π0 , π1 ≥ 0, = 1, a00 , a01 ≥ 0, a10 + a11 = 1, a10 , a11 ≥ 0

b00 + b01 = 1, b00 , b01 ≥ 0, c00 + c01 = 1, c00 , c01 ≥ 0, d00 + d01 = 1, d00 , d01 ≥ 0,

b10 + b11 = 1, b10 , b11 ≥ 0 c10 + c11 = 1, c10 , c11 ≥ 0 d10 + d11 = 1, d10 , d11 ≥ 0.

In phylogenetic algebraic geometry, on the other hand, we allow ourselves the luxury of ignoring inequalities and reality issues. We regard φ as a morphism of complex varieties. The most natural thing to do, for an algebraic geometer, is to work in a projective space. The polynomials φijk are homogeneous with respect to the diﬀerent letters a, b, c, d and π. We can thus change our perspective and consider our map as a projective morphism φ : P3 × P3 × P3 × P3 × P1 → P7 . This morphism is surjective, and it is an instructive undertaking to examine its ﬁbers. To underline the points made in the introduction, let us now cut down on the number of model parameters and replace the domain of the morphism by a natural subset P . For instance, let us deﬁne P by requiring that the four matrices are identical, a00 a01 . Ma = Mb = Mc = Md = a10 a11 Equivalently, P = P3diag × P1 , where P3diag is the diagonal of P3 × P3 × P3 × P3 . The restricted morphism φ|P : P3diag × P1 → P7 is given by the following eight polynomials: φ000 φ001

= =

π0 a400 + π0 a00 a01 a210 + π1 a210 a200 + π1 a310 a11 π0 a300 a01 + π0 a00 a01 a10 a11 + π1 a210 a00 a01 + π1 a210 a211

φ010 φ011

= =

π0 a300 a01 + π0 a00 a01 a10 a11 + π1 a210 a00 a01 + π1 a210 a211 π0 a200 a201 + π0 a00 a01 a211 + π1 a210 a201 + π1 a10 a311

φ100

=

π0 a300 a01 + π0 a201 a210 + π1 a11 a10 a200 + π1 a210 a211

φ101 φ110

= =

π0 a200 a201 + π0 a201 a10 a11 + π1 a11 a10 a00 a01 + π1 a10 a311 π0 a200 a201 + π0 a201 a10 a11 + π1 a11 a10 a00 a01 + π1 a10 a311

φ111

=

π0 a301 a00 + π0 a201 a211 + π1 a11 a10 a201 + π1 a411 .

The image of φ|P lies in the 5-dimensional projective subspace of P7 deﬁned by φ001 = φ010 and φ101 = φ110 . It is a hypersurface of degree eight in this P5 . The deﬁning polynomial of this hypersurface has 70 terms. Studying the geometry of this fourfold is a typical problem of phylogenetic algebraic geometry. For instance, what is its singular locus?

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The deﬁnition of the map φ for an arbitrary tree T with n leaves and an arbitrary number k of states is a straightforward generalization of the n = 3 example given above. It is simply the calculation of the probabilities of independent events along the tree. In general, each coordinate of the map φ is given by a homogeneous polynomial of degree equal to the number of edges of T plus one. If the root distribution is not a parameter, the degree of these polynomials is precisely the number of edges. One staple among the computational techniques for dealing with tree based probabilistic models is the sum-product algorithm. The sum-product algorithm is a clever application of the distributive law that allows for the fast calculation of the polynomials φσ . The basic idea is to factor the polynomials that represent φσ recursively along the tree. For instance, in our example above with homogeneous rate matrix: φ000 = π0 a00 (a00 (a200 ) + a01 (a210 )) + π1 a10 (a10 (a200 ) + a11 (a210 )). This expression can be evaluated with 10 multiplications and 3 additions instead of the initial expression which required 16 multiplications and 3 additions. In Section 3, we will show how these factorizations help in identifying polynomial relations among the φσ , i.e., polynomials vanishing on the image of the morphism φ.

3. Some models and some familiar varieties Most evolutionary models discussed in the literature have either two or four states for their random variables. The number n of leaves (or taxa) can be arbitrary. Computer scientists will often concentrate on asymptotic complexity questions for n → ∞, while for our purposes it would be quite reasonable to assume that n is at most ten. There are no general restrictions on the underlying tree T , but experience has shown that trivalent trees and trees in which every leaf is at the same distance from the root are often simpler. Suppose now that the number k of states, the number n of taxa and the tree T are ﬁxed. The choice of a model is then speciﬁed by ﬁxing a subset P ⊆ CN . The set P comprises the allowed model parameters. Here is a list of commonly studied models: General Markov This is the model P = CN . All the transition matrices Me are pairwise distinct, and there are no constraints on the k 2 entries of Me . The algebraic geometry of this model was studied by Allman and Rhodes [1, 2]. Group Based The matrices Me are pairwise distinct, but they all have a special structure which makes them simultaneously diagonalizable by the Fourier transform of an abelian group. In particular, P is a linear subspace of CN , speciﬁed by requiring that some entries of Me coincide with some other entries. For example, the Jukes-Cantor model for binary states (k = 2)

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a0 a1 stipulates that all matrices Me have the form . The Jukes-Cantor a1 a0 model for DNA (k = 4) is the topic of the next section. For more information on group-based models see [10, 22, 24]. Stationary Base Composition The matrices Me are distinct but they all share the common left eigenvector π = (π1 , . . . , πk ). This hypothesis expresses, for example when k = 4, the assumption that the distribution of the four nucleotides remains the same throughout some evolutionary process. An algebraic study of this model appears in [3]. Reversible The matrices Me are distinct symmetric matrices with the common left eigenvector π = (1, 1, . . . , 1). Again, as before, P is a linear subspace of CN . Commuting The matrices Me are distinct but they commute pairwise. We have not yet seen this model in the biology literature, but algebraists love the commuting variety [14, 19]. It provides a natural supermodel for the next one. Substitution The Me matrices have the form exp(te ·Q) where Q is a ﬁxed matrix. Equivalently, all matrices Me are powers of a ﬁxed matrix A = exp(Q) with constant entries, but where the exponent te is a real parameter. This is the most widely used model in biology (see [11]) but for us it has the disadvantage that it is not an algebraic variety, unless the rate matrix Q has commensurate eigenvalues. Homogeneous The matrices Me are all equal, or they all belong to a small ﬁnite collection. In this model, the number of free parameters is small and independent of the tree, so the parametric inference algorithm of [18] runs in polynomial time. No Hidden Nodes When all nodes are observed random variables then the parameterization becomes monomial, and the model is a toric variety. For the homogeneous model, the combinatorial structure of this variety was studied in [8]. Mixture models Suppose we are given m trees T1 , . . . , Tm (not necessarily disk tinct) on the same set of taxa. Each tree Ti has its own map φi : CN → Cn . The mixture model is given by the sum of these maps, that is, φ = φ1 + · · · + φm . For example, the case T1 = T2 = · · · = Tm and k = 4 may be used to model the fact that diﬀerent regions of the genome evolve at diﬀerent rates. See [12, 13]. Root distribution For any of the above models, the root distribution π can be taken to be uniform, π = (1, 1, . . . , 1), or as a vector with k independent entries.

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Among these models are many varieties which are familiar in algebraic geometry. Segre Varieties These appear as a special case of the model with no hidden nodes. Veronese Varieties These appear as a special case of the homogeneous model with no hidden nodes. The models in [8] are natural projections of Veronese varieties. Toric Varieties The previous two classes of varieties are toric. All group-based models are seen to be toric after a clever linear change of coordinates. The toric varieties of some Jukes-Cantor models will be discussed in the next section. Gr¨obner bases of binomials for arbitrary group-based models are given in [24]. Secant Varieties and Joins Joins appear when taking the mixture models of a collection of models. The secant varieties of a model amounts to taking the mixture of a model with itself. A special case of the general Markov model includes the secant varieties to the Segre varieties [1]. The secant varieties to Veronese varieties [5] appear as special cases of the homogeneous models with hidden nodes. Determinantal Varieties Many of the evolutionary models are naturally embedded in determinantal varieties, because the tree structure imposes rank constraints on matrices derived from the probabilities observed at the leaves. Getting a better understanding of these constraints is important for both theory and practice [9]. The remainder of this section is the discussion of one example which aims to demonstrate that phylogenetic trees arise quite naturally when studying these classical objects of algebraic geometry. Consider the Segre embedding of P1 × P1 × P1 × P1 in P15 . This four-dimensional complex manifold is given by the familiar monomial parameterization pijkl = ui · vj · wk · xl ,

i, j, k, l ∈ {0, 1}.

Its prime ideal is generated by the 2×2-minors of the following three 4×4-matrices: ⎛ ⎞ ⎛ ⎞ p0000 p0001 p0010 p0011 p0000 p0001 p0100 p0101 ⎜p0100 p0101 p0110 p0111 ⎟ ⎜p0010 p0011 p0110 p0111 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝p1000 p1001 p1010 p1011 ⎠ , ⎝p1000 p1001 p1100 p1101 ⎠ , p1100 p1101 p1110 p1111 p1010 p1011 p1110 p1111 ⎛

p0000 ⎜p0001 ⎜ ⎝p1000 p1001

p0010 p0011 p1010 p1011

p0100 p0101 p1100 p1101

⎞ p0110 p0111 ⎟ ⎟. p1110 ⎠ p1111

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These three matrices reﬂect the following three bracketings of the parameterization: pijkl = ((ui · vj ) · (wk · xl )) = ((ui · wk ) · (vj · xl )) = ((ui · xl ) · (vj · wk )). And, of course, these three bracketings correspond to the three binary trees below.

u

v

w

x

u

w

v

x

u

x

v

w

Let X denote the ﬁrst secant variety of the Segre variety P1 ×P1 ×P1 ×P1 . Thus X is the nine-dimensional irreducible subvariety of P15 consisting of all 2×2×2×2tensors which have tensor rank at most 2. The secant variety X has the parametric representation pijkl = π0 · u0i · v0j · w0k · x0l + π1 · u1i · v1j · w1k · x1l . This shows that the secant variety X equals the general Markov model for the tree below.

π

u 1

x v

w

2

3

4

The prime ideal of X is generated by all the 3 × 3-minors of the three matrices above. We write X(12)(34) for the variety deﬁned by the 3 × 3-minors of the leftmost matrix, X(13)(24) for the variety of the 3 × 3-minors of the middle matrix, and X(14)(23) for the variety of the 3 × 3-minors of the rightmost matrix. Then we have, scheme-theoretically, X = X(12)(34) ∩ X(13)(24) ∩ X(14)(23) .

(3.1)

These three varieties are the general Markov models for the three binary trees depicted above. For instance, the determinantal variety X(12)(34) equals the general Markov model for the binary tree below.

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a

u 1

245

b

v

w

x

2

3

4

Indeed, the standard parameterization φ of this model equals pijkl

=

π0 · (a00 u0i v0j + a01 u1i v1j ) · (b00 w0k x0l + b01 w1k x1l ) +π1 · (a10 u0i v0j + a11 u1i v1j ) · (b10 w0k x0l + b11 w1k x1l ).

This representation shows that the leftmost 4 × 4 matrix has rank at most 2, and, conversely, every 4 × 4 matrix of rank ≤ 2 can be written like this. We conclude that the general Markov model appears naturally when studying secant varieties of Segre varieties. It is instructive to redo the above calculations under the assumption u = v = w = x. Then the ambient P15 gets replaced by the four-dimensional space P4 with coordinates

p1 = p0001

p0 = p0000 = p0010 = p0100 = p1000

p2 = p0011 = p0101 = p0110 = p1001 = p1010 = p1100 p3 = p0111 = p1011 = p1101 = p1110 p4 = p1111 . Under these substitutions, all three 4 × 4-matrices reduce to the same 3 × 3-matrix ⎛ ⎞ p0 p1 p2 ⎝p1 p2 p3 ⎠ . p2 p3 p4 The ideal of 2×2-minors now deﬁnes the rational normal curve of degree four. This special Veronese variety is the small diagonal of the Segre variety P1 ×P1 ×P1 ×P1 ⊂ P15 . The secant variety of the rational normal curve is the cubic hypersurface in P4 deﬁned by the determinant of the 3 × 3 matrix. Hence, unlike (3.1), the homogeneous model satisﬁes X = X(12)(34) = X(13)(24) = X(14)(23) .

(3.2)

Studying the stratiﬁcations of Pn −1 induced by phylogenetic models, such as (3.1) and (3.2), will be one of the open problems to be presented in Section 5. k

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First, however, let us look at some widely used models which give rise to a nice family of toric varieties.

4. The Jukes-Cantor model The Jukes-Cantor model appears frequently in the computational biology literature and represents a family of toric varieties which have the unusual property that they are not toric varieties in their natural coordinate system. Furthermore, n while at ﬁrst glance they sit naturally inside of P4 −1 , the linear span of these models involve many fewer coordinates. In this section, we will present examples of these phenomena, as well as illustrate some open problems about the underlying varieties.

Example 4.1. Let T be the tree with 3 leaves below.

b a c 1

2

d 3

We consider the Jukes-Cantor DNA model of evolution, where each random variable has 4 states (the nucleotide bases A,C,G,T) and the root distribution is uniform, i.e., π = (1/4, 1/4, 1/4, 1/4). The transition matrices for the JukesCantor DNA model have the form ⎛ ⎞ a0 a1 a1 a1 ⎜ a1 a0 a1 a1 ⎟ ⎟ Ma = ⎜ ⎝ a1 a1 a0 a1 ⎠ . a1 a1 a1 a0 The transition matrices Mb , Mc , and Md are expressed in the same form as Ma with “a” replaced by b, c, and d respectively. From these matrices and the rooted tree T , we get the map φ : P1 × P1 × P1 × P1 → P63 , where the coordinates of P63 are the possible DNA bases at the leaves. For example,

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pAAA =

1 (a0 b0 c0 d0 + 3a1 b1 c0 d0 + 3a1 b0 c1 d1 + 3a0 b1 c1 d1 + 6a1 b1 c1 d1 ). 4

That is, pAAA is the probability of observing the triple AAA at the leaves of the tree. Since this parameterization is symmetric under renaming the bases, there are many linear relations. pAAA = pCCC = pGGG = pT T T

4 terms

pAAC = pAAG = pAAT = · · · = pT T G pACA = pAGA = pAT A = · · · = pT GT

12 terms 12 terms

pCAA = pGAA = pT AA = · · · = pGT T pACG = pACT = pAGT = · · · = pCGT

12 terms 24 terms

We are left with 5 distinct coordinates. From the practical standpoint, one is often interested in the accumulated coordinates, which are given parametrically as follows: p123 = pAAA + pCCC + pGGG + pT T T = e0 c0 d0 + 3e1 c1 d1 p12 = pAAC + pAAG + · · · + pT T G = 3e0 c0 d1 + 3e1 c1 d0 + 6e1 c1 d1 p13 = pACA + pAGA + · · · + pT GT = 3e0 c1 d0 + 3e1 c0 d1 + 6e1 c1 d1 p23 = pCAA + pGAA + · · · + pGT T = 3e1 c0 d0 + 3e0 c1 d1 + 6e1 c1 d1 pdis = pACG + pACT + · · · + pCGT = 6e1 c1 d0 + 6e1 c0 d1 + 6e0 c1 d1 + 6e1 c1 d1 where e0 = a0 b0 +3a1 b1 and e1 = a0 b1 +a1 b0 +2a1 b1 . Interpreting these coordinates in terms of the probabilistic model: p123 is the probability of seeing the same base at all three leaves, pij is the probability of seeing the same base at leaves i and j and a diﬀerent base at leaf k, and pdis is the probability of seeing distinct bases at the three leaves. Note that the image of φ is a three-dimensional projective variety. This is a consequence of the uniform root distribution in this model. The ﬁber over a generic point is isomorphic to P1 and stems from the fact that it is not possible to individually determine the matrices Ma and Mb . Only the product Ma Mb can be determined. It is easily computed that the vanishing ideal of this model is generated by one cubic with 19 terms. Remarkably, there exists a linear change of coordinates so that this polynomial becomes a binomial. Thus the corresponding variety is a toric variety in the new coordinates. This change of coordinates is given by the Fourier transform, see [24] for details. In these coordinates the parameterization factors:

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q0000 = p123 + p12 + p13 + p23 + pdis = (a0 + 3a1 )(b0 + 3b1 )(c0 + 3c1 )(d0 + 3d1 ) 1 1 1 q0011 = p123 − p12 − p13 + p23 − pdis = (a0 + 3a1 )(b0 + 3b1 )(c0 − c1 )(d0 − d1 ) 3 3 3 1 1 1 q1101 = p123 − p12 + p13 − p23 − pdis = (a0 − a1 )(b0 − b1 )(c0 + 3c1 )(d0 − d1 ) 3 3 3 1 1 1 q1110 = p123 + p12 − p13 − p23 − pdis = (a0 − a1 )(b0 − b1 )(c0 − c1 )(d0 + 3d1 ) 3 3 3 1 1 1 1 q1111 = p123 − p12 − p13 − p23 + pdis = (a0 − a1 )(b0 − b1 )(c0 − c1 )(d0 − d1 ) 3 3 3 3 In the Fourier coordinates, the cubic with nineteen terms becomes the binomial 2 . q0011 q1110 q1101 − q0000 q1111

These Fourier coordinates are indexed by the subforests of the tree, where we deﬁne a subforest of a tree to be any subgraph of the tree (necessarily a forest), all of whose leaves are leaves of the original tree. For instance, the coordinate q0000 corresponds to the empty subtree, the coordinate q1101 corresponds to the tree from leaf 1 to leaf 3 and not including the edge to leaf 2, and the coordinate q1111 corresponds to the full tree on three leaves. In general there are F2n−1 Fourier coordinates for a tree with n leaves, where Fm is the m-th Fibonacci number. Example 4.2. Now we consider an example of the Jukes-Cantor DNA model with uniform root distribution on the following tree T with 4 leaves.

c a 1

d b

e

f

2

3

4

The variety of this model naturally lives in a 44 −1 = 255 dimensional projective space. However, after noting the symmetry of the parameterization, as in the previous example, there are only 15 coordinates in this model which are distinct. After applying the Fourier transform, the parameterization factors into a product, and hence, the variety is naturally described as a toric variety in P14 . However, there are in fact 2 extra linear relations which are not simply expressed as a simple equality of probabilities so that our variety sits most naturally inside a P12 . Note

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that 13 = F2·4−1 , a Fibonacci number, as previously mentioned. We will present the parameterization in these 13 Fourier coordinates. Associated to each of the six edges in the tree is a matrix with two parameters (a0 and a1 , b0 and b1 , etc.) as in the previous example. The Fourier transform is a linear change of coordinates not only on the ambient space of the variety, but also on the parameter space. The new parametric coordinates are given by u0 = a0 + 3a1 ,

u 1 = a0 − a1 ,

v0 = b0 + 3b1 ,

v1 = b0 − b1 , . . .

and so on down the alphabet. To each subforest of the 4 taxa tree T , there is a coordinate qijklmn , where the index ijklmn is the indicator vector of the edges which appear in the subforest. The parameterization is given by the following rule: qijklmn = ui · vj · wk · xl · ym · zn . The ideal of phylogenetic invariants in the Fourier coordinates is generated by polynomials of degrees two and three. The degree two invariants are given by the 2 × 2 determinants of the following matrices: q000000 q000011 , M0 = q110000 q110011 ⎛ q101110 M1 = ⎝q011110 q111110

q101101 q011101 q111101

⎞ q101111 q011111 ⎠ . q111111

(4.1)

The dimensions of these matrices are also Fibonacci numbers. The rows of these matrices are indexed by the diﬀerent edge conﬁgurations to the left of the root and the columns are indexed by the edge conﬁgurations to the right of the root. There are also cubic invariants which do not have nice determinantal representations. They come in two types: q0000jk q1111lm q1111no − q1100jk q1011lm q0111no , qjk0000 qlm1111 qno1111 − qjk0011 qlm1101 qno1110 . The only condition on j, k, l, m, n, o is that each index is actually the indicator function of a subforest of the tree. The variety of the Jukes-Cantor model on a 4 taxa tree has dimension 5, so its secant variety is a proper subvariety in P12 . In applications, the secant varieties of the model are called mixture models. For this model, the secant variety has the expected dimension 11, and so is a hypersurface. Since the matrix M1 has rank 1 on the original model, it must have rank 2 on the secant variety: thus, the desired hypersurface is the 3 × 3 determinant of M1 . Example 4.3 (Determinantal closure). Now consider the Jukes-Cantor DNA model with uniform root distribution on a binary tree with 5 leaves, as pictured:

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Nicholas Eriksson, Kristian Ranestad, Bernd Sturmfels, Seth Sullivant

c

a

d e

b

f g

h

As in Example 4.1, the Fourier coordinates (modulo linear relations) are given by the subforests of T , of which there are 34. In the Fourier coordinates, this ideal is generated by binomials of degree 2 and 3, of the types we have seen in the previous example. While the cubic invariants have a relatively simple description, the quadratic invariants are all represented as the 2 × 2 determinants of matrices naturally associated to the tree. In particular, tools from numerical linear algebra can be used to determine if these invariants are satisﬁed. Since the degree 2 invariants are all determinantal, it seems natural to ask what algebraic set these determinantal relations cut out: that is, what is the determinantal closure of the variety of the Jukes-Cantor DNA model on a ﬁve taxa tree? The ideal of this determinantal closure is generated by the 2 × 2-minors of the four following matrices: q11001111 q00001111 q10111000 q11111000 q01111000

q10110101 q11110101 q01110101

q10110110 q11110110 q01110110

q11111000 q11111011 q00001101 q00001111 q00001110

q10110101 q10110111 q10110110

q11000011 q00000011

q10111011 q11111011 q01111011

q11000000 q11000011

q01110101 q01110111 q01110110

q11001110 q00001110

q10110111 q11110111 q01110111

q01111000 q01111011

q11001101 q11001111 q11001110

q11001101 q00001101

q10111101 q11111101 q01111101

q10011000 q10111011

q11110101 q11110111 q11110110

q11000000 q00000000 q10111110 q11111110 q01111110

q10111111 q11111111 q01111111

q00000000 q00000011

q10111101 q10111111 q10111110

q01111101 q01111111 q01111110

q11111101 q11111111 q11111110

Surprisingly, this ideal is actually a prime ideal, and so the algebraic set is a toric variety. It has dimension 10 and degree 501, whereas this Jukes-Cantor model has only dimension 7. How does the Jukes-Cantor model sit inside its determinantal closure?

5. Problems The main problem in phylogenetic algebraic geometry is to understand the complex variety, i.e., the complex Zariski closure XC = φ(P ),

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251

of a phylogenetic model. This problem has many diﬀerent reformulations, depending on the point of view of the person posing the problem. One problem posed by computational biologists [6, 16] is to determine the “phylogenetic invariants” of the model. Problem 5.1 (Phylogenetic invariants). Find generators of the ideal deﬁning XC . Problem 5.2. Which equations or phylogenetic invariants are needed to distinguish between diﬀerent models? These problems are of particular interest for applications in phylogenetics, where one wishes to ﬁnd which tree gives the evolutionary history of a set of taxa. Some more geometric problems are: Problem 5.3. What are the basic geometric invariants of φ and XC for the various models? • What is the dimension of XC ? • If φ is generically ﬁnite, what is the generic degree? • What is the degree of XC ? • What is the base locus or indeterminacy locus of φ? • What is the singular locus of XC ? Problem 5.4. For a ﬁxed type of model with k states, and number n of leaves (or taxa), consider the set of rooted trees with n leaves and the corresponding n arrangement A of varieties in Ck . Describe the stratiﬁcation of A, where two points in A are in the same strata if they are contained in the intersection of the same models. Is the stratiﬁcation of A the same as the stratiﬁcation of the space of phylogenetic trees (cf. [4])? The tropicalization of a variety is the “logarithmic limit set” of the points on the complex variety. Tropical geometry is the geometry of the min-plus semi-ring. It was shown in [18] that the tropical geometry of statistical models plays a crucial role in parametric inference. Problem 5.5. Determine the combinatorial structure of the tropicalizations of the various models of evolution. In particular, work out parametric inference for the substitution model. Problem 5.6. How does the tropicalization of a mixture model relate to the tropical mixture of the tropicalization of the model: that is, compare the tropicalization of secant varieties and joins to the secant varieties and joins of tropicalizations, see [7].

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Nicholas Eriksson, Kristian Ranestad, Bernd Sturmfels, Seth Sullivant

In practice, it can be diﬃcult to ﬁnd a full set of generators of the ideal of XC . Therefore, we suggest certain subsets of the ideal that may be enough to distinguish between diﬀerent models (as Problem 5.2 asks). We think of these subsets as types of closure operation, for example, XC is the Zariski closure (over C) of XR . We suggest the following closures as possibly easier to ﬁnd and use: Linear closure, the linear span of XC . For work on this problem, see [22]. Quadratic closure deﬁned by the quadratic generators of the ideal. This is closely related to the conditional independence closure from algebraic statistics, which is deﬁned by determinantal quadratic generators, i.e., quadrics of rank 4. Determinantal closure deﬁned by the determinantal polynomials in the ideal. For example, there is a large set of determinantal relations that hold for any of the models deﬁned above. In practice, having large sets of determinantal generators of the ideal is convenient, as determinantal conditions can be eﬀectively evaluated using numerical linear algebra, see [9]. Local closures deﬁned by invariants that each depend only on subtrees of T . Often these give all the invariants for a model, e.g., [24]. Orbit closures applicable if the parameter space has a dense orbit under some group and φ is equivariant. Possible related objects are quiver varieties and hyperdeterminants, see [25]. Note that part of the diﬃculty of studying these closure operations is coming up with a good deﬁnition for them. Problem 5.7. Study the stratiﬁcations induced by the union of the set of “closures” of these varieties for a given model with ﬁxed numbers of leaves (or taxa). From these rather general problems we turn to more speciﬁc, computationallyoriented problems. Many of them are special cases of the general problems above and are concrete starting points for attempting to resolve these more general problems. They also serve as an introduction to the complexity that can arise. Problem 5.8. Consider a tree T with n leaves and consider the subvariety of (C2 )⊗n consisting of all 2 × 2 × · · · × 2-tables P such that all ﬂattenings of P along edges that splits T have rank at most r. Is this variety irreducible? Do the determinants deﬁne a reduced scheme? What is the dimension of this variety? Problem 5.9. Consider the general Markov model on a non-binary tree T with 6 leaves. Is the variety XC equal to the intersection of all models from binary trees on 6 leaves which are reﬁnements of T ? If the answer is yes, does the same statement hold scheme-theoretically? Problem 5.10. Given two trees T and T on the same number of taxa, what are the irreducible components of the intersections of their corresponding varieties?

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Problem 5.11. For all trees with at most eight leaves, compute a basis for the space of linear invariants of the homogeneous Markov model, with and without hidden nodes. What about quadratic invariants? Problem 5.12. What is the dimension of the Zariski closure of the substitution model? Problem 5.13. Classify all phylogenetic models that are smooth. Problem 5.14. Compute the phylogenetic complexity of the group Z2 × Z2 (see [24, Conjecture 28]). Problem 5.15. Study the secant varieties of the Jukes-Cantor binary model for all trees with at most six leaves. Do any of them fail to have the expected dimension? When do determinantal conditions suﬃce to describe these models? Problem 5.16. Let T be the balanced binary tree on four leaves. Compute the Newton polytope (as deﬁned in [18]) of the homogeneous model for DNA sequences.

References [1] E. Allman and J. Rhodes. Phylogenetic invariants for the general Markov model of sequence mutation. Mathematical Biosciences 186 (2003) 133-144. [2] E. Allman and J. Rhodes. Quartets and parameter recovery for the general Markov model of sequence mutation, Applied Mathematics Research Express, to appear. Available at http://abacus.bates.edu/~jrhodes/. [3] E. Allman and J. Rhodes. Phylogenetic invariants for stationary base composition, Journal of Symbolic Computation, to appear. Available at http://abacus.bates.edu/~jrhodes/. [4] L. J. Billera, S. Holmes, and K. Vogtmann. Geometry of the space of phylogenetic trees. Advances in Applied Mathematics 27 (2001) 733-767. [5] M. V. Catalisano, A. V. Geramita, and A. Gimigliano. Higher secant varieties of Segre-Veronese varieties, math.AG/0309399. [6] J. A. Cavender and J. Felsenstein. Invariants of phylogenies: a simple case with discrete states. Journal of Classiﬁcation 4 (1987) 57-71. [7] M. Develin. Tropical secant varieties of linear spaces, 2004, math.CO/0405115, submitted to Discrete and Computational Geometry. [8] N. Eriksson. Toric ideals of homogeneous phylogenetic models, math.CO/0401175, Proceedings of ISSAC, 149-154. ACM Press, 2004. [9] N. Eriksson. Tree construction with singular value decomposition, in preparation, 2004. Available at http://math.berkeley.edu/~eriksson. [10] S. Evans and T. Speed. Invariants of some probability models used in phylogenetic inference. Annals of Statistics 21 (1993) 355-377.

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[11] J. Felsentein. Inferring Phylogenies. Sinauer Associates, Inc., Sunderland, 2003. [12] V. Ferretti and D. Sankoﬀ. Phylogenetic invariants for more general evolutionary models, Journal of Theoretical Biology 173 (1995) 147-162. [13] V. Ferretti and D. Sankoﬀ. A remarkable nonlinear invariant for evolution with heterogeneous rates, Mathematical Biosciences 134 (1996) 71-83. [14] R. Guralnick, and B. Sethuraman. Commuting pairs and triples of matrices and related varieties Linear Algebra Appl. 310 (2000) 139-148. [15] T. Hagedorn. Determining the number and structure of phylogenetic invariants, Advances in Applied Mathematics 24 (2000) 1-21. [16] J. A. Lake. A rate-independent technique for analysis of nucleic acid sequences: evolutionary parsimony. Molecular Biology and Evolution 4 (1987) 167-191. [17] J. M. Landsberg and L. Manivel. On the ideals of secant varieties of Segre varieties, math.AG/0311388, Foundations of Computational Mathematics, to appear. [18] L. Pachter and B. Sturmfels. Tropical geometry of statistical q-bio.QM/0311009, Proceedings Natl. Acad. Sci. USA, to appear.

models,

[19] A. Premet. Nilpotent commuting varieties of reductive Lie algebras. Invent. Math. 154 (2003) 653–683. [20] D. Sankoﬀ and M. Blanchette. Phylogenetic invariants for genome rearrangements. Journal of Computational Biology 6 (1999) 431-445. [21] C. Semple and M. Steel. Phylogenetics. Oxford University Press, Oxford, 2003. [22] M. Steel and Y. Fu. Classifying and counting linear phylogenetic invariants for the Jukes Cantor model. Journal of Computational Biology 2 (1995) 39-47. [23] M. Steel, L. Sz´ekely, P. Erd¨ os, and P. Waddell. A complete family of phylogenetic invariants for any number of taxa under Kimura’s 3ST model. NZ Journal of Botany 13 (1993) 289-296. [24] B. Sturmfels and S. Sullivant. Toric ideals of phylogenetic q-bio.PE/0402015, to appear in Journal of Computational Biology.

invariants,

[25] J. G. Sumner and P. D. Jarvis. Entanglement invariants and phylogenetic branching, q-bio.PE/0402007.

Nicholas Eriksson Department of Mathematics University of California Berkeley, CA 94720-3840 USA Email: [email protected] Kristian Ranestad Department of Mathematics PB 1053 Blindern 0316 Oslo, Norway Email: [email protected]

Phylogenetic algebraic geometry Bernd Sturmfels Department of Mathematics University of California Berkeley, CA 94720-3840 USA Email: [email protected] Seth Sullivant Department of Mathematics University of California Berkeley, CA 94720-3840 USA Email: [email protected]

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Some results on fat points whose support is a complete intersection minus a point Elena Guardo and Adam Van Tuyl

Abstract. This paper is an investigation of some of the numerical invariants associated to a set of fat points Z ⊆ P2 when the support of Z is a complete intersection minus a point. 2000 Mathematics Subject Classiﬁcation: 13D40, 13D02, 13H10, 14A15

1. Introduction Let k be an algebraically closed ﬁeld of characteristic zero and R √ = k[x, y, z]. If I = (F, G) ⊆ R is a complete intersection of type (a, b), and if I = I, then I is the deﬁning ideal of ab points in P2 . We shall say such a set X is a complete intersection of points of type (a, b), and we will denote it by CI(a, b). Complete intersections of points satisfy the Cayley-Bacharach property (see [8]); more precisely: Theorem 1.1. Let X = CI(a, b) ⊆ P2 and let P ∈ X be any point. Then the Hilbert function of Y = X\{P } is given by HY (t) = min{HX (t), |X| − 1} for all t ≥ 0 where HX is the Hilbert function of X. Since the function HX depends only on the type (a, b) of X, we see that HY only depends upon the type as well. The theme of this paper is to understand Theorem 1.1 in the context of fat points. Let us be more speciﬁc about the problem. Let P1 , . . . , Pab be the ab points of the complete intersection X = CI(a, b), and let m ∈ N+ . Then the ideal IZ = IPm1 ∩ · · · ∩ IPmab where Pi ↔ IPi deﬁnes a homogeneous fat point scheme Z ⊆ P2 whose support is the complete m . Since intersection X. Using [18, Lemma 5, Appendix 6] we have that IZ = IX IZ is a power of a complete intersection, HZ depends only upon the type (a, b) and m (see [11] for details). We will sometimes write {CI(a, b); m} for Z.

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Now let P ∈ X be any point and consider the ideal m m IY = IPm1 ∩ · · · ∩ I P ∩ · · · ∩ IPab .

The ideal IY deﬁnes the subscheme Y ⊆ Z formed by removing the fat point (P, m) from Z. Note that the support of Y is a complete intersection minus a point. We shall denote Y by {CI(a, b); m}\{(P, m)}. Because the Hilbert function of Z depends only upon (a, b) and m, it is natural to wonder if an analog of Theorem 1.1 holds for Y ; that is, does the Hilbert function of Y depend only upon HZ ? Examples in [2, 3, 9, 10, 11] show that the answer to this question is no. In keeping with the theme of these Proceedings, the Hilbert function of Y has the unexpected property that diﬀerent constructions of the underlying complete intersection, e.g., whether or not the forms deﬁning CI(a, b) are irreducible, can result in diﬀerent Hilbert functions. As well, examples can be constructed where HY depends upon what point P is removed from the support X. To study the invariants associated to Y it is therefore necessary to introduce extra conditions on the underlying support. To this end, we introduce some relevant notation. Suppose I = (F, G), with deg F = a and deg G = b, deﬁnes a complete intersection of points of type (a, b). If both F and G are irreducible, then we say such a set of points is of type CIgen (a, b), and we sometimes denote this set by Xgen . If F and G are the product of linear forms, i.e., F = L1 · · · La and G = L1 · · · Lb , then we say such a set of points is of type CIgrid (a, b), and we denote sets of point of this type by Xgrid . (The subscript grid is used to denote the fact that the points of X are the points of intersection in the grid formed by the lines deﬁned by the Li s and Lj s.) We write Zgen = {CIgen (a, b); m} to denote the homogeneous fat point scheme with Supp(Zgen ) = Xgen of multiplicity m, and Ygen = {Xgen ; m}\{(P, m)} to be the homogeneous fat point scheme with Supp(Ygen ) = Xgen \{P }. The schemes Zgrid and Ygrid are deﬁned similarly. If X, Z, or Y is written without a subscript, then the results depend only on the type. We describe the results in this paper. In §2, we investigate the invariant α(IY ) := min{i | (IY )i = 0}, the smallest degree of a form passing through Y . We begin by showing that computing dimk (IY )t is equivalent to computing dimk [(Lm ) ∩ IZ ]t+m for an appropriate linear form L. We then use this result to show that if a < b, then α(IY ) = α(IZ ) regardless of the construction of the support. A conjecture on the behavior of α(IY ) when a = b is given at the end of the section. In §3, we restrict to double points, that is, when m = 2. It is shown that when a ≤ b, except in some special cases, Ygen and Ygrid have the same graded Betti numbers. We also describe these numbers in terms of a and b. We now survey known results. Some of the invariants of Y were already known in the case that the complete intersection lies on a curve of small degree, i.e., 1 ≤ a ≤ 3. When a = 1, the support of the set of fat points lies on a line. Thus HY can be determined from a result of Davis and Geramita [7]. If the complete intersection lies on a conic in P2 , i.e., a = 2, then HY follows from Catalisano [5] (in the case of smooth conics) and Harbourne [14] (for arbitrary conics). When a = 3, the support lies on a plane cubic (possibly reducible and nonreduced) in P2 .

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The function HY follows from [15]. As well, [14] contains results on the resolutions of fat points in P2 lying on plane curves of degree at most three. Fat points on other complete intersections in P2 were studied in [2, 3, 9, 10]. The paper [11] initiated an investigation of this problem for complete intersections of points in Pn .

2. The form of smallest degree passing through Y If Z ⊆ P2 is any fat point scheme, then the numerical character α(IZ ) := min{i | (IZ )i = 0} is the smallest degree of a form passing through Z. As described in [13, page 89], if one can compute α(Z) for any Z, then one can compute HW for any speciﬁc fat point scheme W (see also [12] in this volume). Therefore, it is of interest to study α(IY ) if Y = Z\{(P, m)} and (P, m) is any fat point of Z = {CI(a, b); m}. We begin by giving an alternative way to compute dimk (IY )t . Lemma 2.1. Let X = CI(a, b) ⊆ P2 . Fix an m ∈ N+ and P ∈ X, and set Z = {X; m} and Y = Z\{(P, m)}. Then there exists a linear form L ∈ R that satisﬁes the following properties: 1. the line deﬁned by L passes through P but not through any Q ∈ X\{P }. 2. if IX = (F, G), then L |F or G. 3. dimk (IY )t = dimk [(Lm ) ∩ IZ ]t+m for all t ∈ N. Proof. There are an inﬁnite number of lines that pass through P . However, only a ﬁnite number of these lines will also pass through a point Q ∈ X\{P }. Furthermore, only a ﬁnite number of linear forms can divide F or G. So, it is clear that a linear form L ∈ R can be found that satisﬁes (1) and (2). Let L be a linear form that satisﬁes (1) and (2), and consider the following short exact sequence: ×Lm

0 −→ R/(IZ : (Lm ))(−m) −→ R/IZ −→ R/(IZ , Lm ) −→ 0. Because (IZ : (Lm )) = IY , the short exact sequence implies that dimk (IY )t = dimk Rt + dimk (IZ )t+m − dimk (IZ , Lm )t+m for all t ∈ N. Now dimk (IZ , Lm )t+m = dimk (IZ )t+m + dimk (Lm )t+m − dimk ((Lm ) ∩ IZ )t+m . Substituting this expression into the above identity, and also using the fact that dimk (Lm )t+m = dimk Rt then gives us the conclusion dimk (IY )t = dimk ((Lm ) ∩ IZ )t+m for all t ∈ N.

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It follows from Lemma 2.1 that to determine HY , it is enough to determine the Hilbert function of R/((Lm ) ∩ IZ ). Lemma 2.1 will allow us to show that m , it follows that α(IZ ) = ma, α(IY ) = α(IZ ) when a < b. Note that since IZ = IX and because IZ ⊆ IY , α(IY ) ≤ α(IZ ). Theorem 2.2. Let X = CI(a, b) ⊆ P2 . Fix an m ∈ N+ and P ∈ X, and set Z = {X; m} and Y = Z\{(P, m)}. 1. If a < b, then α(IY ) = α(IZ ) = ma. 2. If m ≥ 2 and m(a + 1) − 3 < (m − 1)b, then HY (t) = HZ (t) if and only if t ≤ ma + b − 3, or equivalently, α(IY /IZ ) = ma + b − 2 when we consider IY /IZ as an ideal of R/IZ . Proof. The proof of both statements will be by induction on m. We therefore set Zm := {CI(a, b); m} and Ym := {CI(a, b); m}\{(P, m)} for each m ∈ N+ . (1) If m = 1, then the conclusion follows from Theorem 1.1. So, suppose m ≥ 2. We need to show that dimk (IYm )ma−1 = 0. If L is the linear form of Lemma 2.1, then it shall be enough to show that ((Lm ) ∩ IZm )ma+m−1 = 0 since Lemma 2.1 (3) then implies dimk (IYm )ma−1 = dimk ((Lm ) ∩ IZm )ma+m−1 = 0. So, suppose K ∈ ((Lm ) ∩ IZm )ma+m−1 , and hence K = Lm A ∈ (IZm )ma+m−1 . Because a < b, we have ma + m − 1 = m(a + 1) − 1 ≤ mb − 1 < mb. So (IZm )ma+m−1 = (F m , F m−1 G, . . . , F Gm−1 , Gm )ma+m−1 = F (IZm−1 )(m−1)a+m−1 . We thus have Lm A = F H with H ∈ IZm−1 . Now L does not divide F , so H = Lm H ∈ (IZm−1 )(m−1)a+m−1 ⊆ (IYm−1 )(m−1)a+m−1 . Because L is a nonzero divisor on R/(IYm−1 ), this means that H ∈ (IYm−1 )(m−1)a−1 . By induction, we now have H = 0, and hence K = 0 as desired. (2) (⇒) Let t ≥ ma + b − 2. It follows from Theorem 1.1 that there exists a form H such that H ∈ (ISupp(Y ) )a+b−2 \(IX )a+b−2 . Let K be any linear form that deﬁnes a line that does not pass through any point of X. Then F m−1 HK t−ma−b+2 ∈ (IY )t \(IZ )t , thus giving HY (t) = HZ (t). (⇐) Suppose m ≥ 2 and m(a + 1) − 3 < (m − 1)b, and let L be the linear form of Lemma 2.1. Our goal is to show that ((Lm ) ∩ IZm )t+m = Lm (IZm )t for each t ≤ ma + b − 3. Then by Lemma 2.1 (3) we will have dimk (IYm )t = dimk ((Lm ) ∩ IZm )t+m = dimk Lm (IZm )t = dimk (IZm )t which implies HZ (t) = HY (t) for t ≤ ma + b − 3.

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Suppose m = 2. Since it is clear that Lm (IZm )t ⊆ ((Lm ) ∩ IZm )t+m , we need to show the reverse inclusion. So, suppose K ∈ ((Lm ) ∩ IZm )t+m , and hence K = Lm A ∈ (IZm )t+m . Because t + m ≤ ma + b − 3 + m = m(a + 1) − 3 + b < (m − 1)b + b = mb we have (IZm )t+m = (F m , F m−1 G, . . . , F Gm−1 , Gm )t+m = F (IZm−1 )t+m−a . Hence K = Lm A = F H with H ∈ (IZm−1 )t+m−a ⊆ (IYm−1 )t+m−a . Because L |F , we have H = Lm H . Furthermore, since L is a nonzero divisor on R/(IYm−1 ) we thus have H ∈ (IYm−1 )t−a . Now t−a ≤ (m−1)a+b−3. Since m = 2, (IYm−1 )t−a = (IZm−1 )t−a by Theorem 1.1. So F H ∈ (IZm )t , and thus K ∈ Lm (IZm )t . If m > 2, then the proof is the same as the above argument, except that the induction hypothesis, instead of Theorem 1.1, is used to justify the fact that

(IYm−1 )t−a = (IZm−1 )t−a . Remark 2.3. Theorem 2.2 (1) was ﬁrst conjectured to hold in [3]. Computer evidence suggests that statement (2) can be improved to: if a < b and m ∈ N+ , then HZ (t) = HY (t) if and only if t ≤ ma + b − 3. Notice that the proof of the (⇒) direction of (2) already proves one direction of this result. Furthermore, when m = 1 this statement is true by Theorem 1.1; if m = 2 then this follows from [3, Theorem 4.4]. The Hilbert function HY for m = 3 is studied in [10]. When a = b, the value α(IY ) may be diﬀerent than α(IZ ). This was ﬁrst shown in [3, Proposition 3.6]: Proposition 2.4. Let X = CI(a, a) ⊆ P2 . Let m ∈ N+ , P ∈ X, and set Z = {X; m} and Y = Z\{(P, m)}. If m > a2 − a − 1, then α(IY ) < α(IZ ). Under the hypotheses of the previous proposition, if we set mY := max{m | α(IY ) = α(IZ )}, the above result then implies that mY ≤ a2 − a − 1. Surprisingly, the value of mY seems to depend on the construction of the underlying complete intersection. Using computer evidence generated by CoCoA [6], we have made the following conjecture on the value of α(IY ). Conjecture 2.5. Let X = CI(a, a) ⊆ P2 , and let Y = {X; m}\{(P, m)} for any P ∈ X. 1. α(IYgrid ) = ma if and only if m ≤ (a − 1)2 . That is, mYgrid = (a − 1)2 . 2. α(IYgen ) = ma if and only if m ≤ a2 − a − 1. That is, mYgen = a2 − a − 1. Remark 2.6. B. Harbourne [16] has shown us that if a > 3 and X = Xgen = 2 CIgen (a, a) ⊆ P2 , then a 2−a ≤ mYgen < a2 − a, and if X = Xgrid = CIgrid (a, a), then a − 1 ≤ mYgrid < a2 − a − 2. This conjecture will be further explored in [1].

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3. Graded Betti numbers Let X = CI(a, b) ⊆ P2 . By [3, Theorem 4.4] Ygrid = {Xgrid ; 2}\{(P, 2)} and Ygen = {Xgen ; 2}\{(P, 2)} have the same Hilbert function. However, Example 4.6 in [3] showed that they may not have the same graded Betti numbers. Here we show that Ygen and Ygrid always have the same graded Betti numbers if a ≤ b except in the case that (a, b) = (2, b) with b ≥ 3. We shall require the following lemmas. The ﬁrst is a consequence of Theorem 1.1. Lemma 3.1. Let X = CI(a, b) ⊆ P2 with deﬁning ideal IX = (F, G). Let P ∈ X be any point and set Y = X\{P }. Then there exists a form H with deg H = a+b−2 such that IY = (F, G, H). Furthermore, the graded minimal free resolution of IY has the form 0 → R2 (−a − b + 1) → R(−a) ⊕ R(−b) ⊕ R(−a − b + 2) → IY → 0. We deﬁne ΔH(t) := H(t) − H(t − 1) with H(t) = 0 if t < 0. More generally, Δd H(t) := Δd−1 H(t) − Δd−1 H(t − 1). Lemma 3.2. Let X ⊆ P2 be any zero-dimensional scheme with deﬁning ideal IX and Hilbert function HX . If αt , respectively βt , denotes the number generators, respectively the number of syzygies, of degree t of IX , then −Δ3 HX (t) = αt − βt . Furthermore, we have the bounds max{0, −Δ3 HX (t)} ≤ αt ≤ −Δ2 HX (t) and max{0, Δ3 HX (t)} ≤ βt ≤ −Δ2 HX (t − 1). Proof. See [4] and [17].

We now come to the main theorem of this section. Theorem 3.3. In P2 consider the two schemes of double points Ygen = {CIgen (a, b); 2}\{(P, 2)} and Ygrid = {CIgrid (a, b); 2}\{(P, 2)} with a ≤ b. We write Y to indicate both Ygrid and Ygen . 1. If a = 1 and b > 1, then the resolution of Y has the form 0 → R(−b − 1) ⊕ R(−2b + 1) → R(−2) ⊕ R(−b − 2) ⊕ R(−2b + 2) → IY → 0. 2. If a = 2 and b ≥ 3, then

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(i) the resolution of IYgrid has the form 0 → R(−2b) ⊕ R(−2b − 1) ⊕ R2 (−b − 3) → → R(−4) ⊕ R(−2b) ⊕ R2 (−b − 2) ⊕ R(−2b + 1) → IYgrid → 0. (ii) the resolution of IYgen has the form 0 → R(−2b − 1) ⊕ R2 (−b − 3) → → R(−4) ⊕ R2 (−b − 2) ⊕ R(−2b + 1) → IYgen → 0. 3. If 2 < a < b, then the resolution of IY has the form 0 → R(−a − 2b + 1) ⊕ R(−a − 2b + 2) ⊕ R2 (−2a − b + 1) → → R(−2a)⊕R(−2b)⊕R(−a−2b+3)⊕R(−a−b)⊕R(−2a−b+2) → IY → 0. 4. If 1 < a = b, then the resolution of IY has the form 0 → R3 (−3a + 1) → R3 (−2a) ⊕ R(−3a + 3) → R/IY → 0. Proof. (1) If a = 1 and 1 < b, then Ygen = Ygrid = {CI(1, b − 1); 2}, in which case the result follows from [7]. (2) If a = 2 and b ≥ 3, then, from [2, Proposition 4.7], Ygrid = {CIgrid (2, b); 2}\ {(P, 2)} has a minimal free resolution of type (i). Because Ygen = {CIgen (2, b)\{(P, 2)} lies on an irreducible conic, we can use [5] to verify that the minimal free resolution has the form given in (ii). Therefore Ygrid and Ygen have diﬀerent graded Betti numbers when (a, b) = (2, b) with b ≥ 3. (3) Suppose 2 < a < b. Using [2, Proposition 4.7] Ygrid = {CIgrid (a, b); 2}\{(P, 2)} has the same graded Betti numbers as a partial intersection Y p.i. of type (p, q) where p = (2b, 2b − 2, b, b − 1) and q = (a − 1, 1, a − 1, 1). Hence, the minimal graded free resolution of IYgrid has the form given in (3). Since Ygrid and Ygen are both zero-dimensional schemes of P2 , and from [3] they have the same Hilbert function, to show that they have the same minimal graded Betti numbers, it suﬃces to show that the number of generators of IYgen and their degrees are the same as those of IYgrid . Then the degrees of the syzygies are a consequence of Lemma 3.2. Let F and G denote the two irreducible forms of degree a and b, respectively, that generate CIgen (a, b). We shall consider the cases b ≥ 2a − 2 and b < 2a − 2 separately. If b ≥ 2a − 2, from [3, Theorem 4.4] we get

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Δ3 HYgen (t) =

1 0 −1 0 −1 0 −1 2 0 −2 if a = 3 −1 if a = 3 1 if a = 3 0 if a = 3 0 −1 1 0

t=0 1 ≤ t ≤ 2a − 1 t = 2a 2a + 1 ≤ t ≤ a + b − 1 t=a+b a + b + 1 ≤ t ≤ 2a + b − 3 t = 2a + b − 2 t = 2a + b − 1 2a + b ≤ t ≤ 2b − 1 t = 2b t = 2b + 1, 2b + 2 2b + 3 ≤ t ≤ a + 2b − 4 t = a + 2b − 3 t = a + 2b − 2, a + 2b − 1 t ≥ a + 2b.

Let αi , respectively βi , denote the number of generators, respectively syzygies, in degree i. So, if a = 3, we want to show that α2a = αa+b = α2a+b−2 = 1, α2b = 2, and αi = 0 otherwise, and if a = 3, then we want to show α2a = αa+b = α2a+b−2 = α2b = αa+2b−3 = 1 and αi = 0 otherwise. It is clear that α2a = αa+b = 1 for all a ≥ 3, and αi = 0 for all other i ≤ a + b. Since F 2 , F G ∈ IYgen with deg F 2 = 2a and deg F G = a + b, we can assume that these forms are the minimal generators of degrees 2a and a + b, respectively. By Lemma 3.2 we have 0 ≤ βi ≤ 1 for a + b < i ≤ 2a + b − 2. If βi = 1 in this interval, there must be a syzygy among the generators of degree < 2a + b − 2, i.e., there exist f1 and f2 such that f1 F 2 + f2 F G = 0 with f1 and f2 forms of suitable degrees. But this implies that f1 F + f2 G = 0, i.e., that this syzygy must have degree 2a + b because F and G form a regular sequence. This contradicts the fact that i ≤ 2a + b − 2. So, using Lemma 3.2, βi = αi = 0 for a + b < i < 2a + b − 2, α2a+b−2 = 1, and β2a+b−2 = 0. Let H be the form of degree a + b − 2 of Lemma 3.1. Then F H ∈ IYgen and deg F H = 2a + b − 2. Moreover, since F H is not in the ideal generated by F 2 and F G, we can take F H to be the minimal generator of degree 2a + b − 2. To compute αi for 2a + b − 1 ≤ i ≤ 2b, we again compute βi on this interval. Note that there is a syzygy of degree i among F 2 , F G, F H, that is, f1 F 2 +f2 F G+ f3 F H = 0, if and only if there is a syzygy among F, G, and H of degree i − a, that is, f1 F + f2 G + f3 H = 0. But F, G, H generate ISupp(Ygen ) , and by Lemma 3.1, there are only two minimal syzygies of degree a + b − 1. But this means that β2a+b−1 = 2 and βi = 0 for i = 2a + b, . . . , 2b. Then Lemma 3.2 gives αi = 0 for all i = 2a + b − 1, . . . , 2b − 1, and α2b = 2 if a = 3, and α2b = 1 if a = 3. If a = 3, then we are done since the regularity of IYgen equals a+2b−3 (this can be read oﬀ of the Hilbert function), so there are no generators of higher degree.

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So, suppose a = 3 and compute αi for 2b + 1 ≤ i ≤ a + 2b − 3. Since G2 ∈ IYgen and deg G2 = 2b, we can take G2 to be the minimal generator of degree 2b. Using Lemma 3.2 we have 0 ≤ βi ≤ 1 for 2b + 1 ≤ i ≤ a + 2b − 3. If βi = 1 in this interval, then f1 F 2 + f2 F G + f3 F H + f4 G2 = 0 with f1 , f2 , f3 , f4 forms of suitable degrees. But this means G(f4 G) = F K for some suitable form K. This contradicts the fact that F and G form a regular sequence. Hence, βi = αi = 0 if i = 2b + 1, . . . , a + 2b − 4, βa+2b−3 = 0 and αa+2b−3 = 1. Again, since reg(IYgen ) = a + 2b − 3, there can be no other generators of higher degree, thus completing the case that b ≥ 2a − 2. If b < 2a − 2 then ⎧ t+1 0 ≤ t ≤ 2a − 1 ⎪ ⎪ ⎪ ⎪ 2a 2a ≤t≤a+b−1 ⎪ ⎪ ⎪ ⎪ a + b ≤ t ≤ 2b − 1 ⎨ 3a + b − 1 − t 3a + 3b − 2 − 2t 2b ≤ t ≤ 2a + b − 3 ΔHYgen (t) = ⎪ ⎪ a + 2b − 1 − t 2a + b − 2 ≤ t ≤ a + 2b − 4 ⎪ ⎪ ⎪ ⎪ 1 t = a + 2b − 3 ⎪ ⎪ ⎩ 0 t ≥ a + 2b − 2 . Using an argument similarly to the one given above, we get the conclusion. (4) It was shown in [3] that if a = b, then Ygrid and Ygen have the same Hilbert function, namely: ⎧ t+1 0 ≤ t ≤ 2a − 1 ⎪ ⎪ ⎪ ⎨ 2(3a − t − 1) 2a ≤ t ≤ 3a − 4 ΔHYgrid (t) = ΔHYgen (t) = ⎪ 3 t = 3a − 3 ⎪ ⎪ ⎩ 0 t ≥ 3a − 2. Now arguing as we did above, we get that both Ygrid and Ygen have a minimal resolution of type: 0 → R3 (−3a + 1) → R3 (−2a) ⊕ R(−3a + 3) → R/IY → 0.

Acknowledgments. The authors would like to thank the organizers of “Projective Varieties with Unexpected Properties” for an interesting conference, and for the opportunity to visit beautiful Siena. The second author also would like to acknowledge the ﬁnancial support of Natural Sciences and Engineering Research Council of Canada.

References [1] M. Buckles, E. Guardo, B. Harbourne, A. Van Tuyl, Some results on forms of least degree in ideals of fat points in special position, in preparation. [2] M. Buckles, E. Guardo, A. Van Tuyl, Fat points on a grid in P2 , Le Matematiche (Catania) 55 (2000), 169–189 (2001). [3] M. Buckles, E. Guardo, A. Van Tuyl, Fat points on a generic almost complete intersection, Le Matematiche (Catania) 55 (2000), 191–202 (2001).

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[4] G. Campanella, Standard bases of perfect homogeneous polynomial ideals of height 2, J. Algebra 101 (1986), 47–60. [5] M.V. Catalisano, “Fat” points on a conic, Comm. Algebra 19 (1991), 2153–2168. [6] CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra, Available at http://cocoa.dima.unige.it [7] E.D. Davis, A.V. Geramita, The Hilbert function of a special class of 1dimensional Cohen-Macaulay graded algebras, The curves seminar at Queen’s, Vol. III (Kingston, Ont., 1983), Exp. No. H, 29 pp., Queen’s Papers in Pure and Appl. Math., 67 Queen’s Univ., Kingston, ON, 1984. [8] A.V. Geramita, M. Kreuzer, L. Robbiano, Cayley-Bacharach schemes and their canonical modules, Trans. Amer. Math. Soc. 339 (1993), 163–189. [9] E. Guardo, Schemi di “Fat Points”, Ph.D. Thesis, Universit`a di Messina (2000). [10] E. Guardo, Subschemes of fat points on complete intersections and partial intersections, (2004) Preprint. [11] E. Guardo, A. Van Tuyl, Powers of complete intersections: graded Betti numbers and applications (2004), to appear in Illinois Journal of Mathematics math.AC/0409090 [12] B. Harbourne, The (unexpected) importance of knowing alpha, (2004), in this volume. [13] B. Harbourne, Problems and progress: a survey on fat points in P2 , in Zerodimensional schemes and applications (Naples, 2000), 85–132, Queen’s Papers in Pure and Appl. Math., 123 Queen’s Univ., Kingston, ON, 2002. [14] B. Harbourne, Free resolutions of fat point ideals on P2 , J. Pure Appl. Algebra 125 (1998) 213–234. [15] B. Harbourne, Anticanonical rational surfaces, Trans. Amer. Math. Soc. 349 (1997) 1191–1208. [16] B. Harbourne, personal communication. [17] R. Maggioni, A. Ragusa, Construction of smooth curves of P3 with assigned Hilbert function and generators’ degrees, Le Matematiche (Catania) 42 (1987) 195–209 (1989). [18] O. Zariski, P. Samuel, Commutative Algebra, Volume II. D. Van Nostrand Company, Princeton, NJ, 1960. Elena Guardo Dipartimento di Matematica e Informatica Viale A. Doria, 6 - 95100 - Catania, Italy Email: [email protected] Adam Van Tuyl Department of Mathematical Sciences Lakehead University Thunder Bay, ON P7B 5E1, Canada Email: [email protected]

The

(unexpected)

importance of knowing α

Brian Harbourne∗

Abstract. We show that the determination, for all r ≥ 0, of the Hilbert functions of all fat point subschemes of Pd with support at r generic points is equivalent to determining, for all r ≥ 0, the least degrees α such that the Hilbert functions are positive (and hence to determining the classes of all eﬀective divisors on blow ups of Pd at r generic points). We also use this point of view for d = 2 to show that the following conjecture is, surprisingly, equivalent to the standard conjecture for the Hilbert function of fat points in the plane with generic support: for any reduced irreducible curve C on a blow up of P2 at generic points, we conjecture that C 2 ≥ g − 1, where g is the arithmetic genus of C. 2000 Mathematics Subject Classiﬁcation: 14C25, 13D40

1. Introduction For simplicity, we work over the complex numbers, C. Fix an integer d ≥ 2 (the case of d = 1 being trivial). Let m = (m1 , . . . , mr ) be a ﬁnite sequence of positive integers, let p1 , . . . , pr be generic points of Pd , and consider the fat points ideal I(m, d) generated by all forms in C[x0 , . . . , xd ] which vanish at each point pi to order at least mi . A signiﬁcant open problem is: Problem 1.1. For each ﬁnite sequence m of positive integers, determine the Hilbert function of I(m, d); i.e., for each t, determine the dimension dimC I(m, d)t of the homogeneous component of I(m, d) of degree t. An apparently easier (but still open) problem is: Problem 1.2. For each ﬁnite sequence m of positive integers, determine α(m, d); i.e., the least degree t such that dimC I(m, d)t > 0. A related open problem is: ∗ Acknowledgements: The author thanks the sponsors of the Siena conference for their support and for providing such a pleasant environment for discussing mathematics, and the NSA for its support of his research.

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Problem 1.3. For each integer r > 0, determine the dimension h0 (X(r, d), OX(r,d) (D)) of the complete linear system of eﬀective divisors linearly equivalent to D, for every divisor D on the blow up X(r, d) of Pd at r generic points, p 1 , . . . , pr . Another, seemingly easier, open problem is: Problem 1.4. For each integer r > 0, determine (inside the divisor class group Cl(X(r, d))) the subsemigroup EFF(r, d) of classes of eﬀective divisors on X(r, d). The purpose of this note is to point out the not diﬃcult but perhaps unexpected and not widely appreciated fact that these problems are all equivalent. Using this point of view, the standard conjectural solution (versions of which have previously been given in [S], [Ha1], [G], [Hi] and elsewhere) to these open problems for d = 2 can be reformulated in a very concise way, as we show in Section 3.

2. The problems are equivalent We ﬁrst show that Problems 1.1 and 1.3 are equivalent. Let m = (m1 , . . . , mr ). Then dimC I(m, d)t is equal to h0 (X(r, d), OX(r,d) (D)), where D = tE0 − m1 E1 − · · · − mr Er , E0 is the pullback to X(r, d) from Pd of the class of a hyperplane, and for i > 0, Ei is the class of the exceptional divisor corresponding to the blow up of the point pi . Thus a solution to Problem 1.3 implies a solution to Problem 1.1. Conversely, given any divisor class D, we have D = tE0 − m1 E1 − · · · − mr Er for some integers t and mi . By permuting the integers mi , we obtain m = (m1 , . . . , mr ) with m1 ≥ · · · ≥ mr . Let s be the greatest integer such that ms > 0. Note that mi < 0 implies that −mi Ei is in the base locus of |D|. (To see this, take global sections of 0 → OX(r,d) (D − Ei ) → OX(r,d) (D) → OEi (mi ) → 0, and induct.) We have h0 (X(r, d), OX(r,d) (D)) = h0 (X(r, d), OX(r,d) (tE0 − m1 E1 − · · · − mr Er )) = h0 (X(r, d), OX(r,d) (tE0 − (m1 )+ E1 − · · · − (mr )+ Er )), where, for any integer j, we let (j)+ denote the maximum of j and 0. Since the points are generic, we also have that h0 (X(r, d), OX(r,d) (tE0 − (m1 )+ E1 − · · · − (mr )+ Er )) = h0 (X(s, d), OX(s,d) (tE0 − m1 E1 − · · · − ms Es )) = dimC I(m , d)t , where m = (m1 , . . . , ms ). Thus a solution to Problem 1.1 implies a solution to Problem 1.3. Similarly, a solution to Problem 1.4 implies a solution to Problem 1.2, since, given m = (m1 , . . . , mr ), α(m, d) is the least t such that h0 (X(r, d), OX(r,d) (tE0 − m1 E1 − · · · − mr Er )) > 0, and hence such that tE0 − m1 E1 − · · · − mr Er is the class of an eﬀective divisor. Conversely, tE0 − m1 E1 − · · · − mr Er is the class of an eﬀective divisor if and only if tE0 − (m1 )+ E1 − · · · − (mr )+ Er is, hence if and only if t ≥ α(m , s), where m is as above. Thus a solution to Problem 1.2 implies a solution to Problem 1.4. It is now enough to check that a solution to Problem 1.1 implies a solution to Problem 1.2, and vice versa. Clearly, a solution to Problem 1.1 implies a

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solution to Problem 1.2. Conversely, suppose we want to compute dimC I(m, d)t . Given an integer i ≥ 0, let m(i) denote the sequence (m1 , . . . , mr , 1, . . . , 1) with i additional entries appended, each such additional entry equal to 1. If α(m, d) > t, then clearly dimC I(m, d)t = 0. Otherwise, let j be the least integer such that α(m(j), d) > t. Then dimC I(m(j), d)t = 0, but dimC I(m(i), d)t = dimC I(m, d)t − i for 0 ≤ i ≤ j, since imposing each single additional generic base point to a nonempty linear system drops the dimension of the linear system by exactly 1. Thus dimC I(m, d)t = j, hence a solution to Problem 1.2 implies a solution to Problem 1.1.

3. A new formulation of the standard conjecture for P2 In this section we work on P2 ; i.e., we ﬁx d = 2. We ﬁrst recall the version of the standard conjectural solution to Problem 1.3 for P2 , given in [Ha1]. Conjecture 3.1. Let Xr be the blow up of P2 at r generic points. For each r ≥ 1: if C ⊂ Xr is a reduced, irreducible curve of negative self-intersection, then C 2 = −1 and C is smooth and rational; moreover, if D is an eﬀective nef divisor on Xr , then h1 (Xr , OXr (D)) = 0. In this section we show this conjecture is equivalent to the following: Conjecture 3.2. If C ⊂ Xr is a reduced, irreducible curve on the blow up Xr of P2 at any r ≥ 1 generic points, then C 2 ≥ g − 1, where g is the arithmetic genus of C. It will be useful here and later to keep in mind that h2 (Xr , OXr (C)) = 0 for any eﬀective divisor C, or indeed for any divisor C (such as a nef divisor) such that C · E0 ≥ 0. (To see this, recall that the canonical class on Xr is KXr = −3E0 + E1 + · · · + Er . Since E0 is the class of a line, it is nef, so it follows from E0 · (KXr − C) = −3 − C · E0 < 0 that KXr − C is not eﬀective. Therefore h2 (Xr , OXr (C)) = h0 (Xr , OXr (KXr −C)) = 0.) Thus, by Riemann-Roch, we have h0 (Xr , OXr (C)) = (C 2 − C · KXr )/2 + 1 + h1(Xr , OXr (C)) for any divisor C (such as an eﬀective or nef divisor) with E0 · C ≥ 0. First we verify that Conjecture 3.1 implies Conjecture 3.2, as pointed out by Ciliberto (see [C], Remark 4.10). Let C be a reduced irreducible curve on Xr . If C 2 ≥ 0, then C is eﬀective and nef so h1 (Xr , OXr (C)) = 0 by Conjecture 3.1. Thus 1 ≤ h0 (Xr , OXr (C)) = (C 2 − C · KXr )/2 + 1, so C 2 ≥ C · KXr . Now, by the genus formula, 2C 2 ≥ C 2 + C · KXr = 2g − 2, so C 2 ≥ g − 1. If however C 2 < 0, then, by Conjecture 3.1, C 2 = −1, and C is smooth and rational, so g = 0. Thus C 2 = −1 = g − 1, so again C 2 ≥ g − 1. We now show, conversely, that Conjecture 3.2 implies Conjecture 3.1. If for any reduced, irreducible curve C we have C 2 ≥ g − 1, then clearly C 2 < 0 implies

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C 2 = −1, g = 0 and so C is smooth and rational. (It also follows that C · KXr = −1 and that 1 = h0 (Xr , OXr (C)) ≥ (C 2 − C · KXr )/2 + 1 = 1, and hence h0 (Xr , OXr (C)) = (C 2 − C · KXr )/2 + 1.) So now it is enough to show that every eﬀective nef divisor D on Xr satisﬁes h1 (Xr , OXr (D)) = 0, or, what is by Riemann-Roch the same, that h0 (Xr , OXr (D)) = (D2 − D · KXr )/2 + 1. But this is a consequence of the following lemma. Lemma 3.3. Let D ⊂ Xr be an eﬀective nef divisor. Then Conjecture 3.2 implies that one of the following holds: 1. |D| = |lA| for some reduced irreducible smooth rational curve A with h1 (Xr , OXr (lA)) = 0 = A2 and h0 (Xr , OXr (lA)) = l + 1; 2. |D| = |lA| for some reduced irreducible divisor A with A2 = A · KXr = 0, h0 (Xr , OXr (lA)) = 1 and h1 (Xr , OXr (lA)) = 0; or 3. |D| contains a reduced and irreducible member, and h1 (Xr , OXr (D)) = 0. Proof. First, consider the case that D is reduced and irreducible. By assumption, D2 ≥ g − 1, hence 2D2 ≥ 2g − 2 = D2 + D · KXr , so D2 ≥ D · KXr . Moreover, D is nef, so D2 ≥ 0. If h0 (Xr , OXr (D)) > 1, then a general section of |D − E| is still reduced and irreducible, but h0 (Xr+1 , OXr+1 (D − E)) = h0 (Xr , OXr (D)) − 1, where E is the exceptional curve coming from the blowing up Xr+1 → Xr of an additional generic point, and we identify D with its pullback to Xr+1 . If D2 = 0, then (D − E)2 = −1, hence as we saw above h0 (Xr+1 , OXr+1 (D − E)) = 1, and so 2 = h0 (Xr , OXr (D)) ≥ (D2 − D · KXr )/2 + 1 = 2, so we have h0 (Xr , OXr (D)) = (D2 − D · KXr )/2 + 1, as desired. If D2 > 0, then (D − E)2 = D2 − 1, h0 (Xr+1 , OXr+1 (D − E)) = h0 (Xr , OXr (D)) − 1, and ((D − E)2 − (D − E) · KXr )/2 + 1 = (D2 − D · KXr )/2 + 1 − 1. Thus it is enough to show h0 (Xr+1 , OXr+1 (D − E)) = ((D − E)2 − (D − E) · KXr+1 )/2 + 1. Continuing in this way, we reduce to the case that D2 = 0 (in which case we are, as we have seen, done), or to the case that D2 > 0 but h0 (Xr , OXr (D)) = 1. But in the latter case, 1 = h0 (Xr , OXr (D)) ≥ (D2 − D · KXr )/2 + 1, hence D · KXr ≥ D2 , but D2 ≥ D · KXr , so D2 = D · KXr , and we again have h0 (Xr , OXr (D)) = (D2 − D · KXr )/2 + 1. Thus h1 (Xr , OXr (D)) = 0 if D is reduced and irreducible, so now assume that no member of |D| is reduced and irreducible. Then either: (a) |D| has a ﬁxed component but D is not ﬁxed; (b) |D| is ﬁxed but D is not reduced and irreducible; or (c) |D| is ﬁxed component free, but its general section is not irreducible, which by Bertini’s theorem means that |D| is composed with a pencil. Suppose that |D| has a ﬁxed component; let N be a reduced irreducible component of the ﬁxed part of |D|, but assume D = N . Choose a reduced irreducible component A of the general member of |D − N |. Then D − (A + N ) is eﬀective,

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and we may assume either that A2 ≥ 0, or that A2 < 0 and hence A is a ﬁxed component of |D|. First we show that N 2 ≥ 0. Suppose N 2 < 0 (and hence N 2 = −1 = N · KXr ). Since D is nef, D must have a reduced irreducible component A meeting N positively. As we saw above, h2 vanishes, so h0 (Xr , OXr (A + N )) ≥ ((A + N )2 − (A + N ) · KXr )/2 + 1 = h0 (Xr , OXr (A )) + A · N > h0 (Xr , OXr (A )), which contradicts N being a ﬁxed component. Thus 0 ≤ N 2 and, since N is reduced and irreducible (so nef) and ﬁxed, we have 1 = h0 (Xr , OXr (N )) = (N 2 −N ·KXr )/2+1 and so N 2 = N · KXr . Since N is nef, we see N · A ≥ 0, but h0 (Xr , OXr (A)) = h0 (Xr , OXr (A + N )) ≥ ((A + N )2 − (A + N ) · KXr )/2 + 1 = h0 (Xr , OXr (A)) + A · N , so it follows that A · N = 0. And now we see that we cannot have A2 < 0, since in that case A is a ﬁxed component, and the same argument we used for N implies that we would have A2 ≥ 0. If A2 > 0, then the subspace orthogonal to A must be negative deﬁnite (by Sylvester’s signature theorem and the Hodge index theorem; see Remark V.1.9.1 of [Hrt]), which contradicts N 2 ≥ 0 = A · N . Thus A2 = 0. The same argument with A and N switched shows that N 2 = 0, and so also −N · KXr = 0. But 0 < h0 (Xr , OXr (A)) = (A2 − A · KXr )/2 + 1, so −A · KXr ≥ 0. Now, since N is nef, it is standard [Ha2]; i.e., there is a birational morphism Xr → P2 and a corresponding exceptional conﬁguration E0 , E1 , . . . , Er such that N is a nonnegative integer linear combination of the classes H0 , . . . , Hr , where H0 = E0 , H1 = E0 − E1 , H2 = 2E0 − E1 − E2 , and Hi = 3E0 − E1 − · · · − Ei , for i > 2. Since N · Hr = −N · KXr = 0, we see that N · Hi ≥ 0 for all i. The only nontrivial nonnegative linear combinations N of the Hi with N 2 = N · Hr = 0 are the nonnegative multiples of H9 (thus we see that r must be at least 9). But h0 (Xr , OXr (lH9 )) = 1 for all l ≥ 0, hence, since N is reduced and irreducible, we have N = H9 . Now we show that N = A. We have 0 = N ·A = H9 ·A ≥ Hr ·A = −KXr ·A ≥ 0, thus Ei · A = 0 for all i > 9, hence A is a linear combination of E0 , . . . , E9 , orthogonal to H9 with A2 ≥ 0. The only such classes are the multiples of H9 itself (see, for example, Lemma 2.2 of [LH]). Since A is reduced and irreducible, we have A = H9 , as before. Thus D = lH9 for some l ≥ 2, and we have h0 (Xr , OXr (D)) = 1 and h1 (Xr , OXr (D)) = 0, giving part (2) of the lemma. (This also shows that item (a) above does not occur.) So ﬁnally, suppose D is ﬁxed component free, but does not have a reduced and irreducible general member. Then it must be composed with a pencil. Thus a general member of |D| is a sum D1 + · · · + Dl of reduced irreducible and linearly equivalent curves (hence |D| = |lD1 |), with 2 ≤ h0 (Xr , OXr (D1 )) = (D12 − D1 · KXr )/2 + 1 (hence 2 ≤ D12 − D1 · KXr ), and h0 (Xr , OXr (lD1 )) ≤ l + 1. Therefore, l + 1 ≥ h0 (Xr , OXr (lD1 )) ≥ (l2 D12 − lD1 · KXr )/2 + 1 (hence 2 ≥ lD12 − D1 · KXr and so 2 ≥ (l − 1)D12 + D12 − D1 · KXr ≥ 2 + (l − 1)D12 , which, since D12 ≥ 0, implies D12 = 0 and so 2 = −D1 · KXr and g = 0). Now l + 1 = h0 (Xr , OXr (lD1 )) ≥

(l2 D12 − lD1 · KXr )/2 + 1 = l + 1, and part (1) follows.

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References [C] C. Ciliberto, Geometric aspects of polynomial interpolation in more variables and of Waring’s Problem, Proc. of the Third European Congress of Mathematics, Vol. 1 (Barcelona, 2000), 289-316, Progr. Math. 201, Birkh¨ auser, Basel (2001). [G] A. Gimigliano, On Linear Systems of Plane Curves, Ph. D. thesis, Queen’s University, Kingston, Ontario (1987). [Ha1] B. Harbourne, The geometry of rational surfaces and Hilbert functions of points in the plane, Can. Math. Soc. Conf. Proc. 6 (1986), 95–111. [Ha2] B. Harbourne, Complete linear systems on rational surfaces, Trans. Amer. Math. Soc., 289 (1985), Number 1, 213–226. [Hrt] R. Hartshorne, Algebraic Geometry, Springer-Verlag, GTM 52, (1977), pp. xvi + 496. [LH] M. Lahyane and B. Harbourne, Irreducibility of −1-Classes on Anticanonical Rational Surfaces and Finite Generation of the Eﬀective Monoid, to appear, Pac. J. Math. [Hi] A. Hirschowitz, Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles g´en´eriques, Jour. Reine Angew. Math. 397 (1989), 208-213. [S] B. Segre, Alcune questioni su insiemi ﬁniti di punti in geometria algebrica, Atti Convegno intern. di Geom. Alg. di Torino, 15-33. Brian Harbourne Department of Mathematics University of Nebraska Lincoln, NE 68588-0323 USA Email: [email protected]

Hilbert functions of Gorenstein algebras associated to a pencil of forms Anthony Iarrobino

Abstract. Let R be a polynomial ring in r variables over an inﬁnite ﬁeld K, and denote by D a corresponding dual ring, upon which R acts as diﬀerential operators. We study type two graded level Artinian algebras A = R/I, having socle degree j. For each such algebra A, we consider the family of Artinian Gorenstein [AG] quotients of A having the same socle degree. By Macaulay duality, A corresponds to a unique 2-dimensional vector space WA of forms in Dj , and each such AG quotient of A corresponds to a form in WA - up to non-zero multiple. For WA = F, G , each such AG quotient Aλ corresponds to an element of the pencil of forms (one dimensional subspaces) of WA : given Fλ = F +λG, λ ∈ K∪∞ we have Aλ = R/Ann(Fλ ). Our main result is a lower bound for the Hilbert function H(Aλgen ) of the generic Gorenstein quotient, in terms of H(A), and the pair HF = H(R/Ann F ) and HG = H(R/Ann G). This result restricts the possible sequences H that may occur as the Hilbert function H(A) for a type two level algebra A. 2000 Mathematics Subject Classiﬁcation: 13D40, 14C05

1. Introduction Let R = K[x1 , . . . , xr ] be the polynomial ring in r variables, over an inﬁnite ﬁeld K. We will assume also for simplicity of exposition that char K = 0, but all statements may be extended suitably to characteristic p (see Remark 2.10). We will consider only graded Artinian quotients A = R/I of R, and we denote by m the irrelevant maximal ideal m = (x1 , . . . , xr ). We denote by Ai the i-th graded component of A. Recall that the socle Soc(A) of A satisﬁes Soc(A) = (0 : m) = {f ∈ A | mf = 0},

(1.1)

and the type of A is t(A) = dimK Soc(A). We will denote by j = j(A) the socle degree of A, the largest integer such that Aj = 0, but Aj+1 = 0. An Artinian algebra A = R/I of socle degree j is level, if any of the following equivalent conditions hold i. Soc(A) = Aj ,

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ii. The canonical module Hom(A, K) of A is generated in a single degree, iii. Each Ii , 0 ≤ i ≤ j can be recovered from Ij as follows: for 0 < i ≤ j

Ii = Ij : Rj−i = {f ∈ Ri | Rj−i · f ⊂ Ij }.

(1.2)

Recall that A is Artinian Gorenstein if A is level of type one. Our main objects of study here will be type two level algebras A and their Artinian Gorenstein [AG] quotients.

1.1. Recent results on level algebras. We ﬁrst brieﬂy recall some recent work on graded level algebras. First, the Artinian Gorenstein algebras have been the object of much study. For height three, the structure theorem of Buchsbaum-Eisenbud [BuEi] has led not only to a characterization of the Hilbert functions that may occur, but also to the irreducibility [Di] and smoothness [Kle2] of the family PGOR(H) parametrizing Artinian Gorenstein quotients of R, r = 3, of a given (symmetric) Hilbert function H. A second proof of smoothness follows from results of M. Boij and A. Conca-G. Valla (see [Bj4, CoVa], and [IK, §4.4] for a survey of related topics). One line of study relates punctual subschemes Z of Pr−1 to Gorenstein Artinian quotients of their coordinate rings OZ [Bj2, G, EmI, IK, Kle3]. Classical apolarity, or the inverse systems of Macaulay provide a connection to sums of powers of linear forms, and to a classical Waring problem for forms (see [Te, EmI, G, IK]). When r ≥ 4 PGOR(H) often has several irreducible components, a fact ﬁrst noted by M. Boij [Bj3], and elaborated by others (see [IS, Kle3]). The set of Gorenstein sequences — ones that occur as Hilbert functions of Artinian Gorenstein quotients of R — is not known for r ≥ 4; for r ≥ 5 they include non-unimodal sequences, with several maxima. Level algebras A of types t(A) > 1 are a natural next topic of study after the Gorenstein algebras, particularly in the low embedding dimensions r ≤ 3 or even r = 4, where the families of Artinian Gorenstein quotients are better understood. When r = 2, the family LevAlg(H) is well understood (see [I2, ChGe]): these , families are smooth, of known dimension, and their closures are the union H ≤H LevAlg(H ) of similar strata for termwise no greater Hilbert functions of the same socle degree. When r = 3, and t = 2, tables of possible H for small socle degree j, possible resolutions, and many methods that are more general are given in [GHMS]; this case is also studied by F. Zanello [Za1]. However the possible sequences H are not known when r ≥ 3 even in the case t = 2; and although specialists believe there should be cases where LevAlg(H) for r = 3, t = 2 have several irreducible components, this problem is still open. There has been work connecting these results with the simultaneous Waring problem for binary and ternary forms [Car, CarCh, I2]. Several authors have studied the extremal Hilbert functions for level algebras of given embedding dimension r, type t, and socle degree j [BiGe, ChoI]. The minimal resolutions for compressed level algebras (those having maximum Hilbert

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function given (r, t, j)) are studied in [Bj1, MMN, Za1]. Also Zanello has obtained results about extremal Hilbert functions for level algebras, given the pair Hj−1 and Hj = 2 [Za2]. For r ≥ 3 there is much to be learned about families of level algebras of given Hilbert functions, even when t = 2. Certainly, pencils of curves on P2 have been long a topic of geometric study; however, a stronger connection (but see [ChGe]) has yet to be made between on the one hand the traditional geometrical approach to pencils of curves and their singularities, and on the other hand the study of the level algebras associated to these pencils. In this article we show some inequalities connecting the Hilbert function of type two level algebras, and the Hilbert function of their Artinian Gorenstein quotients. These results had been embedded in the longer preprint [I3], which will now be refocussed on reﬁnements of the numerical results and on parametrization. In section 1.2 we give notation, and brieﬂy state the main results, and in section 1.3 we present further context, including the questions that motivated us. In Section 2 we prove our results and give examples.

1.2. Inverse systems. Let D = K[X1 , . . . , Xr ] denote a second polynomial ring. The ring R acts on D as partial diﬀerential operators: for h ∈ R, F ∈ D h ◦ F = h(∂/∂X1 , . . . , ∂/∂Xr ) ◦ F.

(1.3)

The pairing σj : Rj × Dj → K,

σj (h, F ) = h ◦ F,

(1.4)

is exact. This is the apolarity or Macaulay duality action of R on D [Mac1]. A type t level algebra A = R/I of socle degree j corresponds via the Macaulay duality to a unique t-dimensional vector space WA , WA = {F ∈ Dj | I ◦ F = 0} = {F ∈ Dj | Ij ◦ F = 0}.

(1.5)

Thus WA is the perpendicular space to Ij in the exact duality between Rj and Dj , and R ◦ WA , may be regarded as the dualizing module Aˆ = Hom(A, K) to A. The Hilbert function H(A) is the sequence H(A)i = dimK Ai . We have, for 0 ≤ u ≤ j, ⊥ Ru ◦ WA = Ij−u ⊂ Dj−u , and

(1.6)

H(A)j−u = dimK Ru ◦ WA .

(1.7)

Remark 1.1. A one-dimensional subspace E ⊂ WA corresponds to an Artinian Gorenstein [AG] quotient R/Ann E of the level algebra A, having the same socle degree j as A. We parametrize these spaces E as points of the projective space P(WA ) associated to WA . (See also [Kle1, I1] and [IK, p. 177]). The Hilbert function HE = H(R/Ann E) is evidently semicontinuous: HE > T termwise for some ﬁxed sequence T = (t0 , . . . , tj ) deﬁnes an open subset of P(WA ) since dim Ru ◦ E > tj−u is an open condition. Thus, among the Hilbert functions of Gorenstein quotients of A having the same socle degree j, there is a termwise

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maximum H(Egen ), that occurs for E belonging to an open dense subset of the projective variety P(WA ). When the type of A is two, then WA = F, G is two-dimensional; the onedimensional subspaces constitute a pencil of forms Fλ = F + λG, λ ∈ K ∪ ∞ = P1K = P(WA ) (here we set F∞ = G). Each AG quotient algebra Aλ = R/Ann (F + λG) has the same socle degree j = j(A) as A, and these comprise all the Gorenstein quotients of A having socle degree j(A). Thus, the family Aλ , λ ∈ P1 is the pencil of Artinian Gorenstein quotients associated to the pencil F + λG. We let AF = R/Ann F and AG = R/Ann G, and set HF = H(AF ), HG = H(AG ). We focus here on the type two case, and on the pencil of degree-j homogeneous forms or hypersurfaces, Fλ ∈ Dj and their symmetric Hilbert functions Hλ = H(Aλ ). By (1.7) (Hλ )j−u is the dimension of the space of order u partial derivatives of Fλ . Evidently, the set of Hilbert functions H(Aλ ) that occur is a PGL(r − 1) invariant of the level algebra A. Also, Remark 1.1 implies that there is a termwise maximum value H(Aλgen ), that occurs for an open dense set of λ ∈ P1 . We now state the most important part of our main result, Theorem 2.2. For i = j − u, we let di (F, G) = dimK Ru ◦ F ∩ Ru ◦ G = (HF )i + (HG )i − H(A)i , be the overlap dimension, satisfying di (F, G) = H(R/(Ann F + Ann G))i (see equation (2.4)ﬀ). Theorem 1.2. Let A = R/Ann (F, G) be a type two level algebra of socle degree j. For all pairs (u, i = j − u) satisfying 0 < u < j we have H(A)u − di (F, G) ≤ H(Aλgen )i .

(1.8)

In Theorem 2.4 we give a lower bound for H(Aλgen ) that depends only on H(A).

1.3. Questions and examples: pencils of forms. We oﬀer some questions about pencils of forms and the Hilbert functions they determine, and state their status. This provides some further context for our work, and as well we pose open problems. Question 1.3. What are natural invariants for pencils of forms? i. What sequences H occur as Hilbert functions H(A)? Status: Open for r ≥ 3, even for t = 2, but see [GHMS, Za1]. ii. Is there a sequence H = (1, 3, . . . , 2, 0), such that LevAlg(H) has two irreducible components? Status: Open. The answer to the analogous question is ”no” for embedding dimension r = 2, and ”yes” for r ≥ 4.

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iii. Can we use our knowledge of the Hilbert functions and parameter spaces for Artinian Gorenstein algebras, to study type two level algebras A of embedding dimensions three and four? Status: This has been the main approach to classifying type two level algebras. See [GHMS] and [Za1, Za2], as well as Lemma 2.6, Examples 2.7 and 2.8 below. iv. Given a type two Artinian algebra A, consider the pencil of Gorenstein quotients Aλ = R/Ann (F + λG) having the same socle degree as A. What can be said about the Hilbert functions H(Aλ )? Status: We begin a study here. See also [I3, Za1, Za2]. The Question 1.3 about natural invariants of A connects also with classical invariant theory, but we do not pursue this here: see [DK] and [RS] for analogous connections in the Gorenstein case. The following example illustrates Question 1.3(iv), and as well the main result. Example 1.4. Let r = 2, F = X 4 , G = XY 3 , then HF = (1, 1, 1, 1, 1), HG = (1, 2, 2, 2, 1), the ideal I = Ann (F, G) = Ann F ∩ Ann G = (x2 y, y 4 , x5 ). The type two level algebra A = R/I has Hilbert function H(A) = (1, 2, 3, 3, 2). The dualizing module Aˆ = R ◦ F, G ⊂ D satisﬁes Aˆ = 1; X, Y ; X 2 , Y X, Y 2 ; X 3 , Y 3 , XY 2 ; X 4 , XY 3 . The Gorenstein quotients Aλ satisfy H(Aλ ) = (1, 2, 3, 2, 1) for λ = ∞, 0. This is the maximum possible (so compressed) Hilbert function for a Gorenstein Artinian quotient of R having socle degree 4. The following speciﬁc question arose from a discussion with A. Geramita about the Hilbert functions possible for type two level algebras. It was the starting point of our work here. Question 1.5. Let F, G be two degree-j homogeneous polynomials, elements of D = K[X1 , . . . , Xr ], such that i. F, G together have at least 2r − 2 linearly independent ﬁrst partial derivatives, and ii. F, G together involve all r variables: this is equivalent to the (j − 1)-order partials of F, G spanning X1 , . . . , Xr . Does some linear combination Fλ = F +λG have r linearly independent ﬁrst partial derivatives? We answer this question positively in Corollary 2.5. Here are two examples to illustrate.

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Example 1.6. Let r = 3, j = 4, F = X 4 + Y 4 , G = (X + Y )4 + Z 4 . Then the pencil V = F, G involves all three variables, and these forms together have four linearly independent ﬁrst partials X 3 , Y 3 , (X + Y )3 , Z 3 . For all λ = 0 the form Fλ = F + λG has three linearly independent ﬁrst partials. Example 1.7. Let r = 3, j = 4, F = XZ 3 , G = Y Z 3 . Then V = F, G involves all three variables and these forms together have only 3 = 2r − 3 linearly independent ﬁrst partials. Each Fλ has only 2 linearly independent ﬁrst partials. Thus, the hypothesis in Question 1.5 that V has at least 2r − 2 linearly independent ﬁrst partial derivatives is necessary for the desired conclusion. Our work here is focussed primarily on the following question. Question 1.8. Given Hilbert functions HF = H(R/Ann F ), HG = H(R/Ann G) for two degree-j forms F, G ∈ D, or given H(A), A = R/Ann (F, G), determine the possible Hilbert functions H(Aλ ) for Aλ = R/Ann Fλ , Fλ = F + λG? Are there numerical restrictions on the generic value H(Aλgen )? It is easy to give a partial answer. Evidently, by (2.1), we can’t have two values of λ with H(Aλ )i < H(A)i /2.

(1.9)

We may conclude that small H(Aλ ) are rare, given H(A)! In our main results we show that if HF and HG are small in comparison with H(A), then H(Aλgen ) is large (Theorem 2.2). We then show a lower bound for H(Aλgen ) in terms of H(A) (Theorem 2.4). Several examples illustrate the results (see especially Examples 2.7, 2.8). Example 2.3 gives a pencil Aλ of Gorenstein Artinian quotients not having a minimum Hilbert function; and Example 2.9 gives a compressed type two Artinian level algebra A such that Aλgen is not compressed Gorenstein.

2. Hilbert functions for pencils of forms In this section we state and prove our main results. We ﬁrst give an exact sequence relating AF , AG , and A. We deﬁne R-module homomorphisms ι : A → R/Ann F ⊕ R/Ann G ι(f ) = (f mod Ann F, −f

mod Ann G)

π : R/Ann F ⊕ R/Ann G → R/(Ann F + Ann G) : π(a, b) = (a + b) mod (Ann F + Ann G). Lemma 2.1. Let F, G ∈ Dj determine a type two level Artinian quotient A = R/I, I = Ann (F, G) of R. There is an exact sequence of R-modules ι

π

→ R/Ann F ⊕ R/Ann G − → R/(Ann F + Ann G) → 0, 0→A−

(2.1)

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whose dual exact sequence is π∗

ι∗

0 → R ◦ F ∩ R ◦ G −→ R ◦ F ⊕ R ◦ G −→ R ◦ F, G → 0.

(2.2)

Proof. Since the duality between Rj and Dj is exact, we have (Ann F + Ann G)⊥ = R ◦ F ∩ R ◦ G .

(2.3)

Thus the two sequences are dual. Evidently ι is an inclusion and π is a surjection. The kernel of ι∗ consists of pairs (h1 ◦ F, h2 ◦ G) such that h1 ◦ F − h2 ◦ G = 0; this is evidently the image of π ∗ , so the sequences are exact. We let J = Ann F + Ann G: it depends of course upon the choice of the pair (F, G) ∈ F, G . We denote by H(R/J) = (1, d1 , . . . , dj ) the Hilbert function H(R/J) where di = di (F, G). Thus we have from (2.3), that, letting i = j − u, the integer di measures the overlap in degree i between the inverse systems determined by F and G: di = dimK Ru ◦ F ∩ Ru ◦ G = dimK Ru ◦ F + dimK Ru ◦ G − H(A)i = (HF )i + (HG )i − H(A)i .

(2.4) (2.5) (2.6)

The equalities (2.5), (2.6) are immediate from (1.7) and (2.2). We set, again letting i = j − u, ti = dimK (((Ann F )u ◦ G) ∩ ((Ann G)u ◦ F )),

(2.7)

where ti = ti (F, G) depends on the pair (F, G). Recall from Remark 1.1 that “generic λ” refers to a suitable open dense set of λ ∈ P1 , that is, to all but a ﬁnite number of values of λ. We denote by di , ti the integers di (F, G) and ti (F, G) deﬁned just above. The following main result shows that a small overlap between Ru ◦ F and Ru ◦ G implies a large value for H(Aλgen )i . Theorem 2.2. Let A = R/Ann (F, G) be a type two level algebra of socle degree j. If λ is generic, then for all pairs (u, i = j − u) satisfying 0 < u < j we have H(A)u − di ≤ H(Aλ )i ≤ H(A)u − ti .

(2.8)

The upper bound on H(Aλ ) holds for all λ = 0, ∞. Proof. Fix for now, and through the proof of the claim below, an integer u satisfying 0 < u < j. By “ dim V ” below we mean dimK V . Let Cu ⊂ Ru be a vector subspace complement to (Ann F )u , so Cu ⊕ (Ann F )u = Ru . Let d = di and let e = dim((Ann (F, G))u ): so dim Au = dim Ru − e, and let B ⊂ (Ann F )u be the vector subspace satisfying B = {h ∈ (Ann F )u | h ◦ G ∈ Ru ◦ F }. The homomorphism h → h ◦ G, h ∈ B induces a short exact sequence 0 → (Ann (F, G))u → B → Ru ◦ F ∩ Ru ◦ G ,

(2.9)

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implying dim B ≤ d + e.

(2.10)

Since (Ann F )u ◦ (F + λG) = (Ann F )u ◦ G, we have Ru ◦ (F + λG) = Cu ◦ (F + λG) + (Ann F )u ◦ G.

(2.11)

Claim 1. For generic λ dim Ru ◦ (F + λG) ≥ dim(Cu ◦ F + (Ann F )u ◦ G) = dim Cu + dim(Ann F )u − dim B = dim Ru − dim B ≥ dim Ru − (d + e) = dim Au − d,

(2.12) (2.13)

(2.14)

so dim Ru ◦ (F + λG) ≥ dim Au − d. Proof of claim. The key step is (2.12), which results from (2.11) and deformation. For the space Cu ◦(F +λG)+(Ann F )u ◦G in (2.11) is a deformation of the space Cu ◦ F + (Ann F )u ◦ G on the right of (2.12), and dimension is a semicontinuous invariant. The other steps are straightforward. The claim shows the left-hand inequality in (2.8) for a speciﬁc u. Since P1K is irreducible, the intersection of the dense open subsets of P1K over which the left side of (2.8) is satisﬁed for each u, 0 < u < j, is itself a dense open subset, completing the proof that the left side of (2.8) holds simultaneously for generic λ and all such u. Suppose λ = 0, ∞. Let Cu be a complement in Ru to Ju = (Ann F )u + (Ann G)u . Then we have Ru ◦ (F + λG) = Cu ◦ (F + λG) + (Ann F )u ◦ G + (Ann G)u ◦ F implying dim Ru ◦ (F + λG) ≤ dimk Ru − dim(Ann F )u ∩ (Ann G)u ) − ti = dim Au − ti . This completes the proof of Theorem 2.2.

Example 2.3 (No minimum H(Aλ )). Let r = 3, G = X 8 + Y 4 Z 4 , and F = L81 +· · ·+L85 , where the Li = ai1 X +ai2 Y +ai3 Z are general enough linear forms, elements of D1 . Here “general enough” means that their coeﬃcients {aij ∈ K} lie in , . . . , Lj−u the open dense subset of the aﬃne space A15 such that the powers Lj−u 1 5 are linearly independent in Dj−u and maximally disjoint from Ru ◦ G, 2 ≤ u ≤ 6 (see [I1]). Then we have for 3 ≤ u ≤ 6 , . . . , Lj−u , Ru ◦ F = Lj−u 1 5

(2.15)

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satisfying dimk Ru ◦ F = 5. This determines HF , and we have HF = (1, 3, 5, 5, 5, 5, 5, 3, 1) HG = (1, 3, 4, 5, 6, 5, 4, 3, 1) H(A) = (1, 3, 6, 10, 11, 10, 9, 6, 2) = HF +h HG , where by HF +h HG we mean the sequence satisfying (HF +h HG )i = min{dim Ri , (HF )i + (HG )i }.

(2.16)

Theorem 1 implies that H(Aλgen ) = (1, 3, 6, 10, 11, 10, 6, 3, 1). It is easy to check that there are no other values of λ other than 0, ∞ (corresponding to F, G) such that H(Aλ ) is smaller than H(Aλgen ), and hence no minimum sequence H(Aλ ), since HF and HG are incomparable. Our second main result gives a lower bound for H(Aλgen ) solely in terms of H(A). Theorem 2.4. Let A be a type two level Artinian algebra of socle degree j, and let u, i satisfy 0 < u ≤ i = j − u. Assume that H(A)i ≥ 2H(A)u − 2 − 3δu , where δu ≥ 0 and δu is an integer. Then H(Aλgen )i ≥ H(A)u − δu .

(2.17)

Proof. Assume the hypotheses of the theorem, and suppose by way of contradiction, that for generic λ there is an integer a ≥ 0 satisfying H(Aλ )i = H(A)u − δu − 1 − a.

(2.18)

Take two generic forms F , G in the pencil. Then the overlap between Ru ◦ F and Ru ◦ G (see (2.4),(2.5)) satisﬁes di = 2(H(A)u − δu − 1 − a) − H(A)i ≤ 2H(A)u − 2δu − 2 − 2a − (2H(A)u − 2 − 3δu ) ≤ δu − 2a.

(2.19)

By Theorem 2.2, for generic λ the AG quotient Aλ = R/Ann (F + λG ) satisﬁes H(Aλ )i ≥ H(A)u − (δu − 2a),

(2.20)

a contradiction with equation (2.18). It follows that the assumed equation (2.18) is false, hence H(Aλ )i ≥ H(A)u − δu which is Theorem 2.4.

(2.21)

Note 1. We assumed in the statement and proof of Theorem 2.4 that δu is an integer. Alternatively we could deﬁne δu = (2H(A)u − 2 − H(A)i )/3 and conclude that H(Aλgen )i ≥ H(A)u − δu when δu ≥ 0, and H(Aλgen )i ≥ H(A)u otherwise. In the following corollary we give a positive answer to Question 1.5.

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Corollary 2.5. Let F, G together have at least 2r − 2 linearly independent ﬁrst partial derivatives, and suppose that F, G involve all r variables. Then dimK R1 ◦ (F + λG) = r for generic λ. Proof. Take i = j − 1, u = 1, δ1 = 0 in Theorem 2.4. From the assumptions we have H(A)j−1 ≥ 2r − 2 = 2H(A)1 − 2, which implies by Theorem 2.4 that for generic λ the dimension H(Aλ )j−1 = r. In order to apply Theorem 2.4 most eﬀectively, we use the following result from [GHMS]. For a sequence H = (1, . . . , Hj ), Hj > 0 we denote by H ˆ the reverse sequence H ˆi = Hj−i , 1 ≤ i ≤ j. Recall that an O-sequence is one that is the Hilbert function of some Artinian algebra [Mac2, BrH]. Lemma 2.6. (A. Geramita et al. [GHMS]) Let A be a type t level algebra with ˆ Let A = R/Ann W, W = WA ⊂ Dj , dimK W = t ≥ 2 and let dualizing module A. V ⊂ W be a vector subspace of codimension one. Then there is an exact sequence of R-modules relating the type t − 1 level algebra BV = B/Ann V to A, 0 → C → A → BV → 0

(2.22)

whose dual exact sequence of R submodules of D is 0 → BˆV → Aˆ → Cˆ → 0.

(2.23)

Here Cˆ is a simple R-submodule (single generator). We have for their Hilbert functions H(A) = H(BV ) + H(C), and the reverse sequence H(C)ˆ= (1, . . .) is an O-sequence. Proof. Let F ∈ V span a complement to V in WA . Let the homomorphism τ(F,W,V ) : R → R ◦ F/R ◦ W ∩ R ◦ F → 0 have kernel S. The module Cˆ is isomorphic to R/S. This shows (2.23) and that Cˆ is simple. Example 2.7. Let r = 3 and suppose A is a type two level algebra satisfying H(A) = (1, 3, . . . , 4, 2). Then the pencil WA = F, G ⊂ Dj deﬁning A may be chosen so that HF = (1, 3, . . . , 3, 1) and H(A) ≤ (1, 3, 6, 10, . . . , 6, 3, 1) +h (1, 1, . . . , 1),

(2.24)

(see (2.16) for the sum +h used above). In particular (1, 3, . . . , 8, 4, 2) and (1, 3, . . . , 12, 7, 4, 2) are not sequences possible for the Hilbert function of a level algebra quotient of R = K[x, y, z]. Here in equation (2.24), the sequence (1, 3, 6, . . . , 3, 1) is the compressed Gorenstein sequence of socle degree j (see below).

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Here the Corollary 2.5 implies that we may choose G ∈ WA such that HG = ˆ = (1, 1, . . .), so to be an O-sequence, H(C) ˆ ≤ (1, 3, . . . , 3, 1); it follows that H(C) (1, 1, . . . , 1); this and (2.23) show (2.24). F. Zanello has extended this kind of result, and in [Za1] shows sharp upper bounds for the Hilbert function H(A) for type two level algebra quotients of R in r-variables given H(A)j−1 . Example 2.8. Let r = 3 and let H = (1, 3, 6, 8, 6, 4, 2). Consider ﬁrst W1 = F, G where HF = (1, 3, 5, 7, 5, 3, 1), and G = L6 , L a general enough linear form (element of D1 ), so HG = (1, 1, . . . , 1). Then A1 = R/Ann W1 is easily shown to have Hilbert function H, as, choosing F ﬁrst, and G second, Ru ◦ G = L6−u and is linearly disjoint in general from Ru ◦ G, by the spanning property of the rational normal curve: powers of linear forms span Di , i = j − u (see [I1]). Next, let W2 = F , G where F is a general element of K[X, Y ]6 , and G is a general enough element of K[X + Y, Z]6 . Then HF = HG = (1, 2, 3, 4, 3, 2, 1) and A2 = R/Ann W2 also has Hilbert function H. In either case, Theorem 2.2 implies that H(R/Ann Fλgen ) = H(R/Ann Fλ gen ) = (1, 3, 6, 8, 6, 3, 1). In this example, it is the non-generic elements of each pencil — the “unexpected properties” — that serve to distinguish the pencil. A compressed level algebra of given type t, socle degree j, and embedding dimension r is one having the maximum possible Hilbert function given those integers (see [?, FL, Bj1, MMN]). The following example responds negatively in embedding dimension three to a question of D. Eisenbud, B. Ulrich, and C. Huneke. They asked if a generic socle degree-j AG quotient Aλgen of a compressed type two level algebra A, must also be compressed. This is true for r = 2. Example 2.9. Let r = 3, j = 4, take a, b ∈ K, (a, b) = (0, 0), and set F = X 3 Y +X 2 Z 2 +aXZ 3 +bY Z 3 ,

G = X 3 Z+X 2Y 2 +X 2Y Z+3aXY 2 Z+bY 3 Z.

Here H(A) = (1, 3, 6, 6, 2), and is compressed, but (y 2 − λz 2 ) ◦ (F + λG) = 0 and we have that for generic λ, H(Aλ ) = (1, 3, 5, 3, 1), which is not the compressed sequence (1, 3, 6, 3, 1). Note that in applying Theorem 2.4 with i = 2, j = 4 here we would have 6 = H(A)2 = 2H(A)4−2 − 2 − 3(4/3), so taking δ = 2 in equation (2.17), we would conclude only that H(Aλgen )2 ≥ 6 − 2 = 4. Remark 2.10 (Characteristic p). Assume that K is an inﬁnite ﬁeld of ﬁnite characteristic p. For p > j, the socle degree, there is no diﬀerence in statements. For p ≤ j one must use the divided power ring in place of D, and the action of R on D is the contraction action. With that substitution, the lemmas and theorems here extend to characteristic p. However, in examples, one must substitute divided powers for regular powers — see [IK, Appendix A] for further discussion.

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Acknowledgement. This article began after a conversation with Tony Geramita, in Fall 2002 during a visit to the MSRI Commutative Algebra Year. We noted that in tables he and colleagues had calculated for Hilbert functions of type two algebras (see [GHMS]), that Corollary 2.5 was satisﬁed. I believed there would be a general result, which turned out to be Theorem 2.2. I thank Tony for this discussion, and for helpful comments. I am appreciative to the organizers of the Siena conference “Projective Varieties with Unexpected Properties” for hosting a lively and informative meeting, and the impetus to write a concise presentation of the results; and I thank the referee. Note added in proof. F. Zanello has recently shown upper bounds for the Hilbert function of a generic type c level quotient of a type t level algebra of given Hilbert function H. His Proof uses Theorem 2.4 in an essential way [Za3].

References [BiGe] Bigatti A., Geramita A.: Level algebras, lex segments, and minimal Hilbert functions, Comm. Alg. 33(3), 2003, 1427-1451. [Bj1] Boij M.: Betti numbers of compressed level algebras, J. Pure Appl. Algebra 134 (1999), no. 2, 111–131. [Bj2]

: Gorenstein Artin algebras and points in projective space, Bull. London Math. Soc. 31 (1999), no. 1, 11–16.

[Bj3]

: Components of the space parametrizing graded Gorenstein Artin algebras with a given Hilbert function, Paciﬁc J. Math. 187 (1999), 1–11.

[Bj4]

: Betti number strata of the space of codimension three Gorenstein Artin algebras, preprint, 2001.

[BrH] Bruns W. , Herzog J.: Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics # 39, Cambridge University Press, Cambridge, U.K., 1993; revised paperback edition, 1998. [BuEi] Buchsbaum D., Eisenbud D. : Algebra structures for ﬁnite free resolutions, and some structure theorems for codimension three, Amer. J. Math. 99 (1977), 447–485. [Car] Carlini E.: Varieties of simultaneous sums of powers for binary forms, Matematiche (Catania) 57 (2002), no. 1, 83–97 (2004). [CarCh]

, Chipalkatti J.: On Waring’s problem for several algebraic forms. Comment. Math. Helv. 78 (2003), no. 3, 494–517.

[ChGe] Chipalkatti J., Geramita A.: On parameter spaces for Artin level algebras, Michigan Math. J. 51 (2003), no. 1, 187–207. [ChoI] Cho Y., Iarrobino, A.: Hilbert functions of level algebras, Journal of Algebra 241 (2001), 745–758. [CoVa] Conca A., Valla G.: Hilbert functions of powers of ideals of low codimension, Math. Z. 230 (1999), no. 4, 753–784.

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[Di] Diesel S. J.: Some irreducibility and dimension theorems for families of height 3 Gorenstein algebras, Paciﬁc J. Math. 172 (1996), 365–397. [DK] Dolgachev, I., Kanev V.: Polar covariants of plane cubics and quartics, Advances in Math. 98 (1993), 216–301. [EmI] Emsalem J., Iarrobino A.: Inverse system of a symbolic power I, J. Algebra 174 (1995), 1080-1090. [FL] Fr¨ oberg R., Laksov D.: Compressed algebras, Conf. on Complete Intersections in Acireale, (S.Greco and R. Strano, eds), Lecture Notes in Math. # 1092, Springer-Verlag, Berlin and New York, 1984, pp. 121–151. [G] Geramita, A. V.: Inverse systems of fat points: Waring’s problem, secant varieties of Veronese varieties and parameter spaces for Gorenstein ideals. The Curves Seminar at Queen’s, Vol. X (Kingston, ON, 1995), 2–114, Queen’s Papers in Pure and Appl. Math., 102, Queen’s Univ., Kingston, ON, 1996. [GHMS]

, Harima T., Migliore J., Shin Y. S.: level algebra, to appear, Memoirs A.M.S. .

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[I1] Iarrobino A. : Compressed algebras: Artin algebras having given socle degrees and maximal length, Transactions of the A.M.S., vol. 285, no. 1, pp. 337-378, 1984. [I2] [I3]

: Ancestor ideals of vector spaces of forms, and level algebras, J. Algebra 272 (2004), 530–580. : Pencils of forms and level algebras, in preparation.

[IK]

, Kanev V.: Power Sums, Gorenstein Algebras, and Determinantal Varieties, (1999), 345+xxvii p., Springer Lecture Notes in Mathematics #1721.

[IS]

, Srinivasan H.: Artinian Gorenstein algebras of embedding dimension four: Components of PGOR(H) for H = (1, 4, 7, . . . , 1), J. Pure and Applied Algebra 201 (2005), 62-96.

[Kle1] Kleppe J. O.: Deformations of graded algebras, Math. Scand. 45 (1979) 205–231. [Kle2]

: The smoothness and the dimension of PGOR(H) and of other strata of the punctual Hilbert scheme, J. Algebra 200 (1998), 606–628.

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: Maximal Families of Gorenstein algebras, preprint, 2004, to appear, Transactions A.M.S.

[Mac1] Macaulay F. H. S.: The Algebra of Modular Systems, Cambridge Univ. Press, Cambridge, U. K. (1916); reprinted with a foreword by P. Roberts, Cambridge Univ. Press) [Mac2]

: Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927), 531–555.

[MMN] Migliore J., Mir´ o-Roig R., Nagel U.: Minimal Resolution of Relatively Compressed Level Algebras, Journal of Algebra 284 (2005), 333-370. [RS] Ranestad K., Schreyer, F.-O.: Varieties of sums of powers, J. Reine Angew. Math. 525 (2000), 147–181.

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: H-vectors and socle vectors of graded Artinian algebras, Ph.D. thesis, Queen’s University, 2004.

[Za3]

: Partial derivatives of a generic subspace of a vector space of forms: quotients of level algebras of arbitrary type, preprint, ArXiv Math.AC/0502466, 2005, to appear, Proc. A.M.S. .

Anthony Iarrobino Department of Mathematics Northeastern University Boston, MA 02115, USA Email: [email protected]

Varieties of reductions for gln Atanas Iliev∗and Laurent Manivel

Abstract. We study the varieties of reductions associated to the variety of rank one matrices in gln . In particular, we prove that for n = 4 we get a 12-dimensional Fano variety of Picard number one and index 3, with canonical singularities. 2000 Mathematics Subject Classiﬁcation: 14J45, 20G20, 14L30

1. Introduction This paper is a sequel to [5] and the companion paper [6], where we studied a family of smooth Fano varieties with many remarkable properties. These varieties were constructed as compactiﬁcations of what we called reductions for the four Severi varieties. Recall that the Severi varieties can be deﬁned as the projective planes over the four (complexiﬁed) normed algebras A = R, C, H, O – the reals, the complexes, the quaternions, and the octonions. More precisely, consider the Jordan algebra J3 (A) of A-Hermitian matrices of order 3. The projectivization of the set of rank one matrices in PJ3 (A) is the Severi variety Xa , a homogeneous variety of dimension 2a, where a = 1, 2, 4, 8 denotes the dimension of A. A non singular reduction is deﬁned as a 3-secant plane to Xa passing through the identity matrix I. The projection p from I to the hyperplane PJ3 (A)0 of traceless matrices sends the non-singular reductions to the family of 3-secant lines to the projected Severi variety X a , and the variety of reductions that we studied in [5] is the compactiﬁcation of that family in the Grassmannian of lines in PJ3 (A)0 . We proved that it is a smooth Fano manifold of dimension 3a, Picard number one, and index a + 1. In this paper we consider matrices of rank greater than three, and the corresponding varieties of reductions. For a = 1 they were previously studied by Ranestad and Schreyer [12], who proved that they are smooth up to rank 5, while in rank 6 the tangent cone to a normal slice to the singular locus is, rather remarkably, a cone over the spinor variety S10 . Here we will focus on the case a = 2, which has the interesting feature of being related to diﬀerent, but not less classical problems than the study of Fano varieties. Indeed, a non singular reduction for the ∗ Partially supported by Grant MI-1503/2005 of the Bulgarian Foundation for Scientiﬁc Research

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ˇ n−1 ⊂ Pgln , is the commutative algevariety of rank one matrices X2,n = Pn−1 × P bra of matrices that are diagonal with respect to some basis of Cn – hence a direct connection with the much studied problem of classifying commutative subalgebras of gln . Also, our variety of reductions Red(n) appears as a natural compactiﬁcation of the homogeneous space P GLn /N , where N denotes the normalizer of a maximal torus. For arbitrary n a deep understanding of this compactiﬁcation remains out of our reach: we only establish rather basic properties and raise a number of questions. We mainly prove that Red(n) is smooth in codimension one but always singular for n ≥ 4. Moreover, the canonical divisor of the smooth locus is minus three times the hyperplane divisor – but we don’t know if our varieties of reductions are normal in general. A tempting way to study Red(n) is to consider its tautological ﬁbration, which is birational to Psln . Quite interestingly, the induced rational map from this space to Red(n) is closely related to the geometry of the set of non regular matrices. We only sketch what should be the relevant plethystic transformations, and the connection with the Hilbert scheme of n points in Pn−1 . We can say a lot more when n = 4. We prove that every abelian four dimensional subalgebra of gl4 is in Red(4), which is made of fourteen P GL4 -orbits. Three of these are closed, among which a projective three-space and its dual constitute the singular locus of Red(4). We prove that the tangent cone to a normal slice to each of these singular components is a cone over the Grassmannian G(2, 6) – in particular, Red(4) is normal. Blowing them up, we get a smooth variety in which a maximal torus of P GL4 only has a ﬁnite number of ﬁxed points. This allows us to compute the ranks of the Chow groups of Red(4). We conclude that Red(4) is a rational Fano variety of dimension 12, Picard number one, index 3, with canonical singularities. Of course we expect that the variety of reductions deﬁned for the quaternions have similar properties, the geometry of the Scorza varieties being quite insensitive to the underlying normed algebra (see e.g. [3]).

2. Reductions for gln 2.1. Reductions and abelian algebras. Let Red(n)0 ⊂ G(n − 1, sln ) denote the space of Cartan subalgebras of sln . Recall that P GLn acts transitively on Cartan subalgebras, which are just the algebras of diagonal matrices with respect to some basis. Of course we may (and we will freely) identify them with Cartan subalgebras of gln , one way by adding the identity matrix, the other way by the natural projection p : gln → sln from the identity matrix. From the point of view of reductions, a Cartan subalgebra of gln is seen as ˇ n−1 ⊂ Pgln . a n-secant linear space to the rank one variety Xn = X2,n = Pn−1 × P ∗ Indeed, if such a linear space meets Xn at n distinct points e1 ⊗ e1 , . . . , e∗n ⊗ en ,

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and passes through I, we may suppose that I = e∗1 ⊗ e1 + · · · + e∗n ⊗ en , and then automatically e1 , . . . , en is a basis and e∗1 , . . . , e∗n is the dual basis. Once a Cartan subalgebra a of sln is ﬁxed, we get an isomorphism of Red(n)0 with P GLn /N (a), where the normalizer N (a) is an extension of the maximal torus A ⊂ P GLn whose Lie algebra is a, by the symmetric group Sn . Let Red(n) be the Zariski closure of Red(n)0 in the Grassmannian G(n − 1, sln ). This compactiﬁcation of P GLn /N (a) will be our main object of interest. We call it the variety of reductions for sln (or gln ). First note that Red(n) is a subvariety of the space Ab(n) of abelian (n − 1)dimensional subalgebras of sln . This variety Ab(n) has a simple set-theoretical description as the intersection of G(n−1, sln ) ⊂ PΛn−1 sln with the (projectivised) kernel of the natural map Θ : Λn−1 sln → Λ2 sln ⊗ Λn−3 sln −→ sln ⊗ Λn−3 sln , where the ﬁrst arrow is the natural inclusion, and the second one is induced by the Lie bracket. Beware that this intersection is not transverse, and even not proper already for n = 3, although Red(3) turns out to be smooth. Moreover, Ab(3) = Red(3), and we’ll prove in the second part of this paper that Ab(4) = Red(4). An easy general result is: Proposition 1. The variety of reductions Red(n) is an irreducible component of Ab(n). Proof. The generic element of a maximal torus in sln is a semisimple endomorphism with distinct eigenvalues. Since having distinct eigenvalues is an open condition in sln , containing such an endomorphism is also an open condition in Ab(n). But an abelian subalgebra of dimension n − 1 in sln , which contains an endomorphism with distinct eigenvalues, must be the centralizer of this endomorphism – hence a Cartan subalgebra. This proves our claim.

In fact it is easy to show that Ab(n) = Red(n) for large n. For example, suppose that n = 2m and let L be any subspace of dimension m in Cn . Let a(L) denote the space of endomorphisms whose image is contained in L and whose kernel contains L. Its dimension is m2 , and any (n − 1)-dimensional subspace of a(L) is an abelian subalgebra of sln . Since a generic such subspace determines L uniquely, we get a family of dimension m2 + (2m − 1)(m − 1)2 in A(n), which is strictly bigger than the dimension n(n − 1) of Red(n) as soon as m ≥ 4. A variant leads to the same conclusion for n = 2m − 1 and m ≥ 4. Question A. Does Ab(n) = Red(n) for n = 5 or 6? Remark. Suprunenko and Tyshkevich [14] described explicitly the maximal nilpotent and abelian subalgebras of sl5 and sl6 . It turns out that there is only a ﬁnite number of them up to conjugation, while there exists an inﬁnity in sln , n ≥ 7. In principle this should allow to answer the question above. Indeed, by the Jordan

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decomposition, the semisimple parts of the elements of an abelian subalgebra of sln commute, so that we can ﬁnd a minimal decomposition of Cn preserved by these, and basically, if this decomposition is not trivial, we are reduced to slm with m < n. If the decomposition is trivial, our subalgebra is nilpotent and we can use Suprunenko’s results. The description of the other irreducible components of Ab(n) is certainly an interesting problem. A basic question about the variety of reductions is: Question B. How can we characterize the points of Red(n) among the abelian subalgebras? Remark. A necessary condition for an abelian algebra a ∈ Ab(n) to belong to Red(n), is that the commutative subalgebra of gln , generated by a for the usual matrix product, has dimension at most n (this was already pointed out by Gerstenhaber [4]). But we don’t know any example of an abelian subalgebra in Ab(n) which does not fulﬁll this condition. Our hope is that Red(n) should in general be a much nicer variety than Ab(n) or its other irreducible components, when they exist. For example, we observe that: Proposition 2. The action of P GLn on Ab(n) has ﬁnitely many orbits only for n ≤ 5. Proof. For n ≤ 5 this follows from the work of Suprunenko and Tyshkevich [14]. Now suppose that n ≥ 6, and that n = 2m is even. As above, let L be an m-dimensional and consider the space of endomorphisms a(L). Any (n − 1)dimensional subspace of a(L) is an abelian subalgebra of sln , and a generic such subspace determines L. For P GLn to have a ﬁnite number of orbits in Ab(n), the parabolic subgroup PL of P GLn stabilizing L must have a ﬁnite number of orbits on the open subset G(n − 1, a(L))0 of (n − 1)-dimensional subspaces of a(L) whose generic element has image L. But PL acts on the Grassmannian G(n − 1, a(L)) only through its semisimple part P GLm × P GLm , whose action is equivalent to its natural action on G(n − 1, Mm (C)). The dimension of this Grassmannian is strictly bigger than the dimension of P GLm × P GLm as soon as m ≥ 3, so there must be an inﬁnity of orbits on any open subset. The case of odd n is similar.

In particular, the action of P GLn on Red(n) has ﬁnitely many orbits for n ≤ 5. Question C. Does Red(n) contain inﬁnitely many orbits of P GLn for n ≥ 6?

2.2. Special orbits. A point in Red(n)0 can be described as the centralizer of a regular semisimple element of sln . If we drop the semisimplicity hypothesis, we still get abelian

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(n − 1)-dimensional subalgebras of sln which we call one-regular subalgebras. Such subalgebras belong to Red(n), as follows from the proof of our next result. Proposition 3. The variety of reductions Red(n) contains a unique codimension one orbit Obound . A point in Obound is the centralizer of a regular matrix whose semisimple part has an eigenvalue of multiplicity two. Proof. Indeed, a point in this set Obound is deﬁned by n − 1 points in Pn−1 , plus a plane containing one of the lines, all these spaces being in general position – in particular, P GLn acts transitively on Obound . Counting dimensions, we easily check that its codimension in Red(n) equals one. Now let a be a point of Red(n) − Red(n)0 , and consider a general point x in a. By hypothesis, x is not regular semi-simple. Thus it belongs to the closure of the set of regular non-semisimple elements of sln . But this implies that a belongs to the closure of the set of centralizers of such elements, thus to the closure of Obound . In particular, if a does not belong to Obound , it must belong to a P GLn -orbit of smaller dimension.

Extending this a little bit we can describe other orbits in Red(n). Call an algebra a ∈ Ab(n) two-regular if it can be deﬁned as the common centralizer of two of its elements. The irreducibility of the commuting variety [11] implies: Proposition 4. Any one or two-regular algebra in Ab(n) does belong to Red(n). On the other hand we can describe lots of closed orbits in Ab(n). If we choose a ﬂag of subspaces of Cn , of the form Vi1 ⊂ · · · ⊂ Vip ⊂ Vj0 ⊂ Vj1 ⊂ · · · ⊂ Vjp , and if we consider the set of endomorphisms of Cn mapping Vjk to Vik for k = 1, . . . , p, and mapping Cn to Vj0 and Vj0 to zero, we get an abelian subalgebra of sln , which belongs to Ab(n) when it has the correct dimension, that is, when

n−1= (jk − jk−1 )(il − il−1 ). When the ﬂag varies, we get a closed P GLn -orbit in Ab(n), but it is not clear to us whether it belongs to Red(n) or not. A simple example is the case where our ﬂag reduces to Vj0 , which needs to be either a line or a hyperplane for the dimension condition to be fulﬁlled. We thus ˇ n−1 , which are dual projective Pn−1 and Omin P get two closed orbits Omin spaces. Proposition 5. The orbits Omin and Omin are contained in Red(n).

Proof. Consider the algebra of diagonal matrices with respect to a basis of the form e1 , e1 + te2 , . . . , e1 + ten , and let t tend to zero. An easy computation shows , which is thus contained in that the limit point in G(n − 1, sln ) belongs to Omin

Red(n) – hence Omin as well, by duality.

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2.3. Smoothness. For n ≥ 4, the variety of reductions Red(n) will be singular, but we expect the singular locus to be relatively small. Our main general result in that direction is the following: Proposition 6. The codimension one orbit Obound is contained in the smooth locus of Red(n). In particular Red(n) is smooth in codimension one. Proof. We choose a representative of Obound by ﬁxing a basis e1 , . . . , en of Cn and letting φ1 = e∗2 ⊗ e1 ,

φ2 = e∗1 ⊗ e1 + e∗2 ⊗ e2 ,

φk = e∗k ⊗ ek for 2 < k ≤ n.

We check by an explicit computation that the Zariski tangent space to Ab(n) at this point has dimension n(n − 1). Moreover, a ﬁrst order deformation in Ab(n) (or Red(n)) is given, in matrices, by

ψ1 =

μ ν 0 ··· 0

−θ23 0 0 ··· 0

1 −μ −θ31 ··· −θn1

··· ··· ··· ··· ···

−θ2n 0 0 ··· 0

,

ψ2 =

1 0 −θ31 ··· −θn1

0 1 −θ32 ··· −θn2

−θ13 −θ23 0 ··· 0

··· ··· ··· ··· ···

−θ1n −θ2n 0 ··· 0

,

and for 3 ≤ k ≤ n,

ψk =

0 0 ··· ··· θk1 ··· ···

0 0 ··· ··· θk2 ··· ···

··· ··· ··· ··· θk3 ··· ···

··· ··· ··· ··· ··· ··· ···

θ1k θ2k θ3k ··· 1 ··· −θnk

··· ··· ··· ··· ··· ··· ···

··· ··· ··· ··· −θkn ··· ···

.

(The matrix ψk has non zero coeﬃcients only on the k-th line and k-th column. Note the change of sign after the diagonal 1.) The number of free coeﬃcients is + 4(n − 2) + 2 = n(n − 1), as it should be.

2 n−2 2 We have a very simple geometric description of Obound , which will be useful later. Proposition 7. The closure of Obound is a generically transverse quadric section of Red(n). Proof. The Killing form induces a P GLn –invariant quadric in PΛn−1 sln given by Q(X1 ∧ · · · ∧ Xn−1 ) = det(trace (Xi Xj ))1≤i,j≤n−1 .

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Note that this is the restriction of the quadric in PΛn gln given by (almost) the same formula, the embedding being given by the wedge product with I. Clearly this quadric does not contain Red(n)0 but does contain its boundary. To check that the intersection is generically transverse we compute Q to ﬁrst order on the matrices above. At ﬁrst order, trace ψ12 = 2ν, trace ψ22 = 2, trace ψk2 = 1 for k > 2, and trace ψi ψj = 0 for i = j. Hence Q(ψ1 , . . . , ψn ) = 4ν, which proves our claim.

The tangent space to Red(n) at a generic point a is the image of the adjoint action ad

sln −→ Hom(a, sln /a) = Ta G(n − 1, sln ), whose kernel is the normalizer of a, that is, a itself at the generic point. Note that this makes sense for any a which is its own normalizer, in particular for any point of a regular orbit in Red(n). We deduce that the reduced tangent cone at such a point is linear, of the dimension of Red(n). This does not quite prove that we get a smooth point of Red(n), but we can ask: Question D. Is the set of one-regular subalgebras contained in the smooth locus of Red(n)?

2.4. The canonical sheaf. For a ∈ Red(n)0 , we may identify sln /a with the orthogonal a⊥ of a with respect to the Killing form, hence det Ta Red(n) with ∧top a⊥ , the maximal wedge power. Note that the the maximal torus A in P GLn whose Lie algebra is a acts trivially on this line. We thus get an action of the Weyl group N (A)/A Sn , which is simply given 2 by the sign representation. We deduce that the square KRed(n) 0 of the canonical 0 line bundle of Red(n) , is trivial. Indeed, we can choose an orthonormal basis of a⊥ with respect to the Killing form, and consider the square of the corresponding volume form on Ta Red(n). Since it is left invariant by the stabilizer of a in P GLn , we can translate it by P GLn to get a well-deﬁned non vanishing section ω of 2 KRed(n) 0. Let us compute the vanishing order of this section along the codimension one orbit Obound . To do this we restrict to the following line in Red(n), which meets Obound transversely at t = 0: ⎧⎛ a2 ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎜ta1 a(t) = ⎜ ⎜ ⎪ ⎪ ⎪⎝ ⎪ ⎩

⎞

a1 a2 a3

⎟ ⎟ ⎟, ⎟ ⎠

··· an

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ a1 , . . . , an ∈ C . ⎪ ⎪ ⎪ ⎪ ⎭

What we ﬁrst need is a ﬁrst order deformation of a(t) in Red(n) for each t. We claim that such a deformation is provided by the following matrices:

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ψ1 =

μ t+ν −tθ32 ··· −tθn2

1 −μ −θ31 ··· −θn1

−θ23 −tθ13 0 ··· 0

··· ··· ··· ··· ···

−θ2n −tθ1n 0 ··· 0

1 0 −θ31 ··· −θn1

0 1 −θ32 ··· −θn2

−θ13 −θ23 0 ··· 0

··· ··· ··· ··· ···

−θ1n −θ2n 0 ··· 0

ψ2 =

,

,

and for 3 ≤ k ≤ n,

ψk =

0 0 ··· ··· θk1 ··· ···

0 0 ··· ··· θk2 ··· ···

··· ··· ··· ··· θk3 ··· ···

··· ··· ··· ··· ··· ··· ···

θ1k θ2k θ3k ··· 1 ··· −θnk

··· ··· ··· ··· ··· ··· ···

··· ··· ··· ··· −θkn ··· ···

.

Indeed, the reader can check that for each t, the commutators of these matrices have order two with respect to the local parameters μ, ν, and θkl . This deﬁnes a basis of Ta(t) Red(n), by associating to each local parameter the tangent vector in the corresponding direction. For example, to the parameter μ, we associate the homomorphism ∂/∂μ ∈ Hom(a, sln /a) mapping ψk (t, 0) to ∂ψk /∂μ(t, 0). Explicitly: ∂/∂μ(ψ1 (t, 0)) = e∗1 ⊗ e1 − e∗2 ⊗ e2 , ∂/∂μ(ψk (t, 0)) = 0 for k > 1, ∂/∂ν(ψ1 (t, 0)) = e∗1 ⊗ e2 , ∂/∂ν(ψk (t, 0)) = 0 for k > 1, and so on. Now, what we have to do is to compare this basis with the other basis deﬁned by the adjoint action of a Killing orthonormal basis to a(t)⊥ . For t = 0, let t = τ 2 . Then a(t) ∈ Red(n)0 is the diagonal algebra associated with the basis e1 + τ e2 , e1 − τ e2 , e3 , . . . en of Cn . Its Killing orthogonal has a basis given by e∗1 ⊗ e1 − e∗2 ⊗ e2 , τ e∗2 ⊗ e1 − τ −1 e∗1 ⊗ e2 and the e∗j ⊗ ek , with j = k and j or k is bigger than two. This basis is not quite orthonormal, but the norm of the corresponding volume form does not depend on t. We claim that ∂/∂θjk = ad(e∗j ⊗ ek ), as the reader can check. Moreover, 1 1 ad(τ e∗2 ⊗ e1 − τ −1 e∗1 ⊗ e2 ), ∂/∂ν = ad(e∗1 ⊗ e1 − e∗2 ⊗ e2 ). 2τ 4t Note the factor τ , in agreement with the fact that only the square of the canonical sheaf is trivial on the open orbit. We deduce that the squared volume form ω at a(t) behaves like ∂/∂μ =

ωa(t) (Ctτ )2 ω0 = C 2 t3 ω0

when

t → 0,

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if ω0 denotes the local section of the square of the canonical bundle deﬁned by our local trivialization. Hence ω has a zero of order three along Obound . Since the codimension one orbit is itself a quadric section of Red(n), we deduce: Theorem 8. The canonical sheaf of the smooth locus Red(n)reg is KRed(n)reg = ORed(n)reg (−3), up to two-torsion. To assert that the canonical sheaf of Red(n) is really ORed(n) (−3), we would ﬁrst need to answer the following basic questions. Question E. Is Red(n) normal? Question F. Is the Picard group of Red(n) torsion free? What is its rank? Is it generated by the hyperplane divisor, at least up to torsion? Note that the hyperplane divisor on Red(n)reg is not divisible, since Red(n)reg contains lines and even planes, see Proposition 12.

2.5. Singularities. We devote this section to a local study of Ab(n) and Red(n) around the closed orbit . We choose the point of Omin deﬁned as the space of matrices whose kernel Omin contains and whose image is contained in the hyperplane U = e1 , . . . , en−1 . Locally around that point, an (n − 1)-dimensional subspace of sln is made of matrices of the form A(u) e∗n ⊗ u , α(u) ⊗ en −a(u) where u belongs to the hyperplane U , α is a linear map from U to U ∗ , A a linear map from U to End(U ), and a = trace A. This deﬁnes an abelian subalgebra of sln if and only if the following identities hold: α(u), v A(v)u − a(u)v [A(u), A(v)]w α(v), A(u)w − a(v)α(u), w

= α(v), u , = A(u)v − a(v)u,

(1) (2)

= α(u), w ⊗ v − α(v), w ⊗ u, = α(u), A(v)w − a(u)α(v), w .

(3) (4)

Letting B = A + aI, we can rewrite these identities as α(u), v B(u)v

= α(v), u , = B(v)u,

[B(u), B(v)] = α(u) ⊗ v − α(v) ⊗ u, B(u)t α(v) = B(v)t α(u),

(5) (6) (7) (8)

where B(u)t is the transpose of B(u) and acts on U ∗ . At ﬁrst order, the third set of equations reduces to α(u) ⊗ v = α(v) ⊗ u and implies that α = 0. The second set of equations means that B is mapped to

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zero by the map Hom(U, End(U )) = U ∗ ⊗ U ∗ ⊗ U → Λ2 U ∗ ⊗ U . We thus get (n−1)2 +(n−1)2 (n−2)/2 = n(n−1)2 /2 independent linear equations for the Zariski tangent space. Since this is half the dimension of the ambient Grassmannian, the Zariski tangent space has dimension n(n − 1)2 /2, which is bigger than n(n − 1) as soon as n > 3. Therefore: and Omin are contained in the singular Proposition 9. The minimal orbits Omin locus of Ab(n) for n ≥ 4. in Ab(n). Denote by An the projectivized tangent cone to a normal slice of Omin Equations of this tangent cone are α = 0 and the symmetry condition (6) on B (these equations deﬁne the tangent space), plus the quadratic equations implied by (7):

∀u, v, w ∈ U.

u∧v ∧[B(u), B(v)](w) = 0

(9)

Note that the tangent space is parametrized by the space of morphisms B ∈ Hom(U, End(U )) = U ∗ ⊗ U ∗ ⊗ U satisfying the symmetry condition (6), which implies that in fact they belong to the subspace S 2 U ∗ ⊗ U . This tensor product is the direct sum of two irreducible components, S10...0−2 U and U ∗ . This copy of U ∗ in the tangent space must correspond to the tangent directions to the singular Pn−1 . Since this strata is homogeneous, it is natstrata isomorphic to Omin ural to restrict to the normal slice given by S10...0−2 U , and characterized by the property that the trace of B is identically zero. Therefore, An is the subvariety of PS10...0−2 U , deﬁned by the quadratic equations (9). Let J denote the endomorphism of Cn deﬁned by Jei = ei+1 , where the indexes of the basis vectors are taken modulo n. Let ι denote the inclusion of U in Cn , and π the projection to U along en . Let B(ei ) = πJ i ι. We ﬁrst claim that B belongs to An . Indeed, we have

vk ej+k = vk ei+j+k , B(ei )B(ej )(v) = B(ei ) k+j=n

j+k,i+j+k=n

so that the commutator [B(ei ), B(ej )] is simply given by [B(ei ), B(ej )](w) = wn−i ej − wn−j ei . We immediately deduce that [B(u), B(v)](w) =

i

wn−i ui

⎞ ⎛

wn−j vj ⎠ u, v−⎝ j

and the equations (9) follow. We can be a little more precise: B deﬁnes a tangent direction not only to Ab(n), but really to Red(n). This is because the space of matrices

Varieties of reductions for gln

⎧⎛ 0 ⎪ ⎪ ⎪ ⎪ tx ⎨⎜ 1 ⎜ · · · a(t) = ⎜ ⎪⎜ ⎪ ⎝ txn−2 ⎪ ⎪ ⎩ 2 t xn−1

txn−1 0 ··· ··· ···

··· ··· ··· tx1 t2 x2

tx2 tx3 ··· 0 t2 x1

297 ⎞⎫ x1 ⎪ ⎪ ⎪ ⎪ x2 ⎟ ⎟⎬ ⎟ ··· ⎟ ⎪ xn−1 ⎠⎪ ⎪ ⎪ ⎭ 0

is an abelian (n − 1)-dimensional subalgebra of sln , passing through our prefered , whose generic point is in Red(n)0 (since, for example, if we let point of Omin xi = 0 for i > 0 we obtain a regular semisimple matrix when tn = 1). We thus get a rational curve on Red(n) whose tangent direction at t = 0 is precisely deﬁned by B. Lemma 10. The stabilizer Kn of B in P GLn−1 is ﬁnite. Proof. The Lie algebra of this stabilizer is the space of endomorphisms X ∈ sln such that X.B(u)(v) + B(Xu)(v) + B(Xv)(u) = 0 Let fi = Xei , and

fi+j

∀u, v ∈ U.

= B(ej )fi . We get the conditions fi+j + fi+j + fj+i = 0,

where indices are taken modulo n and with the convention that fn = 0. We deduce that f2 = −2f1+1 , then f3 = 2f1+1+1 − f1+2 . Then the condition that f1+3 + f3+1 = 2f2+2 implies that f1 is a combination of en−1 and en−3 , and the condition that f1+4 + f4+1 = f2+3 + f3+2 leads to f1 = 0. Then by induction fi = 0 for all i, that is, X = 0.

Question G. What is this ﬁnite group Kn ? A consequence of the lemma is that the orbit of the tangent direction deﬁned by B has dimension (n − 1)2 − 1, which is exactly one minus the dimension n(n − in Red(n). This suggests the following 1) − (n − 1) of our normal slice to Omin question: Question H. Does the closure Cn of this orbit coincide with the projectivized in Red(n)? tangent cone to the normal slice to Omin Question I. When is Cn a smooth variety? What can its singularities be? We cannot say much about this compactiﬁcation Cn of P GLn−1 /Kn , which should be an interesting object of study. We’ll prove in the next section that C4 is in fact a familiar object. At least can we say that for n ≥ 4, Cn is not a linear space – and we can therefore conclude: and Omin are contained in the sinProposition 11. The minimal orbits Omin gular locus of Red(n) for n ≥ 4. 0 Omin ? Question J. When does Red(n)sing = Omin

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2.6. Linear spaces and the incidence variety. For n = 3 we proved in [5] that through a general point of Red(3), there passes exactly three projective planes, which are transverse, and maximal. (In fact Red(3) does not contain any linear space of dimension greater than two.) This extends to Red(n) for any n: Proposition 12. Through a general point of Red(n), there passes n2 projective planes, which are transverse, and maximal linear subspaces of Red(n). Proof. A general point of Red(n) is an n-plane E of gln generated by e∗1 ⊗ e1 , . . . , e∗n ⊗ en for some basis e1 , . . . , en of Cn and its dual basis e∗1 , . . . , e∗n . A linear space in G(n − 1, sln ) passing through pE is deﬁned by two subspaces P ⊂ pE ⊂ Q of sln , where P is a hyperplane in pE or pE a hyperplane in Q. In the ﬁrst case, we get a linear space contained in Red(n) if and only if Q is contained in the centralizer of P . Thus P cannot contain any regular element, since otherwise its centralizer would be equal to E. Therefore P must be deﬁned by the condition that two vectors ei and ej belong to the same eigenspace, and its centralizer in sln has dimension n + 1. We thus get n2 linear spaces in Red(n) through pE, which are all projective planes, and clearly transverse. In the second case, Q is generated by pE and some non diagonal endomorphism x, which we can suppose to have zero diagonal coeﬃcients. A hyperplane in Q not containing x is the space of endomorphisms of the form t + μ(t)x, with t ∈ E, for some linear form μ on E. It will be an abelian subalgebra of sln if and only if μ(t)[x, s] = μ(s)[x, t] for all s, t ∈ E. This can hold only if [x, t] = μ(t)y for some y ∈ sln , which implies that x = e∗i ⊗ ej for some i = j. But then μ is ﬁxed up to constant, so our linear space is at most a line and we are back to the ﬁrst case.

Using this fact we can investigate the structure of Red(n) through the incidence variety Zn deﬁned by the diagram

Psln

)

Zn

*

Red(n) ⊂ G(n − 1, sln )

The map π : Zn → Red(n) is a Pn−2 -bundle, the restriction to Red(n) of the tautological vector bundle over G(n − 1, sln ). The projection σ : Zn → Psln is birational. Proposition 13. The map σ : Zn → Psln is an isomorphism exactly above the open subset of regular elements of sln . Recall that an endomorphism is regular if its centralizer has minimal dimension. ¯n This means that there is only one Jordan block for each eigenvalue. The set W of non regular elements is irreducible, with an open subset given by the set of ¯ n is the projection semi-simple elements with a double eigenvalue. In particular, W to sln , of the set Wn ⊂ gln of elements of corank at least two.

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Proof. The ﬁber of σ over x ∈ sln is the space of (n − 1)-dimensional abelian subalgebras a ⊂ sln containing x, hence contained the centralizer c0 (x) = c(x)∩sln . If x is regular, c(x) has dimension n, hence a = c0 (x), so that σ is one-to-one over x. Now suppose that x is a generic non regular endomorphism, so that x is semisimple with distinct eigenvalues, except one with multiplicity two. Let P P1 denote the projective line deﬁned by its two-dimensional eigenspace. Our abelian subalgebra a is deﬁned by an abelian two-dimensional subalgebra of gl(P ), which is either a maximal torus deﬁned by a pair of distinct points in P , or the centralizer of a nilpotent element, deﬁned by one point in P . We conclude that the ﬁber of σ over x is Sym2 P1 P2 . Therefore σ in not one to one over the generic point of ¯ n.

x, and a fortiori over the whole of W ¯ n has codimension three. Since the ﬁber of σ over its generic point Note that W ¯ n , whose has dimension two, we get an exceptional divisor E ⊂ Zn dominating W generic point is a pair (x ∈ t), with t a Cartan subalgebra of sln and x a non regular element of t. The intersection of E with the generic ﬁber p−1 (t) of p, n where t denotes the diagonal torus, is the union of the 2 hyperplanes Hij ⊂ Psln deﬁned by the equations ti = tj , where i < j. Note that the intersections of these hyperplanes are of two diﬀerent types: points in the intersections Hij ∩ Hkl map to points in sln such that the line from that point to the identity matrix I is bisecant to Wn ; points in the intersections Hij ∩ Hjk map to the projection to Psln of the variety of matrices of corank at least three. Since σ is birational, we get an induced rational map ϕ : Psln Red(n) of relative dimension n − 2 which has quite interesting properties. First note that if n . But H. = 1, is the class of a line in a general ﬁber of π, we have E. = 2 hence π ∗ O(1) = n2 H − E. This proves: Proposition 14. The rational map ϕ : Psln → Red(n) is deﬁned by a linear sys¯ n. tem In of polynomials of degree n2 , whose base locus contains W ¯ n a little better. Geometrically, we have Therefore, we need to understand W the following simple description. ¯ n is the projection from the identity matrix, of Proposition 15. The variety W the variety of matrices of corank at least two in Pgln . Proof. By deﬁnition, a matrix X ∈ sln is non regular if and only if some eigenvalue λ has multiplicity m ≥ 2. But then X − λI has corank m and projects to x. The converse assertion is not less obvious.

Note that if X ∈ sln has an eigenvalue with multiplicity two or more, its minimal polynomial has degree less than n – and conversely. We deduce:

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¯ n if and only if Proposition 16. A matrix X ∈ sln belongs to the cone over W X, pX 2 , . . . , pX n−1 are linearly dependent. ¯ n scheme-theoretically? Question K. Does this condition deﬁne W ¯ n of degree smaller than n , since the interFor sure there is no equation of W 2 ¯ n with any Cartan subalgebra of sln is a collection of n hyperplanes. section of W 2

The previous proposition motivates the introduction of a map tn : Λn−1 sln θ

n

S ( 2 ) sln θ(X, pX 2 , . . . , pX n−1 ),

→ →

where we recall that p : gln → sln denotes the natural projection, and we identify sln with its dual through the trace map. More explicitely, if Z1 , . . . , Zn−1 ∈ sln , we have tn (Z1 ∧ · · · ∧ Zn−1 )(X) = det(trace (Zi X j ))1≤i,j≤n−1 . ¯ n. As we have just seen, the base locus of the image of tn is equal to W n−1 sln can in principle be decomposed as a GLn -module The exterior power Λ as follows: in the Grothendieck ring of ﬁnite dimensional GLn -modules, we have the identity Λn−1 sln =

n

(−1)k−1 Λn−k gln .

k=1

These wedge powers, since gln = C ⊗ (Cn )∗ , can be decomposed using the Cauchy formulas, and then the Littlewood-Richardson rule can be used to perform the tensor products. This is not very eﬀective, and in fact the problem of decomposing the exterior powers of the adjoint representation of sln , and more generally of a simple Lie algebra, has been much studied since the pionnering work of B. Kostant [8], see also [1, 13] and references therein. Of course the map tn above has no reason of being injective – we’ll see in the next section that injectivity fails already for n = 4. This leaves quite a number of open questions: n Question L. Does In = IW ¯ n ( 2 )? Does In = Ker Θ? Does In = Im tn ? And can we compute Im tn explicitly ? n

A simple fact to mention about the image of tn is that it certainly contains S n Cn and its dual, embedded through the map sn :

S n Cn ⊗ (det Cn )−1 n v ⊗ (u1 ∧ · · · ∧ un )−1

→ →

n S ( 2 ) sln Pv (X) = v ∧Xv ∧ · · · ∧ X n−1 v/u1 ∧ · · · ∧ un ,

and similarly, the dual map s∗n :

S n (Cn )∗ ⊗ (det Cn ) en ⊗ (f1 ∧ · · · ∧ fn )−1

→ →

n S ( 2 ) sln Pe∗ (X) = e∧t Xe ∧ · · · ∧ t X n−1 e/f1 ∧ · · · ∧ fn .

(It is enough to deﬁne these maps on pure powers, since they generate the space of polynomials. The image of a monomial can be deduced by polarization.)

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For future use, note that we have a commutative diagram S n Cn ||

−→

i

Λn−1 sln α↓↑β

−→

π

S n Cn ||

S n Cn

−→

j

Λn gln

−→

ρ

S n Cn ,

deﬁned as follows. First, the map j is given by j(v n ⊗ e∗1 ∧ · · · ∧ e∗n ) = (e∗1 ⊗ v) ∧ · · · ∧ (e∗n ⊗ v). Second, we deﬁne the projection ρ by letting ρ(X1 ∧ · · · ∧ Xn ) = det(Xi (ej )) ⊗ e∗1 ∧ · · · ∧ e∗n , where e1 , . . . , en is any basis of Cn and e∗1 , . . . , e∗n the dual basis. Note that ρ ◦ j = id. Of course the maps α and β are deﬁned through the decomposition Λn gln = n Λ (CI ⊕ sln ) = Λn−1 sln ⊕ Λn sln . Explicitly, we have α(Y1 ∧ · · · ∧ Yn−1 ) = I ∧ Y1 ∧ · · · ∧ Yn−1 , n 1 (−1)j−1 trace (Xj )X1 ∧ · · · ∧ Xˆj ∧ · · · ∧ Xn . β(X1 ∧ · · · ∧ Xn ) = n j=1 The choice of the constant n1 is such that β ◦ α = id. Finally, we let i = β ◦ j and π = ρ ◦ α. We have π ◦ i = n1 id, since v n ⊗ e∗1 ∧ · · · ∧ e∗n

i

1 n

α

1 n

→ →

(−1)j−1 e∗j , v (e∗1 ⊗ v) ∧ · · · ∧ (e∗j−1 ⊗ v) ∧ · · · ∧ (e∗j+1 ⊗ v) j

e∗j , v (e∗1 ⊗ v) ∧ · · · ∧ I ∧ · · · ∧ (e∗n ⊗ v) j

v ρ

→

1 n

e∗j , v det

e1 v v en

=

1 n

⊗ e∗1 ∧ · · · ∧ e∗n

ej

j

v

e∗j , v v n−1 ej ⊗ e∗1 ∧ · · · ∧ e∗n = j

1 n v ⊗ e∗1 ∧ · · · ∧ e∗n . n

We use this diagram to deﬁne an automorphism τ of Λn−1 sln by (n − 1)! i ◦ π, n and we twist our map tn above by letting tn = tn ◦ τ . The point is that we now have: τ = id − (−1)n−1

Proposition 17. Let t ∈ Λn−1 sln belong to the cone over Red(n). Then the polynomial tn (t) on sln vanishes on the linear subspace t.

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Proof. We just need to prove it for a generic element of Red(n), that is, we may suppose that t corresponds to the space of matrices that are diagonal with respect to some basis e1 , . . . , en . Up to constant, we may therefore let t = (e∗1 ⊗ e1 − e∗2 ⊗ e2 ) ∧ · · · ∧ (e∗n−1 ⊗ en−1 − e∗n ⊗ en ). If x ∈ t is diagonal with eigenvalues x1 , . . . , xn , we immediately get 1 tn (t)(x) = det(xji − xji+1 ) = (−1)n−1 (xi − xj ). i<j

On the other hand, let us compute t0 = i ◦ π(t). First note that t=

n

(−1)j−1 (e∗1 ⊗ e1 ) ∧ · · · ∧ (e∗j−1 ⊗ ej−1 ) ∧ (e∗j+1 ⊗ ej+1 ) ∧ · · · ∧ (e∗n ⊗ en ),

j=1

and since of course I = e∗1 ⊗ e1 + · · · + e∗n ⊗ en , we deduce that α(t) = n(e∗1 ⊗ e1 ) ∧ · · · ∧ (e∗n ⊗ en ), hence π(t) = ne1 · · · en ⊗ e∗1 ∧ · · · ∧ e∗n and

1 j ◦ π(t) = ε(σ)(e∗σ(1) ⊗ e1 ) ∧ · · · ∧ (e∗σ(n) ⊗ en ). (n − 1)! σ∈Sn

From that we could easily compute t0 , but to ﬁnish the computation we prefer to notice that for all Y1 , . . . , Yn−1 , X ∈ gln , the identity (pY1 ∧ · · · ∧ pYn−1 )(pX, . . . , pX n−1 ) = (I ∧ Y1 ∧ · · · ∧ Yn )(I, X, . . . , X n−1 ) holds true, as the reader can easily check. Therefore, β(Y1 ∧ · · · ∧ Yn )(pX, . . . , pX n−1 ) = n

(−1)j−1 (Y1 ∧ · · · ∧ Yˆj ∧ · · · ∧ Yn )(pX, · · · , pX n−1 ) = j=1 n

(Y1 ∧ · · · ∧ Yj−1 ∧ I ∧ Yj+1 ∧ · · · ∧ Yn )(I, X, . . . , X n−1 ).

j=1

Applying this to X ∈ t a diagonal matrix and Yi = e∗σ(i) ⊗ ei , we see that only σ = 1 can contribute. Then we get I = Y1 + · · · + Yn , and we ﬁnally deduce that 1 n tn (t0 )(X) = (xi − xj ). (n − 1)! i<j This completes the proof.

If, as we expect, In = Im tn = Im tn , this proposition gives a coherent identiﬁcation between Red(n) and the image of the rational map deﬁned by that linear system. Indeed, the image of a general point x is the hyperplane of equations ¯ n also vanishing on x. By the proposition, sn (x), seen as a hyperplane of of W equations, does vanish on the centralizer of x, in particular on x itself. Without

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the twist τ , the image of the rational map deﬁned by In is only a translate of Red(n) by a linear automorphism.

2.7. Relations with the Hilbert scheme. The open P GLn -orbit of Red(n) is the space of n-tuples of points in general position in Pn−1 . This is also an open P GLn -orbit in the punctual Hilbert scheme Hilbn Pn−1 , which is thus birational to Red(n). For n = 3 we proved in [5] that there is a morphism Hilb3 P2 → Red(3), in fact a divisorial contraction between these two smooth varieties. We would like to extend this to n ≥ 4. We deﬁne auxiliary morphisms as follows. First, we have a GLn -equivariant map μn :

Λn (S n−1 Cn ) ∧ · · · ∧ en−1 n

en−1 1

−→ →

Λn (S n−1 Cn ) e2 · · · en ∧ · · · ∧ e1 · · · en−1 .

Question M. Is μn an isomorphism for all n? (We know it is for n = 3.) Now deﬁne the SLn -equivariant morphism νn : Λn (S n−1 Cn ) −→ Λn gln en−1 ∧ · · · ∧ en−1 → (e1 ∧ · · · ∧ en−1 ⊗ en ) ∧ · · · ∧ (en ∧ e2 ∧ · · · ∧ en−1 ⊗ e1 ). n 1

Here we identify Λn−1 Cn with (Cn )∗ , hence Λn−1 Cn ⊗ Cn with gln . For example, if e1 , . . . , en are independent and e∗1 , . . . , e∗n is the dual basis, the image tensor is just (e∗1 ⊗ e1 ) ∧ · · · ∧ (e∗n ⊗ en ). This can be identiﬁed (up to scalar, of course), with the linear space generated by the n rank one elements e∗k ⊗ ek of gln , the sum of which is the identity. In other words, we get a point of the open orbit of our reduction variety Red(n). Let Z be a n-tuple of points in general position in Pn−1 . Denote by T (Z) the union of the n2 codimension two linear spaces generated by all the (n − 2)-tuples of points in Z. Let P denote the Hilbert polynomial of this variety T (Z). Once we have chosen homogeneous coordinates adapted to our n-tuple of points, we see ˆi · · · x ˆj · · · xn , so that that the ideal of T (Z) is generated by the monomials x1 · · · x a supplement of IT (Z) (k) has a basis given by all the degree k monomials involving no more than n − 2 indeterminates. We deduce that n−2

nk − 1 P (k) = . j j−1 j=1 The transformation T deﬁnes a rational map from Hilbn Pn−1 to HilbP Pn−1 , which is not a morphism in general. But we may be able to deﬁne a morphism ρn : Hilbn Pn−1 → Red(n) as follows. We ﬁrst map the punctual scheme Z, reduced and in general position, to the linear system |IT (Z) (n−1)| ∈ G(n, S n−1 Cn ) ⊂ PΛn (S n−1 Cn ). Then we apply the linear automorphism μ−1 n , and ﬁnally the linear

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morphism νn to get a point ρn (Z) ∈ PΛn gln . We claim that ρn maps the component Hilbn0 Pn−1 of Hilbn Pn−1 containing the reduced schemes in general position, to the reduction variety Red(n). Indeed, if Z is the union of n points in general position, and if e1 , . . . , en are adapted coordinates, then |IT (Z) (n − 1)| = e1 · · · en−1 , . . . , e2 · · · en , so that ρn (Z) is the subspace of gln generated by e∗1 ⊗ e1 , . . . , e∗n ⊗ en , and belongs to Red(n). Question N. Can ρn be extended to a morphism? If yes, what is the exceptional locus of this morphism? Is ρn a divisorial contraction for n ≥ 4?

3. Reductions for gl4 Its seems rather diﬃcult to answer in full generality the questions we raised in the ﬁrst part of this paper. In this second part we check that (almost) everything works as expected when n = 4. Our ﬁrst result is that Red(4) = Ab(4). More precisely: Proposition 18. The variety of reductions in sl4 , coincides with the space of three-dimensional abelian subalgebras of sl4 . It is made of 14 P GL4 -orbits, exactly ˇ3, three of which are closed: a three-dimensional projective space P3 and its dual P and a variety of complete ﬂags F4 . The proof of this result will occupy the next two sections.

3.1. Classiﬁcation of three dimensional abelian subalgebras of sl4 . First, we have the one-regular abelian subalgebras, whose diﬀerent types are given by the possible sizes of the Jordan blocks of a generic element. We thus get ﬁve regular orbits, with generic Jordan type 1111 (genuine reductions), 211, 22, 31 or 4 (regular nilpotents) (the numbers are just the sizes of the Jordan blocks). We , O9 respectively. denote these orbits by O12 , O11 , O10 , O10 Now suppose that a contains no regular element. If it contains an element of Jordan type 211, a is contained in its centralizer which is a copy of gl2 × gl1 × gl1 , and the blocks from gl2 are generically non regular. But in dimension two this means that they are homotheties, and this leaves only two free parameters, a contradiction. The Jordan type 22 is eliminated for the same reason. If a contains an element of Jordan type 31, it must be contained in gl3 ×gl1 and the blocks from gl3 must be non regular, hence of the form xI + X with X 2 = 0 and we need an abelian plane of such endomorphisms. We know this leaves only two possibilities

Varieties of reductions for gln

(in fact only one up to transposition), ⎛ ⎞ c 0 a 0 ⎜0 c b 0⎟ ⎜ ⎟ ⎝0 0 c 0⎠ or 0 0 0 d

⎛

c b ⎜0 c ⎜ ⎝0 0 0 0

305

⎞ a 0 0 0⎟ ⎟. c 0⎠ 0 d

Hence two orbits O8 and O8 . We are left with the nilpotent abelian algebras containing If there is an element x with a Jordan block of size 3, say ⎛ ⎛ ⎞ 0 1 0 0 a b c ⎜0 0 1 0 ⎟ ⎜0 a b ⎜ ⎟ x=⎜ ⎝0 0 0 0⎠ then y = ⎝0 0 a 0 0 0 0 0 0 e

no regular element. ⎞ d 0⎟ ⎟ 0⎠ f

if y commutes with x. If y is nilpotent, a = f = 0. Since [y, y ] = (de −d e)e∗3 ⊗ e1 , we’ll get a three dimensional abelian algebra if we impose a linear relation between e and d. Up to a change of basis, there are only three cases, d = e, d = 0, e = 0, the last two being exchanged by transposition. Hence three orbits O8 , O7 and O7 . If no element of a has a Jordan block of size 3, then x2 = 0 for every x in a. Suppose that some x has rank two. Every endomorphism commuting with x will preserve its kernel, hence be of the form 2 A AB + BC A B 2 . y= , so that y = 0 C2 0 C Using the commutativity condition, we see that A (and C) must vanish or be proportional to a ﬁxed nilpotent matrix when y varies in a. If A and C are both not identically zero, we get up to a change of basis ⎛ ⎞ 0 a b c ⎜0 0 0 d⎟ ⎟ y=⎜ ad + be = 0. ⎝0 0 0 e ⎠ , 0 0 0 0 This means that d = ze, b = −za for some scalar z. But then a simple change of basis implies that we may suppose that A and C are in fact both identically zero! This means that there is a plane P such that every element of a has P in its kernel and its image in P . In fact this deﬁnes a four-dimensional abelian algebra, of which a is a hyperplane deﬁned by some non zero linear form. This form is deﬁned by some order two matrix, and changing basis gives the usual GL2 × GL2 action by left and right multiplication, with the rank as only invariant. We thus get two orbits O7 (rank two) and O6 (rank one). Finally, suppose that C is identically zero, but not A. Then the condition AB = 0 means that the the image of B is contained in the kernel of A, so that a is the space of traceless endomorphisms with image in a given line. Symmetrically, if A is identically zero, but not C, then a is the space of traceless endomorphisms whose kernel contains a given hyperplane. These two orbits O3 and O3 are ex-

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changed by transposition, they are the minimal orbits denoted Omin and Omin in the ﬁrst part of the paper.

Apart from O8 , O7 , O7 and O7 , all the orbits can be described in terms of geometric datas. For example, O12 is the variety of quadruples of independent points in P3 . A point in O11 is determined by two points and a line in general position, plus a point on the line, and so on. These orbits can therefore be described as open subsets of products of partial ﬂag varieties. A point in O7 or O7 determines a complete ﬂag in P3 , and these orbits are ∗ C -bundles over the complete ﬂag variety F4 . O7 is an aﬃne ﬁbration over the Grassmannian G(2, 4), and O8 an aﬃne ﬁbration over the partial ﬂag variety F1,3 . Here is the list of the 14 orbits with a representative for each. (We omit the condition that the trace must vanish.) The subscript is the dimension.

O12

O10

⎛ a 0 ⎜0 b ⎜ ⎝0 0 0 0 ⎛ a ⎜0 ⎜ ⎝0 0

b 0 a 0 0 c 0 0

⎞ 0 0⎟ ⎟ d⎠ c

0 a b ⎜0 0 a ⎜ ⎝0 0 0 0 0 0

⎞ c b⎟ ⎟ a⎠ 0

⎛ O9

⎛ O8

⎞ 0 0 0 0⎟ ⎟ c 0⎠ 0 d

c ⎜0 ⎜ ⎝0 0

b c 0 0

⎞ a 0 0 0⎟ ⎟ c 0⎠ 0 d

⎛ O11

⎛ O10

O8

O8

⎛ O7

⎞ 0 a b c ⎜0 0 0 a⎟ ⎜ ⎟ ⎝0 0 0 0 ⎠ 0 0 0 0 ⎛

O7

0 ⎜0 ⎜ ⎝0 0

⎞ 0 a b 0 b c⎟ ⎟ 0 0 0⎠ 0 0 0

⎞ a b 0 0 ⎜ 0 a 0 0⎟ ⎜ ⎟ ⎝ 0 0 c 0⎠ 0 0 0 d ⎞ a b c 0 ⎜ 0 a b 0⎟ ⎜ ⎟ ⎝ 0 0 a 0⎠ 0 0 0 d ⎛

⎞ c a⎟ ⎟ b⎠ 0

⎛

⎞ 0 a 0 c b 0⎟ ⎟ 0 c 0⎠ 0 0 d

0 a b ⎜0 0 0 ⎜ ⎝0 0 0 0 0 0 c ⎜0 ⎜ ⎝0 0 ⎛

O7

0 ⎜0 ⎜ ⎝0 0 ⎛

O6

0 ⎜0 ⎜ ⎝0 0

0 0 0 0

⎞ a c 0 b⎟ ⎟ 0 a⎠ 0 0

0 0 0 0

⎞ a b 0 c⎟ ⎟ 0 0⎠ 0 0

F4

Varieties of reductions for gln

⎛ O3

0 a ⎜0 0 ⎜ ⎝0 0 0 0

⎞ b c 0 0⎟ ⎟ 0 0⎠ 0 0

P3

O3

⎛ 0 ⎜0 ⎜ ⎝0 0

307

0 0 0 0

0 0 0 0

⎞ c b⎟ ˇ3 ⎟ P a⎠ 0

3.2. Degeneracies. We want to study which orbits are contained in the closure of which. We will denote O → O if O is included in the boundary of O. First note that if a ∈ O and a ∈ O are one-regular, that is, can be deﬁned as the centralizers of some regular elements x and x , we just need to let x degenerate to x in the open set of regular elements to make a degenerate to a . And letting x degenerate to x is possible as soon as this is compatible with the size of the Jordan blocks. We deduce that O → O as soon as dim O > dim O . More generally, we know that the two-regular orbits in Ab(4) are contained in Red(4). An easy caseby-case check leads to the following conclusion: Lemma 19. The only orbits in Ab(4) which are not two-regular are O7 , O6 , O3 and O3 . We complete the picture by showing that any orbit, with of course O12 excepted, is in the closure of an orbit of larger dimension. This will imply that every three-dimensional abelian subalgebra of sl4 is contained in the closure of the variety of non singular reductions. Actually we prove a little more than needed, in order to deduce the full diagram of degeneracies. O9 → O8 : if we take the representative above of O9 and make the change of basis e1 → te1 , e3 → te3 , we get the abelian algebra of matrices of the form ⎛ ⎞ ⎞ ⎛ 0 a b c 0 t−1 a b t−1 c 2 ⎟ ⎜ ⎜0 0 ta b ⎟ ⎟ = ⎜0 0 t a b ⎟ , ⎜ −1 ⎝0 0 a⎠ 0 0 t a⎠ ⎝ 0 0 0 0 0 0 0 0 0 0 if a = t−1 a, b = b, c = t−1 c. Letting t → 0, we get an abelian subalgebra belonging to O8 . → O8 : if we take the representative above of O10 and make the change of O10 −1 basis e2 → t e2 , we get the abelian algebra of matrices of the form ⎛ ⎞ ⎞ ⎛ a b c 0 a t−1 b c 0 2 ⎜ ⎜0 ⎟ a tb 0⎟ ⎜ ⎟ = ⎜ 0 a t b 0⎟ , ⎝0 a 0⎠ 0 a 0⎠ ⎝ 0 0 0 0 0 d 0 0 0 d

if b = t−1 b. Letting t → 0, we get an abelian subalgebra belonging to O8 . By → O8 . transposition we also have O10

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O8 → O7 : we take the representative above of O8 and make the change of basis e1 → e4 , e2 → e3 , e3 → e2 , e4 → e1 + t−1 e4 , and we let t → 0. O8 → O7 : make the change e3 → t−1 e3 and let t → 0. By transposition we also have O8 → O7 . O7 → O6 : make the change e3 → t−1 e3 and let t → 0 after renormalizing by c = tc. It is not more diﬃcult to see that O7 → O6 and O7 → O6 . O7 → O3 : make the change e2 → t−1 e2 and let t → 0 after renormalizing by a = t−1 a. Transposing, we also get O7 → O3 . Finally, we don’t have O9 → O8 since O9 is nilpotent but not O8 ; neither O8 → O7 because an abelian algebra in O8 maps a ﬁxed hyperplane to a ﬁxed line, while this does not happen for an abelian algebra in O7 ; neither O7 → O3 or O7 → O3 since the matrices in an algebra belonging to O3 vanish on a common line but not on a common plane. We deduce the complete incidence diagram:

O10 ↓ ↓ ↓ O8

O7 ↓ ↓ O3

) *

* ) *

O12 ↓ O11

O9 ↓ O8 O7 ↓ O6

* )

) * )

O10 ↓ ↓ ↓ O8

O7 ↓ ↓ O3

3.3. The linear span of Red(4). Remember that set-theoretically, Red(4) = Ab(4) can be deﬁned as a linear section of G(3, sl4 ) by the kernel of the map Θ : Λ3 sl4 → Λ2 sl4 ⊗ sl4 → sl4 ⊗ sl4 , obtained by composing the obvious inclusion with the commutator Λ2 sl4 → sl4 . With the help of LiE [9], we check that this kernel is kerΘ = S3−1−1−1 C4 ⊕ S111−3 C4 ⊕ S21−1−2 C4 . ˇ 3 and F4 which are the closed orbits Since Red(4) contains three closed orbits P3 , P in the projectivisations of the simple factors of kerΘ, we conclude that the linear

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309

span in PΛ3 sl4 of the abelian subalgebras, is the whole of kerΘ. Its dimension is 35 + 35 + 175 = 245.

3.4. The incidence variety and the induced rational map. Remember the diagram

Psl4

)

Z4

*

Red(4) ⊂ G(3, sl4 )

The map π : Z4 → Red(4) is a P2 -bundle, while the projection σ : Z4 → Psl4 is birational, and an isomorphism above the open set of regular elements of sl4 . The rational map ϕ : Psl4 Red(4) is deﬁned by a linear system I4 of sextics ¯ 4. vanishing on W Proposition 20. The linear system I4 is equal to IW ¯ 4 (6), and to the image of t4 , and to the kernel of Θ. ¯ 4 is generated by Proof. A computation by Macaulay [10] shows that the ideal of W 245 sextics (we thank Marcel Morales for his help in performing this computation). We already know 245 such sextics: the image of s4 , a copy of S 4 C4 ⊗ (det C4 )−1 = S3−1−1−1 C4 , gives 35 of them; the image of s4 gives 35 others, a copy of the dual module; and the image of t4 contains 175 more. Indeed, remember that t4 associates to a triple of matrices Y1 , Y2 , Y3 ∈ sl4 the sextic polynomial ⎞ ⎛ trace (Y1 X) trace (Y2 X) trace (Y3 X) P (X) = det ⎝trace (Y1 X 2 ) trace (Y2 X 2 ) trace (Y3 X 2 )⎠ . trace (Y1 X 3 ) trace (Y2 X 3 ) trace (Y3 X 3 ) Choose two independent vectors u, v ∈ C4 and two independent linear forms α, β vanishing on them. Letting Y1 = α ⊗ u, Y2 = β ⊗ u and Y3 = α ⊗ v, we get the polynomial ⎞ ⎛ α(Xu) β(Xu) α(Xv) P (X) = det ⎝α(X 2 u) β(X 2 u) α(X 2 v)⎠ . α(X 3 u) β(X 3 u) α(X 3 v) Note that this polynomial remains unchanged if we add to v a multiple of u, or to β a multiple of α. This means that, up to constant, this polynomial only depends on the complete ﬂag Cu ⊂ Cu ⊕ Cv = Ker(α) ∩ Ker(β) ⊂ Ker(α). We conclude that the projectivized image of t4 contains a copy of the compete ﬂag manifold F4 . Moreover, since the weights of u, v, β, α in P are 2, 1, 1, 2, the linear span of this ﬂag manifold is a copy of the GL4 -module S21−1−2 C4 , which has dimension 175. We conclude that, as a GL4 -module, 4 4 4 IW ¯ 4 (6) = S3−1−1−1 C ⊕ S111−3 C ⊕ S21−1−2 C = Imt4 .

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This is isomorphic with KerΘ; more precisely, t4 restricts to an isomorphism between KerΘ and Imt4 , since a computation by LiE shows that Λ3 sl4 is multiplicity free.

Once this is established, we can understand the map ϕ geometrically, in particular we can describe the ﬁber of σ over most points x ∈ sl4 , that is, the variety parametrizing the abelian three-dimensional subalgebras of sl4 cointaining x. To state our next result we need to deﬁne several natural subvarieties of Psl4 . ¯ 4 of non regular elements, and the projection We already introduced the variety W ¯ 4 of the rank one variety in Pgl4 . Let X 0 denote the variety of rank one matrices X 4 in Psl4 . Let also Y4 denote the space of matrices in Psl4 which belong to some bisecant line to W4 passing through the identity matrix. A generic point in Y4 is a matrix with two (opposite) eigenvalues, both of multiplicity two, so that Y4 contains an open P GL4 -orbit isomorphic with the space of pairs of skew lines in P3 . Let Y04 denote the complement of the open orbit in Y4 . The points of Y04 are nilpotent matrices, either with two Jordan blocks of size two, or of rank one (hence in X40 ). ¯ 4 and Y4 , whose intersection is X 0 . ¯ 4 is the union of X The singular locus of W 4 Proposition 21. Let x ∈ Psl4 . The ﬁber of σ over x is 1. a point if x is regular, 2. a projective plane if x belongs to W4,reg , 3. the product of two projective planes if x ∈ Y4 − Y40 , ¯4 − X 0. 4. a copy of Red(3) if x ∈ X 4 Proof. If x is regular, the unique three-dimensional subalgebra of sl4 that contains it is its centralizer, thus σ −1 (x) is a point. Note that if moreover x is semisimple, ¯ 4 with the Cartan subalgebra c(x) is the union of six hyperthe intersection of W planes – so that the linear system IW ¯ 4 (6) maps c(x) to one point, as we already know. Now suppose that x is not regular. Since every abelian algebra containing x is certainly included in the centralizer c0 (x), we just need to understand the restriction of ϕ to the linear subspace c0 (x) to be able to determine the image of x by ϕ. ¯ 4 ∪ Y4 , it has three eigenvalues, one of which has If x is not contained in X multiplicity two. Up to conjugation, we may therefore suppose that ⎛ ⎛ ⎞ ⎞ α 0 0 0 ( δ γ 0 0 2 ⎜η ε 0 0 ⎟ ⎜ 0 α 0 0⎟ ⎜ ⎟ ⎟ x=⎜ ⎝ 0 0 β 0 ⎠ ∈ c(x) = ⎝0 0 μ 0⎠ . 0 0 0 ν 0 0 0 γ Let A denote the upper left corner of this matrix M in c(x). For M to belong to ¯ 4 , we have several possibilities: either μ = ν, or μ or ν is an (the cone over) W

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eigenvalue of A, or A must be a homothety. This shows that the linear system IW ¯ 4 (6), restricted to c0 (x), contains a ﬁxed hyperplane and two ﬁxed quadrics, the residual system being generated by the three linear conditions for A to be a multiple of the identity. To resolve the indeterminacies we just need to blow-up the corresponding codimension three linear subspace, and the image of x by ϕ is isomorphic with the ﬁber of that blow-up over x, which is just a projective plane. ¯ 4 −X 0 . Then x has two eigenvalues, one of multiplicity Now suppose that x ∈ X 4 three. The centralizer c(x) is isomorphic with gl3 , and the the linear system IW ¯ 4 (6) restricted to c(x) contains the ﬁxed cubic det(M + tr(M )I) = 0. The residual system is the space of cubics vanishing on the cone with vertex I and base the variety of rank one matrices. This is the system of cubics on sl3 vanishing on the projection of this rank variety, and we know by [5] that its image is nothing but a copy of the reduction variety Red(3).

Remark. This analysis suggests that it could be possible to resolve the indetermi¯ 4 , Y 0 , Y4 , W4 , or nacies of ϕ by blowing up successively the diﬀerent strata X40 , X 4 rather their successive strict transforms – but we have not been able to do that. Also it could be possible to extend this analysis to higher rank: on each strata we can restrict the linear system that should deﬁne ϕ, factor out the ﬁxed components and get a linear system that comes from smaller rank. This also makes sense for a = 2.

3.5. The singular locus. Proposition 22. Red(4)sing = Omin

0

Omin .

Proof. A simple computation shows that O6 is contained in the regular locus (take local coordinates on the Grassmannian, write the commutativity conditions down and get 24 independent linear relations). Since O6 belongs to the closure of any orbit other than the two minimal orbits of dimension 3, which we already know to be singular, there is no other singular orbit.

Recall that we denoted by C4 = A4 the projectivized tangent cone to a normal slice to O3 in Red(4). This is an eight-dimensional variety deﬁned by 15 quadratic equations. Proposition 23. The variety C4 ⊂ P14 is projectively equivalent to G(2, 6). Proof. We deﬁne an equivariant map T from Λ2 S 2 U ∗ to the space of traceless symmetric maps from U to End(U ), by sending an elementary tensor e2 ∧ f 2 to the map B deﬁned by B(u)(u) = (e, u)(f, u) e∧f, 2

∗

u ∈ U,

with the identiﬁcation of Λ U with U . We claim that this map T sends the Grassmannian G(2, S 2 U ∗ ) ⊂ PΛ2 S 2 U ∗ isomorphically on C4 ⊂ PS1,0,−2 U .

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Consider a generic point of G(2, S 2 U ∗ ), that is, a generic pencil of conics in PU P2 . Such a pencil is deﬁned by its base-locus, a set of four points in general position. Choosing homogeneous coordinates for which these four points are [1 : 0 : 0], [0 : 1 : 0], [0 : 0 : 1], [1 : 1 : 1], we get a pencil generated by the reducible conics (x − z)y and (x − y)z. But, by polarization, our map T sends a tensor ee ∧ f f to the map B deﬁned by B(u)(u) = (e, u)(f, u) e ∧f +(e , u)(f, u) e∧f +(e, u)(f , u) e ∧f +(e , u)(f , u) e∧f. Substituting e = x − z, e = y, f ⎛ 3u1 ⎝ −u B(u) = 2 −u3

= x − y, f = z, we get ⎞ −u1 −u1 3u2 −u2 ⎠ − (u1 + u2 + u3 )I. −u3 3u3

Now for two vectors u and v, let δij = ui vj − uj vi . A that ⎛ δ12 + δ13 −3δ12 + δ13 [B(u), B(v)] = ⎝3δ12 + δ23 −δ12 + δ23 3δ13 − δ23 3δ23 − δ13

simple computation shows ⎞ δ12 − 3δ13 −δ12 − 3δ23 ⎠ , −δ13 − δ23

and one can easily check that the image of this matrix is always contained in u, v . We conclude that T maps G(2, S 2 U ∗ ) to C4 , which are both irreducible of dimension 8. Since T is a linear automophism, it restricts to a projective

equivalence between G(2, S 2 U ∗ ) and C4 . By Lemma 10, P GL3 has an open orbit in G(2, 6) = G(2, S 2 C3 ), the space of pencils of plane conics. This is well-known and quite obvious, since a general pencil is determined by its base-locus – four points in general position, and P GL3 acts transitively on such four-tuples. In particular, we deduce that the stabilizer of a general pencil is the stabilizer of its base-locus. This identiﬁes for n = 4 the ﬁnite group we introduced in Lemma 10: Corollary 24. The ﬁnite group K4 is the symmetric group S4 . Proof. Given four points in general position in P2 , there is a unique projective transformation which ﬁxes two of them and exchanges the other two. This implies that the stabilizer in P GL3 of our four-tuple of points is a copy of S4 , and the corresponding pencil of conics has the same stabilizer.

But a more interesting consequence of the previous proposition is: Corollary 25. The variety of reductions Red(4) is normal, with canonical singularities. Proof. Since the Grassmannian G(2, 6) is projectively normal, the cone over it is normal, thus the tangent cone to a singular point of Red(4) is normal as well.

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This implies that Red(4) itself is normal. By Theorem 8 its anticanonical divisor is −KRed(4) = ORed(4) (3), hence eﬀective, and the singularities are then automatically canonical.

Remark. As explained in [7], G(2, 6) is also the projectivized tangent cone to a normal slice to the singular locus of Hilb4 P3 , which is also a P3 , parametrizing double points. What we expect is that the rational map ρ4 : Hilb4 P3 Red(4) constructed in 2.7, is a morphism contracting the divisor in Hilb4 P3 deﬁned as the ˇ 3 , and restricting to closure of linearly dependant four-tuples of points, to O3 P an isomorphism outside this divisor, in particular around the singular locus, which should be mapped to O3 P3 . Therefore the singularities should really be the same, and not just the tangent cones.

3.6. Resolving the singularities. ˜ denote the blow-up of G(3, sl4 ) along the smooth subvarieties O and O . Let G 3 3 Since the tangent cone to Red(4) in a normal slice to each of these orbits is smooth, ˜ is a smooth variety R ˜ with an induced action the strict transform of Red(4) in G of P GL4 . The two exceptional divisors are G(2, 6)-ﬁbrations above copies of P3 . Let T denote a maximal torus in P GL4 . ˜ has only a ﬁnite number of ﬁxed points Proposition 26. The smooth variety R of T . This number is equal to the Euler characteristic ˜ = 193. χ(R) ˜ must dominate a T -ﬁxed point in Red(4). Using our Proof. A T -ﬁxed point in R explicit description of the P GL4 -orbits in Red(4) we can easily determine these ﬁxed points. Indeed, if we choose for T the torus deﬁned by the canonical basis of C4 , we see that an orbit O contains a ﬁxed point only when the corresponding representative is generated by diagonal matrices and matrices of the form e∗i ⊗ ej . Then all the ﬁxed points in the orbit can be deduced from a permutation of the basis vectors. We get the following numbers of ﬁxed points in the diﬀerent orbits: O #OT

O12 1

O11 12

O10 12

O10 0

O9 0

O8 0

O8 12

O8 12

O7 0

O7 0

O7 0

O6 24

O3 4

O3 4

˜ except the eight ones Each of these ﬁxed points gives a unique ﬁxed point in R, in O3 ∪ O3 . For each of these, we need to count the number of normal directions that are ﬁxed by T – that is, the number of T ﬁxed points in the corresponding copy of G(2, 6). It is easy to see that this number is ﬁnite, hence equal to the Euler characteristic of the Grassmannian, that is 15. We thus get 120 ﬁxed points ˜ plus 73 coming from the smooth locus of Red(4). in R, ˜ is That the total number of ﬁxed points equals the Euler characteristic of R then an immediate consequence of the Byalinicki-Birula decomposition [2].

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Corollary 27. Red(4) is rational. ˜ is smooth and has a ﬁnite number of points ﬁxed by a torus action, Proof. Since R it is a compactiﬁcation of a C12 - thus a rational variety, as well as Red(4).

The Byalinicki-Birula decomposition allows to compute the Betti numbers of ˜ For this we need the weights of the T -action on the tangent spaces to R ˜ at the R. ﬁxed points of T . For the 73 ﬁxed points that do not belong to the exceptional divisors of the ˜ (or Red(4), equivalently) projection to Red(4), we compute the tangent spaces to R as limits of tangent spaces at points of the open P GL4 -orbit O12 . Indeed, the tangent space to Red(4) at a point a ∈ O12 , as we have seen, is easily computed as the image of the (injective) map sl4 /a → Hom(a, sl4 /a) = Ta G(3, sl4 ) deﬁned by the Lie bracket. Note that we need only one computation per P GL4 orbit, since the symmetric group S4 acts transitively on the set of T -ﬁxed points in each orbit. Thus only six computations are enough to take care of these 73 ﬁxed points. For the 120 remaining ﬁxed points, we proceed as follows. Consider the point a of O3 deﬁned as at the beginning of 2.5, with n = 4. The splitting of C4 into the sum of the hyperplane U and the line generated by e4 leads to the identiﬁcations Ta G(3, sl4 ) ∪ Ta Red(4)

Hom(∗ ⊗ U, U ∗ ⊗ U ⊕ U ∗ ⊗ ) ∪ Homs (∗ ⊗ U, U ∗ ⊗ U )

where Homs ( ∗ ⊗ U, U ∗ ⊗ U ) := ⊗ S 2 U ∗ ⊗ U ⊂ ⊗ U ∗ ⊗ U ∗ ⊗ U = Hom( ∗ ⊗ U, U ∗ ⊗ U ).

Now, recall that S 2 U ∗ ⊗ U = U ∗ ⊕ S1,0,−2 U . The U ∗ factor corresponds to the tangent directions to the orbit O3 . The other term S1,0,−2 U = ∧2 (S 2 U ∗ ) ⊗ det U is, up to a twist, the ambient space for the Pl¨ ucker embedding of G(2, S 2 U ∗ ), which we identiﬁed with the projectivized tangent cone to Red(4) in the directions nor˜ over this point a of Red(4), are in mal to O3 . Then the ﬁxed points of T in R correspondence with the 15 ﬁxed points of T contained in that Grassmannian. ˜ from those And we deduce the weights of the T -action on the tangent space to R 2 ∗ of the T -action on the tangent space to G(2, S U ), through the previous identiﬁcations. Again, there are enough symmetries for the eﬀective computations to remain tractable. Finally, we choose a general enough one-dimensional subtorus of T , and count the number of negative weights of the restricted action on the tangent spaces to the ﬁxed points : this gives the dimensions of the corresponding strata in the Byalinicki-Birula decomposition. The conclusion is the following:

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˜ are all zero. The even Betti numProposition 28. The odd Betti numbers of R bers are 1, 3, 9, 15, 23, 29, 33, 29, 23, 15, 9, 3, 1. Applying the same arguments as for the proof of Theorem 2.4 in [7], we can deduce the ranks of the Chow groups of Red(4). Indeed, passing from Red(4) to ˜ amounts to replacing two copies of P3 by two G(2, 6)-bundles over them, and R the ranks of the Chow groups are modiﬁed accordingly. We get: Proposition 29. The Chow groups of Red(4) have respective ranks 1, 1, 3, 5, 7, 11, 14, 13, 11, 7, 5, 1, 1. In particular, Red(4) has Picard number one.

References [1] Berenstein A.D., Zelevinsky A.V., Triple multiplicities for sl(r+1) and the spectrum of the exterior algebra of the adjoint representation, J. Algebraic Combin. 1 (1992), 7–22. [2] Bialynicki-Birula A, Some theorems on actions of algebraic groups, Ann. of Math. 98 (1973), 480–497. [3] Chaput P.E., Scorza varieties and Jordan algebras, Indagationes Math. 14 (2003), 169-182. [4] Gerstenhaber M., On dominance and varieties of commuting matrices, Ann. of Math. (2) 73 (1961), 324–348. [5] Iliev A., Manivel L., Severi varieties and their varieties of reductions, to appear in Crelle’s journal. [6] Iliev A., Manivel L., The Chow ring of the Cayley plane, Compositio Mathematica 141 (2005), 146-160. [7] Katz S., The desingularization of Hilb4 P 3 and its Betti numbers, in Zerodimensional schemes (Ravello, 1992), 231–242, de Gruyter, Berlin, 1994. [8] Kostant B., Eigenvalues of the Laplacian and commutative Lie subalgebras, Topology 3 (1965), 147–159. [9] LiE, a software package for Lie group computations, by Arjeh M. Cohen, Marc van Leeuwen and Bert Lisser. [10] Bayer D., Stillman M., Macaulay, a system for computation in algebraic geometry and commutative algebra, available at http://www.math.columbia.edu/ bayer/Macaulay/ [11] Motzkin T.S., Taussky O., Pairs of matrices with property L II, Trans. Amer. Math. Soc. 80 (1955), 387–401. [12] Ranestad K., Schreyer O., The variety of polar simplices, unpublished manuscript.

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[13] Reeder M., Exterior powers of the adjoint representation, Canad. J. Math. 49 (1997), 133–159. [14] Suprunenko D. A., Tyshkevich R.I., Commutative matrices, Nauka i Tehnika, Minsk 1966. Atanas Iliev Institute of Mathematics Bulgarian Academy of Sciences Acad. G. Bonchev street 8, 1113 Soﬁa, Bulgaria. Email: [email protected] Laurent Manivel Institut Fourier UMR 5582 (UJF-CNRS) BP 74, 38402 St Martin d’H`eres Cedex, France. Email: [email protected]

Birational geometry of rationally connected manifolds via quasi-lines Paltin Ionescu

Abstract. This is, mostly, a survey of results about the birational geometry of rationally connected manifolds, using rational curves analogous to lines in Pn (quasi-lines). Various characterizations of a Zariski neighbourhood of a line in Pn are obtained, some of them being new. Also, methods of formal geometry are applied for deducing results of birational nature. 2000 Mathematics Subject Classiﬁcation: 14-02, 14E08, 14E30, 14M20.

0. Introduction The essential role played by rational curves in the birational classiﬁcation of algebraic varieties became clear since the appearence of Mori theory [14, 15]. In 1992, Koll´ ar, Miyaoka and Mori [12] introduced the very useful class of rationally connected manifolds. This class contains important subclasses, such as unirational or Fano manifolds, admits several convenient characterizations and has many good stability properties. One of the characterizations of rationally connected manifolds is the presence of a smooth rational curve having ample normal bundle. A model of a rationally connected projective manifold X is a pair (X, Y ), where Y is a smooth rational curve such that NY /X is ample (cf. [8]). Two models are equivalent, denoted (X, Y ) ∼ (X , Y ), if there is an isomorphism between open Zariski neighbourhoods of Y , respectively Y , sending Y to Y . Rationally connected manifolds are known to have a very complicated birational geometry; see e.g. [9, 10]. Hopefully, a convenient choice of models will simplify the original birational problem. In fact, using Hironaka’s result [6], we see that the birational classiﬁcation (of rationally connected manifolds) is essentially the same as the classiﬁcation of models, modulo the above equivalence relation. Given a rationally connected projective manifold, there is some birational model containing a “quasiline”, i.e. a smooth rational curve having the normal bundle of a line in Pn (cf. [7]). It turns out that the Hilbert scheme of quasi-lines has nice properties; in particular, counting curves through two general points can detect the (uni)rationality of the given manifold. An intermediate step in studying the equivalence of two models

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(X, Y ) and (X , Y ) is the comparison of the formal completions X/Y and X/Y . This allows one to use powerful results from formal geometry due to Hironaka, Hironaka–Matsumura, Hartshorne, Gieseker (see [1] or [4]) in the study of models. This paper is, mostly, a survey based on the works [2], [8] and [7]; on the other hand, it also contains several new results, for instance Proposition 1.8, Theorem 2.8, Proposition 4.11, Theorem 4.16 and Corollary 4.17. Proposition 2.1, Corollary 2.3, Theorem 2.5 and Proposition 4.8 are reﬁnements of results from [2] and [7]. In the ﬁrst section we discuss, informally, models in general. In Theorem 1.3 we prove (cf. [7] and [8]) a reduction result that replaces a given model by one of lower dimension, in the presence of a suitable linear system. We consider deformations of models and recall a ﬁniteness result from [8]. Also, a notion of minimality is brieﬂy disscused. Next we recall from [7] the existence of quasi-lines and explain their use. The second section contains two characterizations of the basic model (Pn , line). The ﬁrst one, Theorem 2.5, cf. [7], is in terms of curves only. The second one, Theorem 2.8 which is new, is in terms of linear systems of “maximal dimension”. We note one consequence of Theorem 2.5, cf. [7]. Rational manifolds containing “big” open subsets of Pn (these are called strongly-rational, cf. [2]) are closed with respect to small deformations. No analogous result is known for rational manifolds. In the third section, following [7], we prove a characterization of rationality in terms of quasi-lines and a useful result on the ascent of rationality. We apply the last to give a simple proof that del Pezzo manifolds of degree 4 are rational. The last section investigates the formal geometry of quasi-lines. We introduce, following [7], a “local number” associated with a quasiline; Theorem 4.2 relates global and local invariants of quasi-lines and has many useful consequences. A computable property of quasi-lines, called regularity, is introduced. A typical application is the following (Corollary 4.7): if two models (X, Y ) and (X , Y ), with Y, Y regular quasi-lines, are formally equivalent (i.e. X/Y X/Y as formal schemes), then they are equivalent. In particular, we get (Proposition 4.8): Let X, X ⊂ Pd+1 , d = 3, 4, 5, be smooth Fano threefolds of degree d and let Y ⊂ X, Y ⊂ X be general conics. If X/Y X/Y , then there exists an isomorphism ϕ : X → X such that ϕ(Y ) = Y . Theorem 4.16 contains a new characterization of the formal completion Pn/line . It may be used to get a new description of the model (Pn , line), Corollary 4.17. We work over the ﬁeld C and we use the standard notation and terminology of Algebraic Geometry, as in [5].

1. Models of rationally connected manifolds Let X denote a complex projective manifold of dimension n (usually n 2). We say that X is rationally connected (cf. [12]) if there is a rational curve passing through any two given points of X. The following theorem summarizes some of the main properties of rationally connected manifolds.

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Theorem A 1. (cf. [12, 3] or [10]) (i) Unirational manifolds and Fano manifolds are rationally connected; (ii) being rationally connected is a birational property and is invariant under smooth deformation; (iii) rationally connected manifolds are simply connected and satisfy i H 0 (X, Ω⊗m X ) = 0 for m > 0 and H (X, OX ) = 0 for i > 0;

(iv) X is rationally connected if and only if there is a smooth rational curve Y ⊂ X such that the normal bundle of Y in X is ample; (v) if n = 2, X is rationally connected if and only if X is rational. Deﬁnition 1.1. A model of a rationally connected manifold X as above, is a pair (X, Y ), where Y ⊂ X is a smooth rational curve with NY /X ample. Isomorphism of models is deﬁned in the obvious way. More importantly, we introduce the following. Deﬁnition 1.2. Two models (X, Y ) and (X , Y ) are equivalent, denoted (X, Y ) ∼ (X , Y ), if there are Zariski open subsets U ⊆ X and U ⊆ X , satisfying Y ⊂ U and Y ⊂ U , and an isomorphism ϕ : U → U , such that ϕ(Y ) = Y . Note that in this case X and X are birationally equivalent. Conversely, assume that X and X are birationally equivalent rationally connected manifolds. Choose a birational map ϕ : X X . By Hironaka’s result (cf. [6]), we may ﬁnd a → X such that ϕ := ϕ ◦ σ is a birational composition of blowing-ups σ : X morphism. Now, we can choose a model (X , Y ) such that ϕ−1 is an isomorphism Y ) and (X , Y ) along Y ; in other words, by letting Y := ϕ−1 (Y ), the models (X, are equivalent. Note, in particular, that proving the rationality of a given rationally Y ) ∼ (Pn , line), connected manifold X, amounts to showing the equivalence (X, for a suitably chosen model of some blow-up X of X. To state our ﬁrst result, we introduce some more notation. If (X, Y ) and (X , Y ) are models and ϕ : X → X is a morphism such that ϕ(Y ) = Y , we write ϕ : (X, Y ) → (X , Y ) and call it a morphism of models. Given a model (X, Y ), we write Y ∼ Y if Y is a general deformation of Y ; in particular, we can consider the new model (X, Y ). Recall that for any model (X, Y ), a theorem due n−1 3 to Grothendieck tells us that NY /X OP1 (ai ), 0 < a1 · · · an−1 . Now we i=1

can state the following: Theorem 1.3. (cf. [8, Theorem 1.12], [7, Theorem 1.7]) Let (X, Y ) be a model n−1 3 such that NY /X OY (ai ) with a1 · · · an−1 and let D be a divisor on X i=1

a such that (D · Y ) := d > 0, with a1 d and dim |D| d. Then there are X blow-up of X, Y ∼ Y , and a diagram of models ϕ Y ) −→ (Z, Y ) → (X, (Pdim |D|−d+1 , l),

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where l is a line, such that: (i) ϕ is surjective with connected ﬁbres; (ii) any smooth ﬁbre of ϕ is rationally connected; (iii) Z := ϕ−1 (l) is smooth; (iv) Y is a section for ϕ|Z . Sketch of the proof. Step 1. We may assume |D| free of ﬁxed components and, replacing Y by a general deformation of it, we may also assume that Y ∩Bs|D| = ∅. Next, we blow-up d−1 points on Y and get a new manifold X . Replace the model (X, Y ) by the model (X , Y ), where Y is the proper transform of Y . We ﬁnd the linear system |D | on X such that: (D · Y ) = 1 and dim |D | 1. Step 2. By step 1, we may assume d = 1 and dim |D| := s 1. After suitable → X, we may also assume Bs|D| = ∅. To prove (i), proceed by blowing-up σ : X induction on s. If s = 1, note that (D · Y ) = 1 shows that the ﬁbres of ϕ|D| are connected. If s 2, (D · Y ) = 1 implies that ϕ is not composed with a pencil, so Bertini’s Theorem applies to ﬁnd a smooth connected member Δ ∈ |D| through We may assume that x, y ∈ Y and we obtain the two general points x, y of X. new model (Δ, Y ). The exact sequence 0 → OX˜ → OX˜ (D) → OΔ (D) → 0 shows that dim |D|Δ | = s − 1 and we may apply the induction hypothesis to deduce (i). (iii) follows by Bertini’s Theorem, since the line determined by ϕ(x), ϕ(y) is general. (iv) is clear. To prove (ii), we use the model (Z, Y ). We see that on a general ﬁbre of ϕ|Z : Z → P1 , two points may be joined by a sequence of rational curves, making use of the same property on Z.

Corollary 1.4. ([7, Corollary 1.8]) Let (X, Y ) be a model and D a divisor on X. n−1 3 OY (ai ), a1 · · · an−1 , d := (D · Y ) > 0, d a1 and Assume that NY /X i=1

dim |D| n + d − 1. Then X is rational. Corollary 1.5. Keep all the other hypotheses of Corollary 1.4 and replace the last inequality by dim |D| n + d − 2. Then X is birational to a conic bundle. Next, we consider deformations of models. Let (X, Y ) be a model. By a deformation of (X, Y ) we mean a commutative diagram Y ⏐ ⏐ q+ T

i

−→ X )p

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where p, q are proper smooth morphisms, i is a closed embedding, T is a connected scheme such that (Xt , Yt ) is a model for each (closed) t ∈ T and (X, Y ) (Xt0 , Yt0 ) for some t0 ∈ T . We say that (X, Y ) and (X , Y ) are deformation equivalent if both appear as ﬁbres of the same deformation. By a polarized model we mean a triple (X, Y, H), where (X, Y ) is a model and H is an ample divisor on X. Let d := (H · Y ). Using Matsusaka’s theorem (cf. [13]) and its reﬁnement from [11] together with Mori theory [15], one can prove the following ﬁniteness result: Theorem 1.6. (cf. [8, Theorem 3.2]) Fix n 2 and d > 0. There are only ﬁnitely many isomorphism classes of polarized models (X, Y, H) such that dim(X) = n and (H · Y ) = d, modulo deformations. Next we discuss a notion of “minimality” for models, cf. [8]. Deﬁnition 1.7. A model (X, Y ) is minimal if any eﬀective divisor D satisfying (D · Y ) = 0 is zero. Note that for any model (X, Y ) the number of prime divisors D with (D · Y ) = 0 is ﬁnite. In dimension two, given a model (X, Y ) there is a unique minimal model (X0 , Y0 ) and a birational morphism ϕ : X → X0 inducing an equivalence of (X, Y ) and (X0 , Y0 ) (here X0 may be singular). Moreover, there is a complete classiﬁcation of the pairs (X0 , Y0 ). See [8, Proposition 1.21] for details. In higher dimensions practically nothing is known about the existence (or uniqueness) of a minimal model in a given equivalence class of models. The following proposition illustrates the use of the minimality property of a given model. Proposition 1.8. Let ϕ : X X be a birational map inducing an equivalence of the models (X , Y ) and (X, Y ). Assume that (X, Y ) is minimal and for some (or any) ample line bundle L ∈ Pic(X) there is L ∈ Pic(X ) which is nef and coincides with ϕ∗ L on the domain of ϕ. Then ϕ is a morphism. be the normalization of the closure of the graph of ϕ, endowed with Proof. Let X → X. We may assume L to be very → X and q : X the natural projections p : X |p∗ L | = E + |q ∗ L| for some ample. We have the equality of linear systems on X, eﬀective divisor E. We ﬁnd easily that (q∗ E · Y ) = 0; the minimality of (X, Y ) implies that codim X (q∗ E) 2. If E > 0, it follows from Hodge Index Theorem, such that q(C) is a point and via suitable slicing, that there exists a curve C ⊂ X (C · E) < 0. We infer that (p∗ L · C) < 0 which contradicts the nefness of L . Thus we proved that E = 0, so p is an isomorphism.

Corollary 1.9. Let (X , Y ) and (X, Y ) be equivalent minimal models. Assume, moreover, that X and X are Fano manifolds. Then (X , Y ) is isomorphic to (X, Y ). Next, we come to a very important question: Given X a rationally connected projective manifold, is there any “convenient” choice of a model (X, Y )? As the

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model (Pn , line) is the basic example, looking at speciﬁc properties of a line in Pn suggests the following deﬁnitions, cf. [2]. Deﬁnition 1.10. (i) Y ⊂ X is called a quasi-line if NY /X

n−1 3 i=1

OP1 (1);

(ii) Y ⊂ X is called an almost-line if Y is a quasi-line and there is a divisor D on X such that (D · Y ) = 1. Example 1.11. Let X be a Fano threefold of index two such that Pic(X) = Z[H], H being the hyperplane section. If Y ⊂ X is a general conic, Y is a quasi-line. This was proved by Oxbury [16]; see also [2, Theorem 3.2] for a more conceptual argument. It is easy to see that X does not contain almost-lines. The following theorem from [7] is essential for the rest of this paper. Theorem 1.12 (existence of almost-lines). Let X be a rationally connected projective manifold. There is X a smooth projective birational model of X, containing an almost-line. Actually, one proves that X is got by a sequence of blowing-ups of X with smooth two-codimensional centers; see [7, Theorem 2.3, Proposition 2.5]. The importance of quasi-lines comes from the following considerations about the Hilbert scheme of such curves. Let Y ⊂ X be a quasi-line. The Hilbert scheme of curves corresponding to the Hilbert polynomial (for a certain polarization) of Y in X is smooth at [Y ]. So [Y ] lies on a unique irreducible component, say H, of this Hilbert scheme. Hence, we can speak about “curves from the family determined by Y ”. Note that this applies to any smooth rational curve with ample normal bundle, and we implicitly used it before, when speaking about Y ∼ Y , a general deformation of Y . We have the universal family of such curves Y and the standard diagram Y ⏐ ⏐ π+

φ

−→ X

H Now ﬁx a point x ∈ Y . Similar considerations apply to the closed subscheme of H corresponding to curves of the family passing through x. We denote by Hx , respectively Yx , the Hilbert scheme of these curves and their universal family. We keep the same notation π and φ for the restriction of the above projections to Yx . Deﬁnition 1.13. Let Y ⊂ X be a quasi-line. (i) The number of quasi-lines from the family determined by Y passing through two general points of X is denoted by e(X, Y ). (ii) The number of quasi-lines from the family passing through one general point of X and tangent to a general tangent vector at that point is denoted by e0 (X, Y ).

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It is easy to see that given a model (X, Y ), e(X, Y ) and e0 (X, Y ) are ﬁnite exactly when Y is a quasi-line. Moreover, e(X, Y ) is nothing but the degree of the projection φ : Yx → X for a general point x ∈ X. One can see also that we always have e0 (X, Y ) e(X, Y ). However, this inequality may be strict. Example 1.14. (cf. [2, Example 2.7]) Take X to be the desingularization of the toric quotient of Pn (n 3) by a cyclic group of order n + 1 and Y the quasi-line that is the image of a line in Pn . We see that e0 (X, Y ) = e0 (Pn , line) = 1 and e(X, Y ) = n + 1. Example 1.15. Let X ⊂ Pd+1 be a smooth Fano threefold of degree d, d = 3, 4, 5 and let Y ⊂ X be a general conic (cf. Example 1.11). If d = 3, we compute e0 (X, Y ) = e(X, Y ) = 6, cf. [7, Proposition 3.2]. If d = 4, we get similarly e0 (X, Y ) = e(X, Y ) = 2. I thank Arnaud Beauville for kindly pointing out this fact to me. If d = 5, we obtain e0 (X, Y ) = e(X, Y ) = 1.

2. The model (Pn , line) The main purpose of this section is to give two diﬀerent characterizations of models equivalent to the basic model (Pn , line). A third characterization will appear in Section 4, using formal geometry. The key to all three characterizations is the following result, which is a reﬁnement of [2, Theorem 4.4]. Proposition 2.1. Let (X, Y ) be a model. The following conditions are equivalent: (i) (X, Y ) ∼ (Pn , line); (ii) there is a divisor D on X such that (D · Y ) = 1 and dim |D| n. Proof. (i) ⇒ (ii) is easy. To prove that (ii) ⇒ (i), we ﬁrst observe that Y does not meet the base locus of |D|. Otherwise, for some point x ∈ Y , we could ﬁnd an element in |D|, which is singular at x. But Y deforms with the point x ﬁxed, so replacing Y by a general deformation of it passing through x, we ﬁnd that (D · Y ) 2. The same argument shows two more things. First, that dim |D| = n; secondly, the rational map ϕ = ϕ|D| , which is deﬁned along Y , is also ´etale along Y . Moreover, ϕ(Y ) is a line in Pn . It follows that the restriction of ϕ to Y is an isomorphism onto a line and Y is a quasi-line. We may further assume, replacing X by a suitable blowing-up, that ϕ : (X, Y ) → (Pn , line) is a morphism of models, ´etale along Y . The following useful general lemma applies to show that ϕ induces

an equivalence (X, Y ) ∼ (Pn , line), as required. Lemma 2.2. Let ϕ : (X , Y ) → (X, Y ) be a morphism of models, with Y, Y quasi-lines, which is ´etale along Y . Then e(X, Y ) = deg ϕ · e(X , Y ). Proof. Fix a point y ∈ Y and let y = ϕ(y ). For x ∈ X a general point, denote by x1 , . . . , xd the points of the ﬁbre ϕ−1 (x), d = deg ϕ. Consider the quasi-lines

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on X equivalent to Y , passing through y and some xi , 1 i d. Their images on X are quasi-lines through y and x, equivalent to Y . The induced map on Chow schemes ϕ∗ : Chy (X ) → Chy (X) is injective when restricted to the open sets parameterizing quasi-lines. This comes from the fact that the considered quasilines on X do not intersect the ramiﬁcation divisor of ϕ. It follows that this

restriction of ϕ∗ is also surjective, whence the desired equality. Corollary 2.3. Let (X, Y ) be a model such that there exists a divisor D on X with the following properties: (i) D is nef and big; (ii) (D · Y ) = 1. Then (X, Y ) ∼ (Pn , line). Proof. Following [2, p. 22]: let d := (Dn ) > 0. Consider the degree-n Hilbert polynomial p(t) := χ(OX (tD)). By duality and Kawamata–Viehweg vanishing theorem we have, for i = 1, . . . , n, p(−i) = χ(OX (−iD)) = (−1)n χ(OX (KX + iD)) = (−1)n h0 (OX (KX + iD)). Since (KX · Y ) −n − 1, we get (KX + iD) · Y < 0, hence h0 (OX (KX + iD)) = 0, i = 1, . . . , n. Therefore p(t) =

d (t + 1) · · · (t + n). n!

On the other hand, p(−(n + 2)) = (−1)n d(n + 1) = (−1)n h0 (OX (KX + (n + 2)D)). Thus h0 (OX (KX + (n + 2)D)) = d(n + 1) n + 1. Put D := KX + (n + 2)D.

It follows that dim |D | n and (D · Y ) = 1, so Proposition 2.1 applies. Corollary 2.4. Let (X, Y ) be a model such that there exists a divisor D on X with the following properties: (i) D is ample; (ii) (D · Y ) = 1. ∼ Then (X, Y ) −→ (Pn , line). It is easy to see that conditions (i) and (ii) imply that (X, Y ) is minimal, so the conclusion follows from Corollary 2.3 and Corollary 1.9. It is amusing to note that Corollary 2.4 (which follows also from adjunction theory) implies the (well-known) fact that Pn is closed with respect to (smooth) small deformations. If n = 2, (X, Y ) ∼ (Pn , line) means that there is a birational morphism ϕ : X → P2 such that Y is the pull-back of a line. For any n 3, there are models (X, Y ) ∼ (Pn , line) such that there is no birational morphism ϕ : X → Pn . In fact, there are models (X, Y ) ∼ (Pn , line) such that there is no divisor D on X which is nef and big with (D · Y ) = 1; cf. [2, Example (4.7)], and [8, Example (2.8)].

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Now, we can state the ﬁrst characterization of the models (X, Y ) equivalent to (Pn , line). It uses only properties of the family of curves determined by Y . The result is a more precise form of Theorem 4.2 of [7]. Theorem 2.5. Let (X, Y ) be a model. The following assertions are equivalent: (a) (X, Y ) ∼ (Pn , line); (b) (i) Y is a quasi-line, (ii) e(X, Y ) = 1, and (iii) there exists a point x ∈ Y such that, for any Y ∼ Y with x ∈ Y , it follows that x is a smooth point of Y . We shall see in the next section that (b)(i) and (b)(ii) together imply that X is rational. However, there are examples showing that condition (b)(iii) is essential for the validity of Theorem 2.5 (see Example 4.19). Proof. Following [7, Theorem 4.2], we sketch a proof that (b)(i), (b)(ii) and (b)(iii) together imply that (X, Y ) ∼ (Pn , line), the other implication being easier. We recall the basic diagram Yx ⏐ ⏐ π+

φ

−→ X

Hx where Hx is the Hilbert scheme of curves from the family determined by Y , passing through x ∈ Y . π has a section E; condition (b)(iii) implies that E is an eﬀective Cartier divisor on Yx (see [7, Lemma 4.3]). Let σ : X → X be the blowing-up of x ∈ X, with E ⊂ X its exceptional divisor. We get a birational morphism φ : Yx → X such that φ = σ ◦ φ . Next we remark that, if F is a general ﬁbre of π and y ∈ F \ (F ∩ E), then φ is a local isomorphism at y. We take H ⊂ E Pn−1 a general hyperplane; we look at D := φ((φ|E ◦ s ◦ π)∗ H), where s : Hx → E is the inverse of the isomorphism π|E : E → Hx . We ﬁnd that x is a smooth point of the eﬀective divisor D. A general deformation of Y through x meets D only at x and the intersection is transverse, so (D · Y ) = 1. Next, for a ﬁxed point x ∈ Y satisfying (b)(iii), let us denote by |Dx | the linear system constructed above. We have dim |Dx | n − 1. Either dim |Dx | n or dim |Dx | = n − 1 and x ∈ Bs|Dx |. In the ﬁrst case, we may apply Proposition 2.1 to conclude. If the last case happens for any x ∈ Y satisfying the open condition (b)(iii), it follows that Pic(X) is uncountable. But X is rationally connected, so H 1 (X, OX ) = 0 and Pic(X) has to be countable. This is a contradiction.

The following deﬁnition ﬁrst appeared in [2]. Deﬁnition 2.6. A projective manifold X is strongly rational, if there is a model (X, Y ) such that (X, Y ) ∼ (Pn , line). Equivalently, X contains an open subset U which is isomorphic to an open subset V ⊆ Pn such that codim Pn (Pn \ V ) 2.

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Theorem 2.7. (cf. [2, Proposition 3.10], [7, Theorem 4.5]) Let X be a projective manifold. The following properties of X are closed with respect to (smooth) small deformations: (i) X contains a quasi-line (respectively an almost-line); (ii) X contains a quasi-line Y and e(X, Y ) = 1; (iii) X is strongly rational. The proof of (iii) follows from the characterization of strongly rational manifolds given in Theorem 2.5. Regarding the last point in Theorem 2.7, we recall that it is an open problem whether or not small deformations of rational manifolds remain rational. The example of cubic fourfolds in P5 seems to suggest that the answer should be negative. The positive result in Theorem 2.7(iii) illustrates how the use of models may simplify problems of birational nature. The second characterization of models (X, Y ) which are equivalent to (Pn , line) is via an extremality property of linear systems. It has been conjectured in [8, Conjecture 2.3]. Theorem 2.8. Let (X, Y ) be a model and D a divisor on X. Let d = (D · Y ). We have: − 1; (i) dim |D| n+d n (ii) (X, Y ) ∼ (Pn , line) if and only if equality holds in (i), for some linear system |D| with d > 0. Proof. (i) (cf. [8, Proposition 2.1]) We may assume d > 0. Consider the d-th jet Consider also the natural map u : bundle of OX (D), denoted Jd (D). H 0 (X, OX (D)) ⊗ OX → Jd (D) which sends a section to its d-th jet. We claim that u is generically injective, which implies the desired inequality. Let s ∈ H 0 (X, OX (D)) be a non-zero section and let D := (s)0 . If x ∈ Y ∩ Supp(D ), we may deform Y to Y by keeping x ﬁxed, so that the local intersection number (Y · D )x be deﬁned. If ux (s) = 0, we get (Y · D )x > d = (Y · D), so s has to be zero. Thus u is injective in the open set swept out by the deformations of Y having ample normal bundle. − 1. Denote by μx (D) the multiplicity at a (ii) Assume that dim |D| = n+d n point x ∈ X of the eﬀective divisor D. For x ∈ Y , let Λx ⊆ |D| be the linear system Λx = {D ∈ |D| | μx (D ) d} = {D ∈ |D| | μx (D ) = d}. We have

dim Λx

n+d n+d−1 n+d−1 − −1= − 1. d d−1 d

→ X be the blowing-up of X at x and let E be its exceptional Let σ : X for the proper transform of D ∈ Λx . Consider the exact sequence divisor. Write D → OPn−1 (d) → 0. 0 → OX (σ ∗ (D) − (d + 1)E) → OX (D)

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O (σ ∗ (D) − (d + 1)E)) = 0, so dim |D| dim |OPn−1 (d)| = We have H 0 (X, X n+d−1 n+d−1 has no base-points − 1; but dim |D| = dim Λx − 1. So |D| d d · Y ) = 0. It follows along E. If Y is the proper transform of Y , we also ﬁnd: (D PN , whose image is of dimension that we have a rational map ϕ := ϕ :X |D|

n − 1 (because its restriction to E is ﬁnite onto ϕ(E)) and contracts the proper transform Y of a general deformation of Y . By generic smoothness, there is only which means exactly one such curve Y passing through the general point of X, that e(X, Y ) = 1. Note, in particular, that Y has to be a quasi-line. So we have checked the ﬁrst two conditions in Theorem 2.5(b). Let us verify the third, too. is base-points free, see [7, p. We may assume, after suitable blowing-up, that |D| 1068]. Let Y ∼ Y , x ∈ Y and consider its proper transform, Y . We may ﬁnd such that no irreducible component of Y is contained in ∈ |D| an element D ). It follows that there is some D ∈ Λx such that (D · Y )x is deﬁned. Supp(D But we have μx (D ) = d which forces Y to be smooth at x. The conclusion follows now by applying Theorem 2.5.

Note that, for d = 1, Theorem 2.8 reduces to Proposition 2.1 and was used in the proof, via Theorem 2.5. The following corollary is a new characterization of Pn (or of Veronese varieties) in terms of models. Corollary 2.9. (cf. [8]) Let (X, Y ) be a minimal model. The following are equivalent: (i) (X, Y ) (Pn , line); (ii) there is a divisor D on X such that d = (D·Y ) > 0 and dim |D| n+d n −1. A diﬀerent proof of Corollary 2.9, based on Mori’s characterization of Pn as the only projective manifold with ample tangent bundle [14], was given in [8, Proposition 2.10].

3. (Uni)rationality via quasi-lines The next proposition, taken from [7], shows how one may use quasi-lines for detecting the (uni)rationality of a given rationally connected projective manifold. Proposition 3.1. (cf. [7, Proposition 3.1]) Let (X, Y ) be a model with Y a quasiline. (i) If e0 (X, Y ) = 1, then X is unirational. (ii) If e(X, Y ) = 1, then X is rational.

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Proof. Let x ∈ Y be a ﬁxed point. Consider the standard diagram Y⏐x ⏐ π+ Hx

φ

−→ X

given by the family of curves determined by Y and passing through x. π has a section E = φ−1 (x). Consider also σ : Blx (X) → X, the blow-up of X at x and let E be its exceptional divisor. The rational map σ −1 ◦ φ is deﬁned at a general point of E and maps E to E. The condition e0 (X, Y ) = 1 means exactly that the restriction of the rational map σ −1 ◦ φ to E gives a birational isomorphism to E. But Yx is birationally a conic bundle for which E provides a rational section. So Yx is birational to E × P1 and (i) follows. To see (ii), observe that the restriction map σ −1 ◦ φ : E E is dominant, being generically ﬁnite. Since e(X, Y ) = 1, σ −1 ◦ φ is birational and by Zariski Main Theorem, it follows that its restriction to E is also birational. Thus Yx is rational and so is X.

Deﬁnition 3.2. Let X be a rationally connected projective manifold. We denote by e(X) the minimum of e(X , Y ) for all models (X , Y ), where σ : X → X is a composition of blowing-ups with smooth centers and Y is a quasi-line in X . Theorem 3.3. (cf. [7, Theorem 3.4]) Let X be a rationally connected projective manifold. (i) e(X) is a birational invariant of X; (ii) X is rational if and only if e(X) = 1. Proof. (i) Let ϕ : X1 X2 be a birational isomorphism between two rationally connected projective manifolds. Let σ : X → X2 be a composition of blowingups and let Y ⊂ X be a quasi-line such that e(X2 ) = e(X , Y ). Let μ = σ −1 ◦ ϕ : X1 X . Take ρ : X → X1 , a composition of blowing-ups such that μ ◦ ρ : X → X is a birational morphism. Let Y ⊂ X be the inverse image by μ ◦ ρ of a general deformation of Y . It follows e(X2 ) = e(X , Y ) = e(X, Y ) e(X1 ). The opposite inequality follows by symmetry. (ii) e(Pn ) = 1, so e(X) = 1 if X is rational, by (i). The converse follows from Proposition 3.1 .

The following result on the ascent of rationality was found in the context of Theorem 1.3. Theorem 3.4. (cf. [7, Theorem 1.3]) Let X be a projective variety and |D| a complete linear system of Cartier divisors on it. Let D1 , . . . , Ds ∈ |D| and put Wi := D1 ∩ · · · ∩ Di for 1 i s. Assume that Wi is smooth, irreducible of dimension n − i, for all i. Assume moreover that there is a divisor E on W := Ws and a linear system Λ ⊂ |E| such that: (i) ϕΛ : W Pn−s is birational, and (ii) |D|W − E| = ∅. Then X is rational.

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Proof. Induction on s. We explain the case s = 1, the general case being completely similar. So, let W ∈ |D| be a smooth, irreducible Cartier divisor such that ϕΛ : W Pn−1 is birational for Λ ⊂ |E|, E ∈ Div(W ) and |D|W − E| = ∅. Replacing X by its desingularization, we may assume that X is smooth. As W is rational, it is rationally connected, so we may ﬁnd some smooth rational curve Y ⊂ W with NY /W ample. We have (Y ·E) > 0 and from (ii) we deduce (Y ·D) > 0. From the exact sequence of normal bundles we get that NY /X is ample, so X is rationally connected. In particular, H 1 (X, OX ) = 0. The exact sequence 0 → OX → OX (D) → OW (D) → 0 shows that dim |D| = dim |D|W | + 1 dim |E| + 1 n. = We may choose a pencil (W, W ) ⊂ |D|, containing W , such that W|W E0 + E1 , with E0 0 and E1 ∈ Λ. By Hironaka’s theory [6], we may use blowingups with smooth centers contained in W ∩ W , such that after taking the proper transforms of the elements of our pencil, to get: (a) Supp(E0 ) has normal crossing; (b) Λ is base-points free (so ϕ : W → Pn−1 is a birational morphism). Further blowing-up of the components of Supp(E0 ) allows to assume E0 = 0 so D|W is linearly equivalent to E. Using the previous exact sequence and the fact = 1, that H 1 (X, OX ) = 0, it follows that Bs|D| = ∅. Finally, (Dn ) = (D|W )n−1 W

so ϕ is a birational morphism to Pn . Example 3.5. (cf. [7, Example 1.4]) Let X ⊂ Pn+d−2 be a non-degenerate projective variety of dimension n 2 and degree d 3, which is not a cone. Then X is rational, unless it is a smooth cubic hypersurface, n 3. If X is singular, by projecting from a singular point we get a variety of minimal degree, birational to X. So X is rational. If X is not linearly normal, X is isomorphic to a variety of minimal degree. Hence we may assume X to be smooth and linearly normal. One sees easily that such a linearly normal, non-degenerate manifold X ⊂ Pn+d−2 has anticanonical divisor linearly equivalent to n − 1 times the hyperplane section, i.e. they are exactly the so-called “classical del Pezzo manifolds”. They were classiﬁed by Fujita in a series of papers; see [9] for a survey of his argument. As Fujita’s proof is quite long and diﬃcult, we show how Theorem 3.4 may be used to prove directly the rationality of X if d 4. Consider the surface W obtained by intersecting X with n − 2 general hyperplanes. Note that W is a nondegenerate, linearly normal surface of degree d in Pd , so it is a del Pezzo surface. As such, W is known to admit a representation ϕ : W → P2 as the blowing-up of 9 − d points. Let L ⊂ W be the pull-back via ϕ of a general line in P2 . It is easy to see that L is a cubic rational curve in the embedding of W into Pd . So, for d 4 L is contained in a hyperplane of Pd . This shows that the conditions of the Theorem 3.4 are fullﬁled for X, |D| being the system of hyperplane sections. We also see that Theorem 3.4 is sharp, as the previous argument fails exactly for the case of cubics.

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4. Formal geometry of quasi-lines If Y is a closed subscheme of the scheme X, the theory of formal functions of X along Y was developed by Zariski and Grothendieck as an algebraic substitute for a complex tubular neighbourhood of Y in X. We denote by X/Y the formal completion of X along Y , which is the ringed space with topological space Y and sheaf of rings inv lim n OX /I n , I being the sheaf of ideals deﬁning Y in X. All results from formal geometry we shall need may be found in either R. Hartshorne’s classic [4], or in the recent comprehensive monograph by L. B˘ adescu [1]. Let us recall that according to Hironaka–Matsumura one can deﬁne the ring of formal rational functions of X along Y , denoted by K(X/Y ). In good cases, it is a ﬁeld containing K(X), the ﬁeld of rational functions of the variety X. We say Y is G2 in X if K(X/Y ) is a ﬁeld and the degree [K(X/Y ) : K(X)] is ﬁnite. An important result due to Hartshorne implies that if Y and X are projective manifolds and NY /X is ample, Y is G2 in X. We say Y is G3 in X if Y is G2 and the inclusion K(X) ⊂ K(X/Y ) is an equality. The following deﬁnition (cf. [7]) is similar to the one in Deﬁnition 3.2; here for a given quasi-line we consider its ´etale neighbourhoods instead of its Zariski neighbourhoods. Deﬁnition 4.1. Let Y ⊂ X be a quasi-line. The number e (X, Y ) is the minimum of e(X , Y ), where X is a projective manifold, Y ⊂ X is a quasi-line and f : (X , Y ) → (X, Y ) is a morphism of models, ´etale along Y . The following theorem, based on results due to Hartshorne and Gieseker, is essential for the sequel. Theorem 4.2. (cf. [7, Theorem 5.4]) If Y ⊂ X is a quasi-line then e(X, Y ) = e (X, Y ) · [K(X/Y ) : K(X)]. Proof. One ﬁrst observes that (see Lemma 2.2), if f : (X , Y ) → (X, Y ) is a morphism of models, with Y ,Y quasi-lines, ´etale along Y , then we have e(X, Y ) = deg f ·e(X , Y ). Next we choose an f as above, such that e(X , Y ) = e (X, Y ) and we claim that Y is G3 in X . Since Y is G2 in X , we may apply a very useful construction due to Hartshorne–Gieseker (see [1]) to get a morphism of models g : (X , Y ) → (X , Y ) as above, such that deg g = [K(X/Y ) : K(X )] and Y is G3 in X . By the previous step and the deﬁnition of e (X, Y ), it follows that is G3 in X . [K(X/Y ) : K(X )] = 1, i.e. Y The diagram associated to f , K(X) ⏐ ⏐ +

−→

K(X )

−→ K(X/Y )

∼

K(X/Y ) ⏐ ⏐ +

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shows that deg f = [K(X/Y ) : K(X)]. Note that the right vertical isomorphism comes from the fact that f , being ´etale along Y induces an isomorphism between

X/Y and X/Y . Corollary 4.3. (cf. [7, Corollary 5.5]) Let (X, Y ) and (X , Y ) be models with ∼ (X, Y ) = e (X , Y ). Y, Y quasi-lines. If X/Y → X/Y as formal schemes, then e Corollary 4.4. (cf. [7, Corollary 5.6]) Let Y ⊂ X be a quasi-line. Then e0 (X, Y ) e (X, Y ) e(X, Y ). Deﬁnition 4.5. We say that a quasi-line Y ⊂ X is regular if e (X, Y ) = e(X, Y ). Corollary 4.6. (cf. [7, Corollary 5.7]) Let Y ⊂ X be a quasi-line. Y is regular if and only if Y is G3 in X. If e0 (X, Y ) = e(X, Y ), then Y is regular. Note, as a very special case, that a quasi-line Y ⊂ X with e(X, Y ) = 1 is G3. This generalizes the fact, ﬁrst noticed by Hironaka in the sixties, that a line in Pn is G3. Corollary 4.7. Let (X, Y ), (X , Y ) be models with Y , Y regular quasi-lines. If ∼ X/Y −→ X/Y as formal schemes, then (X, Y ) ∼ (X , Y ). The following proposition generalizes [7, Corollary 5.12]. Proposition 4.8. Let X, X ⊂ Pd+1 be smooth Fano threefolds of degree d, d = ∼ 3, 4, 5 and let Y ⊂ X, Y ⊂ X be general conics. If X/Y −→ X/Y , then there exists an isomorphism of models ϕ : (X, Y ) → (X , Y ). For a proof, combine Example 1.15 with Corollaries 4.6 and 4.7. Deﬁnition 4.9. Y ⊂ X is a line if Y is a regular almost-line. An ordinary line in Pn is clearly a line in the sense of Deﬁnition 4.9, whence the terminology. Using Theorem 1.12 and the Hartshorne–Gieseker construction one sees that the following holds: Given a rationally connected projective manifold X, we may ﬁnd a generically ﬁnite morphism X → X such that X contains a line. The next question looks interesting: Question 4.10 (existence of lines). Given a rationally connected manifold X can we ﬁnd a (smooth, projective) birational model of X containing a line? Note that almost-lines Y ⊂ X with e(X, Y ) = 1 are lines by Corollary 4.6. The following proposition will allow us to construct other examples of lines.

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Proposition 4.11. In the setting of Theorem 1.3 assume moreover that d = 1 and Y is G3 in Z. Then Y is G3 in X. and apply the Hartshorne–Gieseker construcProof. Assume that Y is not G3 in X Y ). We ﬁnd the morphism of models f : (X, Y ), tion to the model (X, Y ) → (X, ´etale along Y and of degree > 1. Use Bertini’s Theorem to infer that, for a general s line l ⊂ P , Z := (ϕ ◦ f )−1 (l) = f −1 (Z) is smooth and irreducible. We ﬁnd that is disconnected, having Y f −1 (Y ) ⊂ Z as a connected component. A fundamen tal result due to Hironaka–Matsumura (see [1]) asserts that f −1 (Y ) is G3 in Z, because Y is G3 in Z. Now, it is easy to see that a G3 subscheme of a projective manifold is connected (see [1]). This is a contradiction, so Y is G3 in X.

Corollary 4.12. In the setting of Corollary 1.5, assume moreover that d = 1. Then Y is G3 in X. We may apply the preceding proposition, noting that, Z being a surface, any curve with positive self-intersection on it is G3. Example 4.13. Let X be the blowing-up of a smooth cubic threefold in P4 with center an ordinary line. X carries a conic-bundle structure ϕ : X → P2 ; if Y ⊂ X is a section for ϕ of self-intersection one, Y is a line. To see this, notice that Y is an almost-line and apply Corollary 4.12. The following question seems very interesting (especially if it has an aﬃrmative answer), but looks diﬃcult. Question 4.14. Let Y ⊂ X be a line. Is it true that e(X, Y ) is a birational invariant of X (i.e. an invariant of K(X/Y ))? The following example shows that the answer is negative if we only assume Y to be a regular quasi-line. Example 4.15. Let X ⊂ P5 be a smooth complete intersection of two quadrics and let Y ⊂ X be a general conic, cf. Example 1.15. From Corollary 4.6 we deduce that Y is a regular quasi-line. However, X being rational, a positive answer to Question 4.14 would imply e(X, Y ) = 1. The next result, inspired by Proposition 2.1, gives a characterization of the formal completion Pn/line . Theorem 4.16. Let (X, Y ) be a model. (i) There is some L ∈ Pic(X/Y ) such that deg L|Y = 1. (ii) For any such L, we have h0 (X/Y , L) n + 1. (iii) X/Y is isomorphic to Pn/line if and only if there is an L ∈ Pic(X/Y ) such that deg L|Y = 1 and h0 (X/Y , L) = n + 1.

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Proof. Denote by Y (i), i 0, the i-th inﬁnitesimal neighbourhood of Y in X. We have the standard exact sequence 0 → S i+1 (NY∨/X ) → OY (i+1) → OY (i) → 0. (i) The above sequence yields the truncated exponential sequence 0 → S i+1 (NY∨/X ) → OY∗ (i+1) → OY∗ (i) → 1. For any curve Y , we have H 2 (Y, S i+1 (NY∨/X )) = 0, so, by taking cohomology, we get surjections Pic(Y (i + 1)) → Pic(Y (i)) → 0

for i 0.

Therefore we may lift OP1 (1) to Pic(X/Y ) = inv limn Pic(Y (n)). (ii) Let Li := L|Y (i) for i 0. The ﬁrst exact sequence above, tensored by L, gives 0 → S i+1 (NY∨/X ) ⊗ L0 → Li+1 → Li → 0. We deduce easily h0 (L1 ) n + 1,

h0 (Li+1 ) h0 (Li ) for i 1.

As H 0 (X/Y , L) = inv limn H 0 (Y (n), Ln ), the conclusion follows. (iii) One implication is obvious. To see the other, we remark that the hypothesis h0 (X/Y , L) = n + 1 and the preceding exact sequences yield that h0 (Y (i), Li ) = n + 1 for i 1, and each Li is spanned by global sections. Now it is easy to see, using exact sequences as above and induction on i, that Li induces an isomorphism of schemes between Y (i) and the i-th inﬁnitesimal neighbourhood of a line in Pn . These isomorphisms are compatible, so they patch together to give the desired ∼

isomorphism X/Y −→ Pn/line . The next corollary is the third promised characterization of the model (Pn , line). Corollary 4.17. The following conditions are equivalent for a model (X, Y ): (a) (X, Y ) ∼ (Pn , line); (b) (i) there is some L ∈ Pic(X/Y ) such that deg L|Y = 1 and h0 (X/Y , L) n + 1; (ii) Y is regular. Note that condition (b)(i) implies that Y is a quasi-line, so that (b)(ii) makes sense. To see that (b)(i) and (b)(ii) imply (a), combine Theorem 4.16 and Corollary 4.7. Proposition 4.18. (cf. [8, Proposition 4.2], [7, Lemma 5.9]) Let (X, Y ) be a model with Y a quasi-line. Let E be a vector bundle on X such that E|Y = OP1 (a) ⊕ · · · ⊕ OP1 (a) ⊕ OP1 (a + 1) for some a ∈ Z. Let X be P(E) and let π : X → X be the projection. Then:

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(i) there is a quasi-line Y ⊂ X such that π : (X , Y ) → (X, Y ) is a morphism of models; (ii) e(X, Y ) = e(X , Y ). Example 4.19. (i) The model (X, Y ) from Example 1.14 satisﬁes condition (b)(i) from Corollary 4.17, but Y is not regular: we have e (X, Y ) = 1 and e(X, Y ) = n+1. (ii) Consider the model (Pn , line) and apply Proposition 4.18 to E = TPn to ﬁnd the new model (X = P(TPn ), Y ). (X , Y ) satisﬁes (b)(ii) of Corollary 4.17, but does not satisfy (b)(i). Indeed Y is regular since e(X , Y ) = e(Pn , line) = 1 by Proposition 4.18(ii). If (b)(i) would be fulﬁlled, we would have (X , Y ) ∼ (P2n−1 , line). But (X , Y ) is easily seen to be minimal (see [8, Lemma 4.4]), so (X , Y ) would be isomorphic to (P2n−1 , line), which is clearly absurd. Note that (X , Y ) also provides an example verifying the ﬁrst two conditions of Theorem 2.5(b), but not the third.

Acknowledgements We thank the organizers of the Siena Conference for inviting us. Their dedication resulted in a very enjoyable mathematical and social encounter, in one of the most beautiful surroundings one can dream of. We beneﬁtted from two one-month visits, ﬁrst at the University of Genova in September 2003 and the other at Universit´e Lille 1 in April 2004, while working on this paper. We thank Lucian B˘ adescu, Mauro Beltrametti and Jean d’Almeida for their kind invitations and for making our stay very enjoyable. We acknowledge partial ﬁnancial support from Contract CNCSIS no. 33079/2004.

References ˘descu, Projective Geometry and Formal Geometry, Monograﬁe Matematy[1] L. Ba czne, vol. 65, Birkh¨ auser, 2004. ˘descu, M.C. Beltrametti, P. Ionescu, Almost-lines and quasi-lines on [2] L. Ba projective manifolds, in Complex Analysis and Algebraic Geometry, in memory of Michael Schneider, de Gruyter, 2000, pp. 1–27. ´ [3] F. Campana, Connexit´ e rationnelle des vari´et´es de Fano, Ann. Sci. Ecole Norm. Sup. 25 (1992), 539–545. [4] R. Hartshorne, Ample Subvarieties of Algebraic Varieties, Lecture Notes in Math., vol. 156, Springer, 1970. [5] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math., vol. 52, Springer, 1977.

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[6] H. Hironaka, Resolution of singularities of an algebraic variety over a ﬁeld of characteristic zero, Ann. of Math. 79 (1964), 109–326. [7] P. Ionescu, D. Naie, Rationality properties of manifolds containing quasi-lines, Internat. J. Math. 14 (2003), 1053–1080. [8] P. Ionescu, C. Voica, Models of rationally connected manifolds, J. Math. Soc. Japan 55 (2003), 143–164. [9] V.A. Iskovskikh, Y.G. Prokhorov, Algebraic Geometry V. Fano Varieties, Encyclopedia of Math. Sci., vol. 47, Springer, 1999. ´r, Rational Curves on Algebraic Varieties, Ergebnisse der Math. ihrer [10] J. Kolla Grenzgebiete, vol. 32, Springer, 1996. ´r, T. Matsusaka, Riemann–Roch type inequalities, Amer. J. Math. 105 [11] J. Kolla (1983), 229–252. ´r, Y. Miyaoka, S. Mori,Rationally connected varieties, J. Algebraic [12] J. Kolla Geom. 1 (1992), 429–448. [13] T. Matsusaka, Polarised varieties with a given Hilbert polynomial, Amer. J. Math. 94 (1972), 1027–1077. [14] S. Mori, Projective manifolds with ample tangent bundles, Ann. of Math. 110(1979), 593–606. [15] S. Mori, Threefolds whose canonical bundles are not numerically eﬀective, Ann. of Math. 116 (1982), 133–176. [16] W.M. Oxbury, Twistor spaces and Fano threefolds, Quart. J. Math. Oxford 45 (1994), 343–366. Paltin Ionescu Department of Mathematics, University of Bucharest 14 Academiei Str., RO–70109 Bucharest, Romania and Institute of Mathematics of Romanian Academy P.O. Box 1–764, RO–70700 Bucharest, Romania Email: [email protected]

On the discriminant of spanned line bundles Antonio Lanteri and Roberto Mu˜ noz

Abstract. Let X be an irreducible smooth complex projective variety of dimension n, L a non-trivial spanned line bundle on X, V ⊆ H 0 (X, L) a vector subspace of global sections spanning L with dim(V ) = N + 1. The discriminant variety D(X, V ) parameterizes the singular elements of |V |. By Bertini’s theorem one can write dim(D(X, V )) = N − 1 − k with k a non-negative integer. An upper bound on k in terms of N , n, and the dimension of the ﬁbers of the map deﬁned by V is established. Moreover, triplets (X, L, V ) for which k is near the maximum are classiﬁed. 2000 Mathematics Subject Classiﬁcation: 14J40, 14N05, 14C20, 14F05

Introduction Let X be an irreducible smooth complex projective variety of dimension n. Take L a line bundle on X and a linear system |V | ⊆ |H 0 (X, L)| with dim(|V |) = N . We deﬁne the discriminant locus of the triplet (X, L, V ) as D(X, V ) = {H ∈ |V | : H is singular} and we focus on its dimension. Without any hypothesis on |V | there is not an expected dimension for D(X, V ). The ﬁrst natural hypothesis to put is that V is free, i. e., the global generation of L by V . Under this assumption dim(D(X, V )) < N by Bertini’s theorem, hence we can write dim(D(X, V )) = N − 1 − k where k ≥ 0. Generically k = 0 and it is natural to investigate triplets with k > 0. Moreover, with the hypothesis of V spanning L, we obtain a morphism φV : X → PN and the projective geometry of the embedded variety φV (X) ⊂ PN enters into the picture. When |V | gives an embedding D(X, V ) is just the dual variety of X ⊂ PN and this subject has been considered by several authors, for example [4], [6], [7], [12], [13], [14] and others. A nice survey can be [15]. In [10], [11] and [9] D(X, V ) is studied assuming that L is simply ample with V spanning L. In both contexts the known results are far from giving a complete classiﬁcation: mostly they produce upper bounds on k and classify extremal cases for this bound. In this paper we skip the hypothesis of ampleness or very ampleness of L. In this more general setting we give an upper bound on k in terms of N , n, and the dimension of the ﬁbers of the morphism φV deﬁned by V . Concretely, in Lemma

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1.2 we prove that k ≤ min{N, n − F } where F is the maximal dimension of a ﬁbre of φV . Then we deal with the classiﬁcation problem for k near the maximum. If k = N , then D(X, V ) is empty and φV : X → PN is a surjective smooth morphism (1.4). When k = n we have F = 0 and φV is ﬁnite onto Pn . Then L is ample and we can use previous results to assure that (X, L) = (Pn , OPn (1)). Case k = n − 1 gives rise to a completely new situation. In fact, when L is ample and V spans L, k cannot be n − 1 [10], Theorem 2.8 (if φV is an embedding this is a consequence of the so-called Landman’s parity theorem). Dropping the ampleness assumption, k can be n − 1 and we completely classify this case. The proof goes by induction starting from surfaces, see (2.2). For n = 2 we rely on the complete classiﬁcation of triplets (X, L, V ) as above satisfying the condition c2 (J1 (L)) = 0 (see (2.1)). In higher dimensions we need to split the analysis into two cases according to whether n ≥ N (see (3.2)) or n < N (see (4.1)). In particular, it turns out that k = n − 1 if L is big. This generalizes [10], Theorem 2.8.

0. Background material 0.0. Consider X, an irreducible smooth complex projective variety of dimension n. We use standard notation in algebraic geometry. In particular, KX will denote the canonical bundle of X. Let L be a non-trivial line bundle on X and suppose that there exists a vector subspace of global sections V ⊆ H 0 (X, L) spanning L. Let dim(V ) = N + 1 and let φV : X → PN be the morphism deﬁned by V . The discriminant locus of the triplet (X, L, V ) is D(X, V ) := {H ∈ |V | : H is singular}. Consider the correspondence Y := {(x, [s]) ∈ X × |V | : j1 (s)(x) = 0}, where j1 (s) denotes the ﬁrst jet of the section s ∈ V . Note that D(X, V ) is the image of Y via the second projection of X × |V |. Thus D(X, V ) is an algebraic variety in |V | = PN ∨ . In general, D(X, V ) is reducible ([10], [9]). Moreover, if φV (X) = PN then the dual variety (φV (X))∨ is a non-empty irreducible subvariety of D(X, V ). We have dim(D(X, V )) < N , by Bertini’s theorem and then we can write dim(D(X, V )) = N − 1 − k where k ≥ 0 is called the defect of (X, L, V ). If L ample and V spans L then the general element of |V | is smooth and irreducible. Without the ampleness hypothesis we also have to consider linear systems whose elements are all reducible. Let us show an example. Example 0.1. Let X = C × P1 , where C is an irreducible smooth curve of genus g, and let f : X → P1 be the second projection. Take L = f ∗ OP1 (2) and V = f ∗ H 0 (P1 , OP1 (2)). In fact N = 2, since φV is f followed by the Veronese embedding of P1 in P2 given by |OP1 (2)|. Furthermore, D(X, V ) consists of all elements of the form 2Ca , where Ca = f −1 (a). So D(X, V ) = f ∗ D(P1 , H 0 (P1 , OP1 (2))). Note that k = 0 in Example 0.1. In the particular case k > 0 we can prove the following

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339

Lemma 0.2. Let (X, L, V ) be a triplet as in section 0.0. If k > 0 then the general element of |V | is irreducible. Proof. In fact, as a consequence of the second Bertini theorem, the general element of |V | is irreducible unless |V | is composed with a pencil, that is, dim(φV (X)) = 1. In this case, since k > 0 and φV (X)∨ ⊂ D(X, V ), we see that φV (X) ⊂ PN is a curve whose dual variety is not a hypersurface; but this means φV (X) = P1 . Let X → Γ → P1 be the Stein factorization of φV , where Γ is a smooth curve, the ﬁrst morphism has connected ﬁbres, and the second morphism ν : Γ → P1 is ﬁnite. If ν is an isomorphism then all the ﬁbres of φV are connected and so the general one is irreducible, being smooth. If ν is not an isomorphism, then it is necessarily ramiﬁed, hence there is an element in |V | which is non-reduced. In particular, dim(D(X, V )) = 0 and so k = 0, a contradiction.

Let us recall here the deﬁnition of the jumping sets, which is meaningful also without L being ample. In fact, for i ≤ n one can deﬁne Ji = {x ∈ X : rk(d(φV )x ) ≤ n − i}. In the present setting we can prove a result similar to [10], Theorem 2.7. Lemma 0.3. Let (X, L, V ) be a triplet as in section 0.0 such that k > 0, and let J1 (L) be the ﬁrst jet bundle of L. Then there is an exact sequence ⊕(k+1)

0 → OX

→ J1 (L) → Q → 0,

where Q is a vector bundle of rank n−k on X. In particular, c1 (Q) = c1 (J1 (L)) = KX + (n + 1)L; moreover, cn−r+1 (J1 (L)) = 0 for r ≤ k. Proof. Since dim(D(X, V )) = N − 1 − k then the general k-linear space of |V | does not intersect D(X, V ). As in [10], p. 211, choose s0 , . . . , sk ∈ V spanning a Pk in |V | not cutting D(X, V ). Then the jets j1 (s0 ), . . . , j1 (sk ) span a (k+1)-dimensional subspace of the ﬁbre of J1 (L) everywhere, hence they deﬁne an injective vector ⊕(k+1) bundle morphism OX → J1 (L), whose cokernel is a rank n − k vector bundle over X. This concludes the proof.

1. On the dimension of the discriminant Since φV is not necessarily ﬁnite then we can state the following Lemma 1.1. Let v ∈ φV (X) be such that dim(φ−1 V (v)) = f > 0 and consider (v) an f -dimensional irreducible component of φ−1 Y ⊂ φ−1 V V (v). Then: (1.1.1) Y ⊂ Jf ; (1.1.2) if Ji = ∅ then k ≤ n − i;

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(1.1.3) if φV (Jf ) = φV (X) (in particular if dim(φ−1 V (v)) = f for all v ∈ φV (X)) and k = n − f then φV (X) = Pn−f . Proof. Take x ∈ Y and denote by |V − x| (respectively |V − Y |) the linear system deﬁned by the global sections of V vanishing at x (respectively at Y ). It holds that |V − x| = |V − Y |. In particular, for every section s ∈ V vanishing at x, the derivatives of s along tangent vectors to Y at x also vanish automatically. Then x ∈ Jf . This proves (1.1.1). If Ji = ∅ and N − 1 − (n − i) ≥ 0 then for x ∈ Ji it holds that |V − 2x| = PN −1−s ⊂ D(X, V ) with s ≤ n − i and so k ≤ n − i as it is stated in (1.1.2). If N − 1 − (n − i) < 0 then k ≤ N ≤ n − i. To prove (1.1.3), let φV (Jf ) = φV (X) and k = n − f . Take v ∈ φV (X) general. By hypothesis we can choose xv ∈ φ−1 V (v) ∩ Jf = ∅. If N − 1 − (n − f ) ≥ 0 then ∅ = |V − 2xv | = PN −1−(n−f ) ⊆ D(X, V ). Moreover, due to the equality k = n − f , |V −2xv | is an irreducible component of D(X, V ) and |V −2xv | = |V −2xw | for v, w general points in φV (X) (and xw ∈ φ−1 V (w)∩Jf ). Let s ∈ V be a non-trivial section such that j1 (s)(xv ) = 0. As s vanishes at xv , it vanishes at any point of φ−1 V (v). (w). Then s( = 0) But |V − 2xv | = |V − 2xw | so that s vanishes at any point of φ−1 V vanishes at the general point of X, a contradiction. Therefore N − 1 − (n − f ) < 0. In particular N − 1 − (n − f ) = −1, being the dimension of a linear system. This gives N = n − f and D(X, V ) = ∅. From φV (X)∨ ⊆ D(X, V ) it follows that

φV (X) = Pn−f . As a consequence of Lemma 1.1 we obtain the following upper bound on the defect. Lemma 1.2. Let (X, L, V ) be a triplet as in section 0.0 and let F be the maximal dimension of a ﬁbre, i.e., F = max{dim(φ−1 V (v)) : v ∈ φV (X)}. Then 0 ≤ k ≤ n0 , where n0 = min{N, n − F }. Proof. The inequality k ≤ N is a consequence of N − 1 − k ≥ −1. The inequality k ≤ n − F follows from (1.1.1) and (1.1.2).

The following example shows that any possible value of k is attained. Example 1.3. Take N < n, let E be a vector bundle on PN of rank n + 1 − N , and let π : X = P(E) → PN be the projection. For the triplet (X, L = π ∗ (OPN (1)), V = π ∗ H 0 (PN , OPN (1))) we have D(X, V ) = ∅. In particular dim(D(X, V )) = −1 and so k = N . A general result can be stated in the case D(X, V ) = ∅. Proposition 1.4. Let (X, L, V ) be as in section 0.0 then D(X, V ) = ∅ if and only if φV is a smooth surjective morphism onto PN . Proof. If D(X, V ) = ∅ then φV (X)∨ = ∅ and so φV (X) = PN . In particular, N ≤ n. Moreover Ji = ∅ for i ≥ n − N + 1. If not, by (1.1.2), N = k ≤ n − (n −

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341

N + 1) ≤ N − 1 a contradiction. Then we conclude by [8], III, Proposition 10.4. To prove the converse, let φV (X) = PN . Then φV (X)∨ = ∅. So, if D(X, V ) = ∅

then Ji = ∅ for i ≥ n − N + 1. But this contradicts the smoothness of φV . In particular, k ≤ n, by Lemma 1.2. So it is natural to consider the problem of classifying triplets near this bound. Let us start with the case k = n. Theorem 1.5. Let (X, L, V ) be a triplet as in section 0.0 such that k = n. Then (X, L, V ) = (Pn , OPn (1), H 0 (Pn , OPn (1))). Proof. With the same notation as above, Lemma 1.2 implies n ≤ N and F = 0. Then φV is a ﬁnite morphism onto Pn by (1.1.3). In particular, N = n = k, hence D(X, V ) = ∅. Then Proposition 1.4 says that φV is an isomorphism. Alternatively, we can note that L is ample and then the assertion follows from [10], Theorem 2.8.

Next possibility is k = n − 1. For a triplet (X, L, V ) as in section 0.0 such that k = n − 1 we obtain c2 (J1 (L)) = . . . = cn (J1 (L)) = 0 by Lemma 0.3. Let us write an explicit formula for the second Chern class.

1.6. Let (X, L, V ) be a triplet as in section 0.0. Then c2 (J1 (L)) =

n(n − 1) 2 ∗ L + n(KX + L)L + c2 (TX ), 2

∗ is the cotangent bundle of X [3], Lemma 1.6.4. where TX The following lemma splits case k = n − 1 into two diﬀerent subcases.

Lemma 1.7. Let (X, L, V ) be a triplet as in section 0.0 such that k = n − 1 > 0. Then, either (1.7.1) N ≥ n, F = 1 and φV is generically ﬁnite, or (1.7.2) N = n − 1 and φV (X) = Pn−1 . Proof. By Lemma 1.2 we know that F ≤ 1. If F = 0 then L is ample and we can use [10], Theorem 2.8 to get a contradiction. Hence F = 1. If φV is not generically

ﬁnite then φV (X) = Pn−1 by (1.1.3), that is, N = n − 1.

2. Surfaces If n = 1, then k = 0 except for (P1 , OP1 (1)). So the ﬁrst case to consider when k = n − 1 is n = 2. Let us consider the weaker condition of the vanishing of c2 (J1 (L)).

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Lemma 2.1. Let (X, L, V ) be a triplet as in section 0.0 with n = 2 and c2 (J1 (L)) = 0. Then one of the following holds: (2.1.1) |V | is composed with a pencil, that is, dim(φV (X)) = 1 and the general element of |V | is reducible; (2.1.2) (X, L, V ) = (P2 , OP2 (1), H 0 (P2 , OP2 (1))); (2.1.3) there exist a smooth curve B of genus q and a morphism p : X → B, whose general ﬁbre is P1 ; (2.1.4) L2 = 0, g = 1 and φV (X) = P1 . Proof. If (2.1.1) does not hold, then the general element C of |V | is smooth and irreducible. By adjunction formula, taking into account section 1.6 we get L2 + 4(g − 1) + e(X) = 0,

(2.1.5)

where e(X) is the Euler-Poincar´e characteristic of X and g ≥ 0 stands for the genus of C. If g = 0 we can use [1], Proposition V.4.3. If L2 = 0 then φV is the map p of (2.1.3). If L2 > 0 then e(X) = 4 − L2 . On the other hand, since X is rational, 2 2 . Therefore L2 = KX − 8. Since L2 > 0 Noether’s formula gives e(X) = 12 − KX 2 2 2 then KX > 8 and so X = P . If g > 1 or g = 1 and L > 0 then e(X) < 0 and so X is a ruled surface by the theorem of Castelnuovo–de Franchis [2], Theorem X.4. If g = 1 and L2 = 0 then e(X) = 0. Since the general C ∈ |V | is a smooth irreducible curve of genus one the exact sequence 0 → OX → OX (L) → OC (L|C ) → 0 gives h (X, L) ≤ 1 + h0 (C, L|C ) ≤ 2. On the other hand h0 (X, L) ≥ 2 since |V |

spans L. Thus V = H 0 (X, L) and N = 1. 0

We can now put the stronger hypothesis k = n − 1. Theorem 2.2. Let (X, L, V ) be a triplet as in section 0.0 such that n = 2 and k = 1; then either: (2.2.1) X is a geometrically ruled surface π : X = PP1 (E) → P1 , where E is a rank 2 vector bundle over P1 , L = π ∗ OP1 (1) and V = π ∗ H 0 (P1 , OP1 (1)), or (2.2.2) X = C × P1 , where C is a smooth curve of genus ≥ 1, L = π2∗ OP1 (1) and V = π2∗ H 0 (P1 , OP1 (1)), where π2 stands for the second projection. Proof. c2 (J1 (L)) = 0 by Lemma 0.3. Then we can follow Lemma 2.1. By Lemma 0.2 we can exclude (2.1.1) and then g := g(L) ≥ 0. Moreover, k = 1 rules out (2.1.2). Let g = 0. Since we are not in (2.1.2) it follows that L2 = 0 and φV = p, as we have seen in the proof of Lemma 2.1. Note that D(X, V ) = ∅, since k = n − 1. Then, by (1.7.2) the morphism φV exhibits X as a P1 -bundle over P1 . Thus (X, L, V ) is as in (2.2.1). So we can continue the analysis assuming g ≥ 1. Suppose (2.1.3) holds and let C be a general element of |V |. Since g ≥ 1 the map p|C : C → B is a surjection, hence g = g(C) ≥ g(B) = q, by the Riemann–Hurwitz

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343

theorem. We claim that g = q. We have χ(OX ) = 1 − q, since X is ruled. So, taking into account (2.1.5), Noether’s formula gives 2 KX = 12χ(OX ) − e(X) = 12(1 − q) − 4(1 − g) + L2 = 8(1 − q) + 4(g − q) + L2 . 2 ≤ 8(1 − q). On the other hand, since X is ruled (and not P2 ), we know that KX 2 2 Therefore g = q and L = 0. Moreover, KX = 8(1 − q), which says that X is a minimal surface, hence a P1 -bundle over B. Note that L2 = 0 with V spanning L implies that φV has a 1-dimensional image. Hence (1.7.2) holds. In particular, N = n − 1 = 1 and φV (X) = P1 . So we have a morphism φV : X → P1 . Now, if g > 1 then the map p|C : C → B is an isomorphism by Riemann– Hurwitz. Then, for the general (hence for every) ﬁbre F of p we have CF = 1. Therefore the morphism (p, φV ) : X → C × P1 is an isomorphism. This gives (2.2.2). If g = 1, then χ(OX ) = 0. Note that χ(OX ) = deg(φV ∗ ωX/P1 ), cf. [1], p. 162. So we can apply [1], Theorem III.17.3 to conclude that φV is locally trivial (the same conclusion follows also from [1], Theorem III.15.4 recalling that D(X, V ) = ∅). At this point, since X is algebraic, it follows from [1], Theorem V.5.4 that X, as an elliptic ﬁbre bundle over P1 has to be C × P1 . This gives (2.2.2) again. Finally, suppose that (2.1.4) holds. Then g = 1, L2 = 0 and e(X) = 0. Moreover D(X, V ) = ∅. Recalling (1.7.2) we conclude that φV : X → P1 is a relatively minimal elliptic ﬁbration (in the sense of [1], p. 92). Since all ﬁbres are smooth, the canonical bundle formula (cf. [1], Corollary V.12.3) gives KX = φ∗V L, where L = OP1 (χ(OX ) − 2). Furthermore, χ(OX ) = 0, by Noether’s formula. Thus φ∗V L = −2L, i. e., KX = −2L. This implies that X is a minimal surface; otherwise, were E a (−1)-curve on X, then we would get −1 = KX E = −2LE, a contradiction. On the other hand, let H be any ample line bundle on X. Since H meets the ﬁbers of φV , we get KX H = −2LH < 0, which shows that KX is not nef. Therefore X is ruled, being minimal, due to the key lemma in the theory of surfaces. So we are again as in case (2.1.3).

Remark 2.3. Note that the discussion of case (2.1.3) made in Theorem 2.2 in order to prove that φV (X) = P1 does not depend on the assumption k = 1. Let us point out the following easy corollary of Theorem 2.2 and Theorem 1.5, that is a ﬁrst step toward an inductive classiﬁcation for k = n − 1 and n ≤ N . Corollary 2.4. Let (X, L, V ) be a triplet as in section 0.0 with n = 2 and n ≤ N . Then k > 0 implies k = 2 and (X, L, V ) = (P2 , OP2 (1), H 0 (P2 , OP2 (1))).

3. Higher dimensional varieties: case n ≤ N The induction suggested in the previous paragraph will need the following:

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3.1. Let (X, L, V ) be a triplet as in section 0.0 such that the general element X1 ∈ |V | is smooth and irreducible, and consider (X1 , L1 , V1 ), where L1 = L|X1 and V1 is the image by the restriction to X1 of the elements of V . The following commutative diagram shows that the Chern classes of J1 (L1 ) are equal to those of J1 (L)|X1 :

0

→

OX1

→

0

→

OX1

→

0 ↓ ∗ TX (L)|X1 ↓ J1 (L)|X1 ↓ L1 ↓ 0

0 ↓ ∗ → TX (L1 ) 1 ↓ → J1 (L1 ) ↓ = L1 ↓ 0

→ 0 → 0

(3.1.1)

Now we can prove the following Theorem 3.2. Let (X, L, V ) be a triplet as in section 0.0 such that 3 ≤ n ≤ N . If k ≥ n − 1 then (X, L, V ) = (Pn , OPn (1), H 0 (Pn , OPn (1))). In particular k = n. Proof. In view of Theorem 1.5 we can suppose that k = n − 1. We will show that this leads to a contradiction. Let (X, L, V ) be a triplet as in section 0.0 with N ≥ n > 3 and dim(D(X, V )) = N − 1 − (n − 1). Consider s1 , . . . , si general sections in V deﬁning Xi ⊂ X. Denote Li = L|Xi and Vi the image of V by the restriction of global sections to Xi . For simplicity we write ni = n − i = dim(Xi ). We claim that (3.2.1) Xi is smooth and irreducible for 1 ≤ i ≤ n − 2. From (1.7.1) we know that φV is generically ﬁnite. By Bertini’s theorem and Lemma 0.2 we know that X1 is smooth and irreducible, L1 is spanned and V1 spans L1 . Moreover, since s1 is general in V then φV1 is generically ﬁnite. Suppose the claim is false and consider the minimal 1 < i ≤ n − 2 such that Xi does not verify the claim. Since Xi−1 is smooth and irreducible and the choice of s1 , . . . , si ∈ V is general then Li−1 is spanned, Vi−1 spans Li−1 and φVi−1 is generically ﬁnite. Thus Xi is smooth, by the Bertini theorem. So, due to our assumption, Xi is not irreducible; but then |Vi−1 | is composed with a pencil, by the second Bertini theorem. Since φVi−1 is generically ﬁnite, this in turn implies dim(Xi−1 ) = dim(φVi−1 (Xi−1 )) = 1, that is, n − i + 1 = ni−1 = 1 a contradiction. Next, we claim that (3.2.2) (Xn−2 , Ln−2 , Vn−2 ) = (P2 , OP2 (1), H 0 (P2 , OP2 (1))). By (3.2.1) the triplet (Xn−2 , Ln−2 , Vn−2 ) is as in section 0.0 and c2 (J1 (Ln−2 )) = 0 due to the remark after Theorem 1.5 and an inductive application of section 3.1. The claim follows from Lemma 2.1. In fact (3.2.1) rules out (2.1.1). We can also exclude (2.1.4) because in the present case we know that Nn−2 := N − (n − 2) ≥ 2 = nn−2 , while in case (2.1.4) it should be Nn−2 = 1. Finally, (2.1.3) is ruled

On the discriminant of spanned line bundles

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out in the same way, since Remark 2.3 would imply Nn−2 = 1, a contradiction. Therefore (Xn−2 , Ln−2 , Vn−2 ) is as in (3.2.2). (3.2.3) For i ≥ n − 2 there are no non-trivial eﬀective divisors in |KXi + (ni + 1)Li |. If there is an eﬀective non–trivial divisor D ∈ |KXi + (ni + 1)Li | then r := dim(D ∩ Xni −3 ) ≥ dim(D) − codimX (Xni −3 ) = ni − 1 − (ni − 3) = 2. Note that (Ln−2 )|Xi = Lni i −2 . Thus, by adjunction and (3.2.2) we obtain (KXi + (ni + 1)Li )Lini −2 = . . . = (KXn−3 + 4Ln−3 )L2n−3 = (KXn−2 + 3Ln−2 )Ln−2 = 0. Therefore 0 = DLni i −2 = (DLini −3 )Li . This means that φVi has r–dimensional ﬁbers with r ≥ 2, contradicting (1.7.1). Then (3.2.3) holds. We have h1 (X, OX ) = 0 and h0 (X, OX (KX +(n+1)L)) = 0. If KX +L ≥ 0 then KX +(n+1)L > 0. Hence, by (3.2.3), h0 (X, OX (KX +L)) = hn (X, OX (−L)) = 0. Moreover hi (X, OX (−L)) = 0 for i = 1, . . . , n − 1 because L is nef and big. Then, from the exact sequence 0 → OX (−L) → OX → OX1 → 0 we get h (X, OX ) = h (X1 , OX1 ). Iterating this argument we obtain h1 (X, OX ) = h1 (Xn−2 , OXn−2 ) = 0. Now, if h0 (X, OX (KX + (n + 1)L)) = 0 then KX + (n + 1)L is trivial by (3.2.3). As in Lemma 0.3, since k = n − 1, we have the following exact sequence 1

1

⊕n → J1 (L) → Q → 0, 0 → OX

where, now, Q = OX (KX + (n + 1)L) = OX . Since h1 (X, OX ) = 0 this sequence ⊕(n+1) splits, and then J1 (L) = OX . In particular this implies that D(X, V ) = ∅, a contradiction (e. g., this also follows from [5], Theorem 3.1). Therefore h0 (X, OX (KX + (n + 1)L)) = 0. Finally, from the exact sequence 0 → OX (KX + nL) → OX (KX + (n + 1)L) → OX1 (KX1 + (n1 + 1)L1 ) → 0 and the fact that L is nef and big, we obtain h0 (X1 , OX1 (KX1 + (n1 + 1)L1 )) = 0. Iterating this argument we get h0 (Xn−2 , OXn−2 (KXn−2 + 3Ln−2 )) = 0, which is a contradiction in view of (3.2.2).

4. Higher dimensional varieties: case n > N In this section we classify triplets (X, L, V ) as in section 0.0 such that n ≥ 3, k = n − 1 and n > N . Recall that N = n − 1 and φV (X) = Pn−1 , by Lemma 1.7. Then D(X, V ) = ∅. Theorem 4.1. Let (X, L, V ) be a triplet as in section 0.0 such that n ≥ 3 and k = n − 1; then either: (4.1.1) X is a P1 -bundle over Pn−1 , i. e., π : X = P(E) → Pn−1 , where E is a rank 2 vector bundle over Pn−1 , L = π ∗ OPn−1 (1) and V = π ∗ H 0 (Pn−1 , OPn−1 (1)), or

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(4.1.2) X = C×Pn−1 , where C is a smooth curve of genus ≥ 1, L = π2∗ OPn−1 (1) and V = π2∗ H 0 (Pn−1 , OPn−1 (1)), where π2 stands for the second projection. Proof. We prove the theorem for n = 3. The general case follows by induction working on Xh = φ−1 V (h), h ∈ |OPn−1 (1)|. As pointed out before, we know that φV (X) = P2 and D(X, V ) = ∅. Let us deﬁne X = φ−1 V () for ∈ |OP2 (1)|. Note that any element of |V | = φ∗V |OPn−1 (1)| is an X . First, note that 2 (4.1.3) Xp = φ−1 V (p) is a smooth curve for every p ∈ P by Proposition 1.4. Moreover (4.1.4) Xp and Xq are isomorphic, for every p, q ∈ P2 . To see this, consider = p, q . If (2.2.1) holds, then Xp and Xq are P1 . Otherwise, X is canonically isomorphic to C × , i. e., via φV |X , and then Xp ∼ = Xq ∼ = C; this shows (4.1.4). Let us call ι : X → C × this isomorphism. Fix p0 ∈ P2 and set C = Xp0 . If g(C) = 0, then all ﬁbres of φV : X → P2 are P1 . We can easily see (by a local version of the argument we are usig below) that X is a P1 -bundle over P2 and (4.1.1) holds. Consider now the case g(C) ≥ 1. We can deﬁne a morphism f : X → C in the following way. Take x ∈ X. If x ∈ C, deﬁne f (x) = x. If x ∈ C, then consider the line x = p0 , φV (x) and deﬁne f (x) = π1 (ιx (x)), where π1 is the ﬁrst projection of C × . The morphism (f, φV ) : X → C × P2 is an isomorphism. Take (c, p) ∈ C × P2 . If p = p0 , then c ∈ Xp0 ⊂ X is the only preimage of (c, p0 ). If p = p0 , consider the line = p0 , p . Then X ∼ = C × via ι , and the point (c, p) ∈ X ⊂ X is the only preimage of (c, p).

ι−1 Note that Theorem 4.1 (just like Theorem 2.2) is an eﬀective result, i. e., D(X, V ) = ∅ in both cases, hence k = n − 1. As shown in Theorem 2.2, if n = 2 all the elements in |V | are necessarily isomorphic. For n ≥ 3 this is still true in case (4.1.2), but not in case (4.1.1), as shown by the following example, that is a particular case of Example 1.3. Example 4.2. Let E be a non-uniform vector bundle on P2 , and take π : X = P(E) → P2 . For the triplet (X, L = π ∗ (OP2 (1)), V = π ∗ H 0 (P2 , OP2 (1))) we have D(X, V ) = ∅. Consider a general line 0 ⊂ P2 and a jumping line 1 ⊂ P2 ; then P(E|0 ) and P(E|1 ) are two non-isomorphic elements of |V |. Acknowledgements. During the preparation of this paper the ﬁrst author has been partially supported by the MIUR of the Italian Government in the framework of the PRIN “Geometry on Algebraic Varieties” (Coﬁn 2002). The research of the second author has been partially supported by the MCYT project BFM200303971. The second author would also like to thank the GNSAGA-INdAM project “Classiﬁcation of Special Varieties” for partial support, and the University of Milan for making this collaboration possible. Both authors are grateful to the organizers for the nice atmosphere enjoyed during the Conference in Siena. They are also grateful to the referee for suggesting Proposition 1.4.

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References [1] W. Barth, C. Peters and A. Van de Ven, Compact complex surfaces, A Series of Modern Surveys in Math. 4, Springer–Verlag (1984). [2] A. Beauville, Surfaces Alg´ebriques Complexes, Ast´erisque 54 (1978). [3] M. C. Beltrametti and A. J. Sommese, The Adjunction Theory of Complex Projective Varieties, De Gruyter Expositions in Math. 16 (1995). [4] M. C. Beltrametti, M. L. Fania and A. J. Sommese, On the discriminant variety of a projective manifold, Forum Math. 4 (1992), 529–547. [5] S. Di Rocco and A. J. Sommese, Line bundles for which a projectivized jet bundle is a product, Proc. Amer. Math. Soc. 129 (2001), 1659–1663. [6] L. Ein, Varieties with small dual varieties I, Invent. Math. 86 (1986), 63–74. [7] L. Ein, Varieties with small dual varieties II, Duke Math. J. 52 (1985), 895–907. [8] R. Hartshorne, Algebraic Geometry, Graduate Texts in Math. 52, Springer-Verlag (1977). [9] A. Lanteri and R. Mu˜ noz, Varieties with small discriminant variety, Trans. Amer. Math. Soc. (to appear). [10] A. Lanteri, M. Palleschi and A. J. Sommese, On the discriminant locus of an ample and spanned line bundle, J. Reine Angew. Math. 477 (1996), 199–219. [11] A. Lanteri, M. Palleschi and A. J. Sommese, Discriminant loci of varieties with smooth normalization, Comm. Algebra 28 (2000), 4179–4200. [12] A. Lanteri and D. Struppa Projective 7-folds with positive defect, Compositio Math. 61 (1987), 329–337. [13] R. Mu˜ noz, Varieties with low dimensional dual variety, Manuscripta Math. 94 (1997), 427–435. [14] R. Mu˜ noz, Varieties with degenerate dual variety, Forum Math. 13 (2001), 757– 779. [15] E. A. Tevelev, Projectively dual varieties, J. of Math. Sci. (N. Y.) 117 (2003) 4585–4732. Antonio Lanteri Dipartimento di Matematica “F. Enriques” Universit` a Via C. Saldini 50, I-20133 Milano, Italy Email: [email protected] Roberto Mu˜ noz Departamento de Matem´ aticas y F´ısica Aplicadas y Ciencias de la Naturaleza Universidad Rey Juan Carlos an, 28933-M´ ostoles, Madrid, Spain C. Tulip´ Email: [email protected]

Multisecant lines to projective varieties Atsushi Noma∗

Abstract. For a nondegenerate projective variety X ⊆ PN of dimension n and degree d, the existence of a secant line L with the length l(X ∩ L) of the intersection X ∩ L “large” is a strong condition on X, which reﬂects on the Castelnuovo-Mumford regularity of X. It is well-known that the bound l(X ∩ L) ≤ d − N + n + 1 holds true for smooth varieties X. The purpose here is to extend this bound in two ways and to classify scrolls and smooth varieties X having a secant line L with l(X ∩ L) reaching the bound or close to the bound, which reﬁnes and extends the results of Bertin and Kwak. 2000 Mathematics Subject Classiﬁcation: 14N15, 14N30, 14N05

Introduction Let X n ⊆ Pn+e be a non-degenerate (i.e., not contained in any hyperplane) projective variety of dimension n, degree d, and codimension e over an algebraically closed ﬁeld k of char k = 0. A k-plane L (1 ≤ k ≤ e) is said to be m-secant to X if X ∩ L is ﬁnite of length at least m, i.e., +∞ > l(X ∩ L) := length(OX∩L ) ≥ m. If we simply say a k-plane L is secant to X, it means that L is m-secant to X for some m > 0. The purpose here is to give an upper bound on l(X ∩ L) for a secant k-plane L and to classify projective varieties with secant lines in a boundary case. In this direction, it is well-know that l(X ∩ L) ≤ d − e + 1 holds if X is smooth and L is a line (see [1]). Kwak [8] showed that l(X ∩L) ≤ d−e+k holds if X is smooth and L is a curvilinear secant k-plane, i.e., X ∩ L locally lies on a smooth curve. Moreover smooth projective varieties X with secant lines L such that l(X ∩ L) = d − e + 1 or d − e were studied by Bertin [1] and Kwak [8]. Here we extend these results. Our ﬁrst result (Theorem 1.1) is the inequality l(X ∩ L) ≤ d − e + k for a locally Cohen-Macaulay variety X and a secant k-plane L deﬁned over an algebraically closed ﬁeld of arbitrary characteristic. If X is not locally Cohen-Macaulay, this inequality does not necessarily hold (Example 1.2). If L is a line, it is believed that this inequality holds for any, not necessarily locally Cohen-Macaulay, projective variety. But this problem looks still open (see also Remark 2.3). If X is smooth ∗ Partially supported by Japan-Korea Basic Scientiﬁc Cooperation Program, Japan Society for the Promotion of Science.

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Atsushi Noma

and L is a curvilinear secant k-plane, we have a slightly better bound than that in Theorem 1.1: in Theorem 2.4, we show that if l(X ∩ L) = d − e + k − δ and e − k ≥ δ + 1 for some δ, then the sectional genus of X does not exceed δ, which extends the result of [1], Theorem 1 and [8], (3.5). This, together with the theory of sectional genus ([3], [7]), immediately implies that if δ is also small in the above condition, X is a scroll over a curve with genus g(X) ≤ δ, or a variety with h1 (OX ) = 0 whose structure is already known in detail. Thus the problem of classifying these projective varieties with secant lines is reduced to studying their embeddings into projective spaces. In §3, we classify scrolls X with secant lines L such that either l(X ∩ L) = d − e + 1 (Theorem 3.4), or l(X ∩ L) = d − e + 1 − g and e ≥ g − 2 for the sectional genus g of X (Theorem 3.6). On the other hand, each embedding of a variety X with h1 (OX ) = 0 having a secant line is given by some linear projection (Lemma 4.4). These projections can be described in detail, but this result will appear elsewhere. Consequently, we have the classiﬁcation of smooth projective varieties X with secant lines L such that l(X ∩ L) = d − e + 1 or d − e (Theorem 4.1 and Theorem 4.3). This work was partially done while I was visiting Korea Advanced Institute of Science and Technology. I thank the Institute, especially Professor Sijong Kwak for their hospitality.

1. Upper bound on the length of intersection with secants In this section, we work over an algebraically closed ﬁeld k of arbitrary characteristic. Let X ⊆ Pn+e be a nondegenerate projective variety of dimension n, codimension e and degree d deﬁned over k. Let L ⊆ Pn+e be a secant k-plane (1 ≤ k ≤ e) to X. Let πL : Pe+n \ L → Pe+n−k−1

(1.0.0)

¯ we denote the closure of πL (X \ L). We think be the projection from L. By X n+e as the space of 1-quotients of an (n + e + 1)-dimensional k-vector space of P V . Let W ⊆ V be the subspace of linear forms deﬁning L in Pn+e = P(V ). The image of W ⊗ OX → OX (1) is IX∩L/X ⊗ OX (1) for the ideal sheaf IX∩ of X ∩ L in X. We have the following commutative diagram 0 → 0 →

W ⊗ OX ↓α IX∩L/X ⊗ OX (1)

→ →

V ⊗ OX ↓ OX (1)

→ →

(V /W ) ⊗ OX ↓ OX∩L (1)

→

0

→

0.

(1.0.1)

Theorem 1.1. (char k ≥ 0) Assume that X is locally Cohen-Macaulay. Then l(X ∩ L) := length(OX∩L ) ≤ d − e + k.

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351

Proof. We prove by descending induction on k. When k = e, we have l(X ∩ L) = d by B´ezout’s theorem for locally Cohen-Macaulay schemes (see for example [2], (1.4.4), (1.4.2)). Thus we assume k < e. The ﬁbre (πL |X\L )−1 (πL (z)) over πL (z) for a point z ∈ Pe+n is (L, z ∩ X) \ L. Here L, z denotes the linear span of L and z in Pn+e . Then dimL, z ∩ X ≤ 1, since L ∩ X is ﬁnite and L ⊆ L, z is of ¯ = n − 1 or n. codimension 1. Consequently dim X ¯ When dim X = n, for a general point z ∈ X \ L, X ∩ L, z is ﬁnite of length at least l(X ∩ L) + 1. By induction, l(X ∩ L, z ) ≤ d − e + k + 1. Hence l(X ∩ L) ≤ d − e + k. ¯ = n − 1, take a point x ∈ X ∩ L. From α in (1.0.1), we have the When dim X surjection W → IX∩L/X ⊗ OX (1) ⊗ k(x) ∼ = IX∩L/X /mx IX∩L/X for the maximal ideal mx ⊆ OX,x . Associated with this surjection we have the linear subspace M := P(IX∩L/X /mx IX∩L/X ) ⊆ P(W ) = Pe−k+n−1 of dimension at least n − 1 ¯ since since dim X = n and dim X ∩ L = 0. Thus there exists a point z¯ ∈ M \ X, ¯ X = P(W ) is nondegenerate. Let W be the kernel of the quotient W → IX∩L/X /mx IX∩L/X → k corresponding to z¯. In Pn+e , W (⊆ V ) deﬁnes the linear subspace L := L, z ¯ we have of dimension k + 1 for some z ∈ Pn+e with z¯ = πL (z). Since z¯ ∈ X, −1 z ) = ∅ and hence X ∩L is ﬁnite. By induction, l(X ∩L ) ≤ d−e+k+1. (πL |X\L ) (¯ On the other hand, since IX∩L /X ⊗ OX (1) is the image of W ⊗ OX → IX∩L/X ⊗ OX (1), we have IX∩L/X /IX∩L /X ⊗ k(x) = 0. Thus l(X ∩ L) + 1 ≤ l(X ∩ L ). Consequently, l(X ∩ L) ≤ d − e + k.

Example 1.2. If X is not locally Cohen-Macaulay, the inequality in Theorem 1.1 does not necessarily hold: Let C be the smooth rational curve given by P1 → P4 , [s, t] → [s4 , s3 t, st3 , t4 , 0]. Let X be the cone over C with vertex v := [0, 0, 0, 0, 1]. For the coordinates x1 , x2 , x3 , x4 , x5 of P4 , the homogeneous ideal of X in P4 is (x1 x4 − x2 x3 , x21 x3 − x32 , x2 x24 − x33 , x22 x4 − x23 x1 ). Then deg X = 4, and for L := V+ (x1 , x4 ), X ∩ L = {v} as set. On the other hand, l(X ∩ L) = 5 > d − e + 2.

2. Reduction to dimension one: X ∩ L curvilinear case Keep the notation and assumptions as in §1. But from now on, we assume that k is of characteristic zero. This assumption is essential to prove Lemma 2.1 and Theorem 2.4. A secant k-plane L is said to be curvilinear if so is X ∩ L, i.e., each component of X ∩ L lies on a smooth curve. Note that any secant line is curvilinear. If X is smooth and X ∩ L is curvilinear, then we can reduces X to a curve as in [1] and [8], (2.1):

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Lemma 2.1. ([8], (2.1)) Assume that X is smooth and that L is a curvilinear k-plane for 1 ≤ k < e. If X is of dimension n ≥ 2 and H is a general hyperplane containing L, then X ∩ H is a nondegenerate smooth (irreducible) variety. Con¯ of the image πL (X \ L) of the linear projection πL from L sequently the closure X n+e−k−1 has dimension n. to P Proof. First X ∩ H is smooth at every x ∈ X ∩ L, since general H does not contain the projective tangent space Tx (X) ⊆ Pn+e by the curvilinearity of X ∩ L. Moreover (X \ L) ∩ H is smooth by Bertini’s theorem in characteristic zero [6], (III,10.9.2). Therefore X ∩H is smooth, and hence irreducible, since it is connected by Lefshetz’s Theorem. For the nondegeneracy of X ∩ H, see for example [2], (3.5.8) or [5], (18.10). To see the second part, we have only to prove that πL |X\L is generically ﬁnite. We may assume dim X = 1 by the ﬁrst part. Since X is ¯ is not a point, as required. nondegenerate and irreducible, X

Example 2.2. If X ∩ L is not curvilinear or if X is not smooth, the conclusion of Lemma 2.1 is not true. (1) Let L be a projective tangent space Tx (X) to the Veronese surface X in P5 . The projection πL : P5 \ L → P2 induces a rational map of X(∼ = P2 ) to P2 , whose image is a conic and each ﬁber is a conic, the image of a line in P2 through x. (2) Consider a rational normal curve C of degree d in a d-dimensional linear subspace M of Pd+1 (d ≥ 3). Let X ⊆ Pd+1 be the cone over C with vertex v ∈ Pd+1 \ M . Let L be a general line through v. Then l(X ∩ L) = 2. A hyperplane section X ∩ H for general H ⊇ L is the union of d lines through v, and ¯∼ X = C. Remark 2.3. (1)In the last part of [1], Theorem 1, it is stated that an extremal secant line L to a nondegenerate projective variety X of degree d and codimension e (i.e., a line with l(X ∩ L) = d − e + 1) meets only at smooth points of X, but this is not true as we have seen in (2) of Example 2.2. This fault comes from the wrong application of Bertini’s Theorem to the proof of [1], Lemma 1.1. A generic member of a linear system is possibly reducible if the system is composite with a pencil, as in (2) of Example 2.2. By the same reason, the proof of the ﬁrst part of Theorem 1 in [1] works only for smooth projective varieties, and the proof for singular varieties looks still incomplete. (2) The examples (1) and (2) in Example 2.2 illustrate why we should consider ¯ = dim X − 1 in the proof of (1.1) (cf. [8], (1.6)). the case dim X Theorem 2.4. (cf. [1], Theorem 1; [8], (3.5)) Let X ⊆ Pn+e be a smooth nondegenerate projective variety of dimension n and degree d, and let L be a curvilinear secant k-plane for 1 ≤ k < e. If l(X ∩ L) = d − e + k − δ and e − k ≥ δ + 1 for some δ, then g(X) ≤ δ. Here g(X) denotes the sectional genus of X, the genus of a smooth curve obtained by successive hyperplane sections of X (see [3], (2.1); [7] §1).

353

Multisecant lines to projective varieties

Proof. By Lemma 2.1, we may assume that X is a nondegenerate smooth curve. ¯ = πL (X \ L)− of the For the projection πL : Pe+1 \ L → Pe−k , the closure X ¯ image is also a nondegenerate curve, and hence deg X ≥ e − k. Then πL induces ¯ and we have a morphism π : X → X ¯ ≥ deg π · (e − k). d − l(X ∩ L) = e − k + δ = deg π · deg X ¯ = e − k + δ ≥ 2δ + 1. Thus Since e − k ≥ δ + 1, we have deg π = 1 and d¯ := deg X ¯ ¯ If e − k = 1, then δ = 0 g(X) does not exceed the arithmetic genus pa (X) of X. 1 ¯ ¯ ¯ ≤ δ. If e − k ≥ 3, by and P = X. If e − k = 2, since d = 2 + δ ≤ 3, then pa (X) Castelnuovo’s bound [4], (3.7), then ¯ e − k) := m(d¯ − m + 1 ((e − k) − 1) − 1) ¯ ≤ γ(d, (2.4.1) pa (X) 2 4 5 ¯ d−1 . Then m = 1 or 2. Moreover m = 2 if and only if e − k − 1 = where m = (e−k)−1 ¯ ¯ ≤ δ. δ. In both cases, γ(d, e − k) = δ and hence g(X) ≤ pa (X)

¯ and π : X → X ¯ Remark 2.5. If δ = g(X) in Theorem 2.4, then g(X) = pa (X) is isomorphic for n = 1.

3. Multisecants to scrolls 3.0. Let X = PC (E) ⊆ Pn+e = P(V ) (n ≥ 2, e ≥ 2) be an n-dimensional nondegenerate scroll over a smooth projective curve C of genus g with projection τ : PC (E) → C. Here we say that PC (E) ⊆ Pn+e is a scroll if every ﬁbre over C is linear in Pn+e . Assume L = P(V /W ) ⊆ P(V ) is a secant k-plane (1 ≤ k < e) with l(X ∩ L) = d − e + k − δ for some δ. Hence dim W = n + e − k. We claim that if we take the push-forward τ∗ of (1.0.1) in this case and if we set F = Im(W ⊗ OC → E),

Z = τ∗ OX∩L (1),

K = Ker((V /W ) ⊗ OC → Z)

(3.0.1)

then we have the following commutative diagram with the exact rows and surjective columns: 0 → 0 →

W ⊗ OC ↓ F

→

V ⊗ OC ↓ → E

→

(V /W ) ⊗ OC ↓ → Z

→

0 (3.0.2)

→ 0.

Note that Z ∼ = OX∩L (1) since X ∩ L is ﬁnite. Consequently, we have 0 → 0 →

F , F

→ →

F ⊕ (V /W ) ⊗ OC ↓ E

→

(V /W ) ⊗ OC ↓ → Z

→

0 (3.0.3)

→ 0.

¯ = πL (X \ L)− is the image of PC (F) → P(W ), Geometrically, this means that X and X is a subbundle of PC (F ⊕ (V /W ) ⊗ OC ).

354

Atsushi Noma

To see (3.0.2), we have only to show that for any P ∈ X and p = τ (P ) ∈ C, W ⊗ OC,p → Ep → Z ⊗ OC,p → 0

(3.0.4)

is exact. Let e1 , . . . , en ∈ Ep be a basis over OC,p such that P = (0 : · · · : 0 : 1) in the ﬁbre P(E ⊗ k(p)) and let t ∈ OC,p be a local parameter. With this notation, we look at the map W ⊗ OX → OX (1), which consists of W ⊗ OC → E and τ ∗ E → OX (1). By Gaussian elimination, we have generators w˜i = tai eμi + (ai ≥ 0, μi ≥ i, f ij (t) ∈ OC,p ) of the image of W ⊗ OC,p → Ep so j>μi fij (t)ej that τ ∗ w˜i are mapped to tai xμi + j>μi fij (t)xj in OX (1) for the inhomogeneous coordinates xi := ei /en . Since X ∩ L is ﬁnite, we know that {w˜i } consists of n elements, μi = i, and a1 = · · · = an−1 = 0. Hence Z ⊗ OC,p ∼ = OC /(tan ) and (3.0.4) is exact. As a consequence, we also know that L is curvilinear. We consider the ﬁbre Fp of PC (F) → C over p ∈ C as a subspace of P(W ). Let U ⊆ W be the subspace of linear forms vanishing on ∩p∈C Fp ⊆ P(W ). Then, by Lemma 3.1 below, the image Fn−l of U ⊗ OC → F is ample of rank n − l and F is decomposed into ⊕l , F∼ = Fn−l ⊕ OC = Fn−l ⊕ (W/U ) ⊗ OC ∼

where

l = dim W/U.

(3.0.5)

Note that dim U = dim W −l = n+e−k−l and c1 (Fn−l ) = c1 (F) = d−l(X ∩L) = e−k+δ. Consequently letting G be the cokernel of Fn−l → E, we have the following diagram: 0 → 0 →

Fn−l , Fn−l

→ →

Fn−l ⊕ (V /U ) ⊗ OC ↓ E

→

(V /U ) ⊗ OC ↓ → G

→

0 (3.0.6)

→ 0.

Then the surjection (V /U ) ⊗ OC → G˜ := G ∨∨ ∼ = G/ tor(G) induces an embedding ˜ → Pl+k = P(V /U ). Since two ﬁbres of PC (G) ˜ → C never meet in Pl+k , we PC (G) have l ≤ k + 1.

(3.0.7)

Lemma 3.1. Let F be a rank-n bundle over a smooth curve C with a surjective map from a trivial bundle W ⊗ OC . Let Fp ⊆ P(W ) be the ﬁbre of PC (F) over p ∈ C. Let U ⊆ W be the linear forms vanishing on ∩p∈C Fp ⊆ P(W ). Then we have a decomposition F ∼ = F ⊕ (W/U ) ⊗ OC with an ample bundle F and a surjection U ⊗ OC → F induced from W ⊗ OC → F. Proof. First note that F is ample if and only if PC (F) → P(W ) is ﬁnite, which is equivalent to ∩Fp = ∅. In general, the natural map from W/U ⊗ OC to the cokernel of U ⊗ OC → F is bijective. Then the induced map F → W/U ⊗ OC is split and hence we have F ∼ = F ⊕(W/U )⊗OC and U ⊗OC → F . By construction,

∩p∈C P(F ⊗ k(p)) = ∅ in P(U ) and hence F is ample.

355

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3.2. Under the same assumption as in (3.0), assume k = 1. Then l = 0, 1 or 2 by (3.0.7). We claim that if l ≥ 1 then G in (3.0.6) is torsion-free unless l = 1, C ∼ = OP1 (1) ⊕ k(p) for some p ∈ P1 . Indeed, if G is not torsion= P1 and G ∼ free at p ∈ C, then dim P(G ⊗ k(p)) ≥ l. Hence Supp(tor(G)) is a one point. If ˜ = 1 or if dim P(G ⊗ k(p)) = l + 1, then P(G ⊗ k(p)) ∩ PC (G) ˜ in P(V /U ) is deg PC (G) ˜ strictly bigger than P(G ⊗k(p)) as a scheme, which contradicts the assumption that ˜ = 1 and dim G ⊗ k(p) = PC (E) → P(V ) is an embedding. Consequently deg PC (G) 1 ˜ ∼ ∼ 2. Hence l = 1, C = P , G = OP1 (1), and G = OP1 (1) ⊕ OP1 ,p /(ta ) for a local parameter t at p ∈ P1 . To see a = 1, take a line bundle OP1 (b) (b > 0) and lift the map K := Ker(V /U ⊗ OP1 → G)(∼ = K) → (V /U ) ⊗ OP1 to an injection K → OP1 (b) ⊕ (V /U ) ⊗ OP1 whose cokernel is a bundle E . By Lemma 3.3, we have a scroll X := PP1 (E ) ⊆ P(V ) for V = H 0 (OP1 (b) ⊕ (V /U ) ⊗ OP1 ). By construction, PP1 (G ⊗ OP1 ,p /(ta )) ⊆ PP1 (E ) ∩ P(V /U ) ⊆ P(V ). Thus if a ≥ 2 then the tangent space Tx (X ) ⊆ P(V ) at every point x ∈ X over p ∈ P1 is P(V /U ) = P2 , which contradicts the ﬁniteness of Gauss maps (see for example [9], (2.8)). Thus a = 1. Consequently we have: (3.2.1). If k = 1 and l = 1, then C is a plane curve and G ∼ = OC (1) := OP2 (1)|C by (3.0.6), or C ∼ = OP1 (1) ⊕ k(p) for some p ∈ P1 . = P1 and G ∼ (3.2.2). If k = 1 and l = 2, then C ∼ = OP1 (1)⊕2 . Indeed, by = P1 and G ∼ assumption, S := PC (G) → P(V /U ) is a scroll of dimension 2 with canonical sheaf ωS ∼ = OS (−2) ⊗ τ ∗ (det G ⊗ ωC ). On the other hand, ωS ∼ = OS (−4 + deg S) by adjunction. Consequently C ∼ = OP1 (1)⊕2 . = P1 and G ∼ Lemma 3.3. Let F be a very ample vector bundle on a projective variety Y . Let X be a subvariety of P := PY (F ⊕OY⊕m ). Set V = H 0 (Y, F ⊕OY⊕m ). The composite ϕ : X ⊆ P → P(V ) is an embedding if and only if so is ϕ¯ : X ∩P(OY⊕m ) → Pm−1 := P(H 0 (Y, OY⊕m )). Proof. Since P → P(V ) is an embedding oﬀ Pm−1 ×P(V ) P ∼ = P(OY⊕m ), we have only to show that ϕ is an immersion if and only if so is ϕ. ¯ For x ∈ X ∩ P(OY⊕m ), let mx be the maximal ideal of OX,x . Let u1 , . . . , ul be local equations of Pm−1 in P(V ) at ϕ(x). In the diagram 0

→

0

→

(u1 , . . . , ul ) ↓ (u1 , . . . , ul ) + m2x /m2x

→ →

OP(V ),ϕ(x) ↓α OX,x /m2x

→ →

OPm−1 ,ϕ(x) ↓β OX,x /(u1 , . . . , ul ) + m2x

α is surjective if and only if so is β, which proves our lemma.

→

0

→

0,

Theorem 3.4. Let X = PC (E) ⊆ Pn+e = P(V ) (n, e ≥ 2) be a nondegenerate n-dimensional scroll over a smooth projective curve C. Then X has a secant line L = P(V /W ) ⊆ P(V ) with l(X ∩ L) = d − e + 1 if and only if C = P1 and one of the following holds for an ample vector bundle Fi with rank Fi = i > 0 and c1 (Fi ) = e − 1: (1) E = Fn−2 ⊕ OP1 (1)⊕2 , V = H 0 (E), and L is a line in P(H 0 (OP1 (1)⊕2 )) ⊆ P(V ).

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Atsushi Noma

(2) E = Fn−1 ⊕ OP1 (2), V = H 0 (E), and L is a line in P(H 0 (OP1 (2))) ⊆ P(V ). (3) E = F˜n−1 ⊕ OP1 (1), V = H 0 (E), where F˜n−1 ﬁts into 0 → Fn−1 → F˜n−1 → k(p1 ) → 0 for some p1 ∈ P1 . The line L corresponds to H 0 (E) → H 0 (k(p1 )⊕ k(p2 )) for the direct sum of F˜n−1 → k(p1 ) and a surjection OP1 (1) → k(p2 ) with some p2 = p1 ∈ P1 . for a negative (4) E is the quotient bundle of an injection K → Fn ⊕ OP⊕2 1 (i.e., dual of ample) bundle K of rank 2, V = H 0 (Fn ⊕ OP⊕2 1 ), and L = ⊕2 ⊕2 0 P(H (OP1 )) ⊆ P(V ) such that the induced surjection OP1 → E/Fn gives an ⊕2 embedding of PP1 (E) ∩ PP1 (OP⊕2 1 ) ⊆ PP1 (Fn ⊕ OP1 ) into L. Proof. We prove the “only if” part. By Theorem 2.4, we have C = P1 . From (3.0), especially (3.0.5), we have an ample bundle Fn−l of rank n − l for some l (0 ≤ l ≤ 2) with c1 (Fn−l ) = e − 1 and a subspace U ⊆ H 0 (Fn−l ) with surjection U ⊗ OP1 → Fn−l such that Fn−l and U ﬁt into (3.0.6) and (3.0.5). By comparing dimension, U = H 0 (Fn−l ) and W = H 0 (F). When l = 2, from (3.2.2), we have G = OP1 (1)⊕2 and hence E = Fn−2 ⊕ OP1 (1)⊕2 , V /U = H 0 (G), and V = H 0 (E), which is (1). When l = 1, by (3.2.1), G ∼ = OP1 (2) or OP1 (1) ⊕ k(p1 ) for some p1 ∈ P1 . Hence 0 V /U = H (G). For the former, we have (2). For the latter, let F˜n−1 be the kernel of E → OP1 (1) = G ∨∨ . By comparing E → G ∨∨ and E → G, we have an exact sequence 0 → Fn−1 → F˜n−1 → k(p1 ) → 0. Consequently F˜n−1 is ample and hence E ∼ = Fn−1 ⊕ OP1 ) → E induces an exact = F˜n−1 ⊕ OP1 (1). The map F(∼ sequence 0 → OP1 → OP1 (1) → k(p2 ) → 0 for some p2 ∈ P1 . In this case p1 = p2 and Z = k(p1 ) ⊕ k(p2 ), because dim Z ⊗ k(p) ≤ 1 for all p ∈ P1 by the ﬁniteness of X ∩ L. Since F˜n−1 (−1) is generated by global section, by changing the splitting of E, E → Z is the direct sum of F˜n−1 → k(p1 ) and OP1 (1) → k(p2 ), which is (3). When l = 0, the kernel K of (V /W ) ⊗ OP1 → Z in (3.0.2) has no global section and hence it is negative. By (3.0.6), (V /W ) ⊗ OP1 → Z gives embedding PP1 (E) ∩ PP1 ((V /W ) ⊗ OP1 ) → L. Hence we have (4). Conversely the “if” part for (1), (2) and (3) is clear. For (4), we apply Lemma 3.3 for (3.0.3).

Corollary 3.5. Let X = PP1 (E) ⊆ Pn+e = P(V ) (n, e ≥ 2) be a nondegenerate n-dimensional rational scroll with a secant line L = P(V /W ) ⊆ P(V ) such that l(X ∩ L) = d − e + 1 − δ for some δ ≥ 0. Then one of the following holds for some ample bundle Fi with rank Fi = i > 0 and c1 (Fi ) = e + δ − 1 and for some vector subspace U of H 0 (Fi ) of codimension δ: (1) E = Fn−2 ⊕ OP1 (1)⊕2 , V = U ⊕ H 0 (OP1 (1)⊕2 ) ⊆ H 0 (E)(=: V˜ ), and L ⊆ P(H 0 (OP1 (1)⊕2 )). (2) E = Fn−1 ⊕ OP1 (2), V = U ⊕ H 0 (OP1 (2)) ⊆ H 0 (E)(=: V˜ ), and L ⊆ P(H 0 (OP1 (2))) ⊆ P(V ). (3) E = F˜n−1 ⊕ OP1 (1), where F˜n−1 ﬁts into 0 → Fn−1 → F˜n−1 → k(p1 ) → 0. V = U ⊕ H 0 (OP1 (1) ⊕ k(p1 )) ⊆ H 0 (E)(=: V˜ ) and L corresponds to H 0 (E) →

Multisecant lines to projective varieties

357

H 0 (k(p1 ) ⊕ k(p2 )) for the direct sum of F˜n−1 → k(p1 ) and a surjection OP1 (1) → k(p2 ) with p2 = p1 ∈ P1 . for a negative (4) E is the quotient bundle of an injection K → Fn ⊕ OP⊕2 1 ⊕2 0 ˜ ) ⊆ H (F ⊕ O bundle K of rank 2, V = U ⊕ H 0 (OP⊕2 1 n P1 )(=: V ), and ⊕2 ⊕2 0 L = P(H (OP1 )) such that the induced surjection OP1 → E/Fn gives an ⊕2 embedding of PP1 (E) ∩ PP1 (OP⊕2 1 ) ⊆ PP1 (Fn ⊕ OP1 ) into L. Consequently X and L in P(V ) are the isomorphic images of the linear projection ˜ to X with center of X ⊆ P(V˜ ) = Pn+e+δ and a (d − e − δ + 1)-secant line L ˜ P(V /V ). Proof. The same argument as in Theorem 3.4 works in this case, and we have (1), (2), (3) or (4). To see V , we note that c1 (Fn−l ) = e + δ − 1 and h0 (Fn−l ) = n − l + e − 1 + δ = dim U + δ.

Theorem 3.6. Let X = PC (E) ⊆ Pn+e = P(V ) (n ≥ 2) be a nondegenerate ndimensional scroll over a smooth projective curve C of genus g ≥ 1. Assume that e ≥ g + 2. Then X has a secant line L with l(X ∩ L) = d − e + 1 − g if and only if n = 2, C is a plane curve, E is an extension of OC (1) := OP2 (1)|C by a line bundle F1 of degree e − 1 + g (i.e., 0 → F1 → E → OC (1) → 0 is exact), V = H 0 (E), and L is a line in P(H 0 (OC (1))) ⊆ P(V ). Proof. We prove the “only if” part. From (3.0), especially (3.0.5), we have an ample bundle Fn−l of rank n − l for l = 0 or 1 with c1 (Fn−l ) = e − 1 + g and a subspace U ⊆ H 0 (Fn−l ) of dimension n+e−1−l with a surjection U ⊗OC → Fn−l . Then (X , OX (1)) := (PC (Fn−l ), OPC (Fn−l ) (1)) is a polarized variety with delta genus Δ(X , OX (1)) ≤ g and deg(X ) = e − 1 + g ≥ 2g + 1. If n − l ≥ 2, by [3], (I.3.5), then h1 (OX ) = 0 = g, contradiction. Hence l = 1 and n = 2. By (3.2.1) and (3.0.6), C is a plane curve and 0 → F1 → E → OC (1) → 0 is exact. Moreover, U = H 0 (F1 ) and dim V /U = h0 (OC (1)). Hence V = H 0 (E) and

L ⊆ P(H 0 (OC (1))). Conversely, the “if” part follows from Lemma 3.3.

4. Classiﬁcation in the boundary case Theorem 4.1. Let X ⊆ Pn+e (n, e ≥ 2) be a nondegenerate n-dimensional smooth projective variety of degree d. Then X has a secant line L with l(X ∩L) = d − e+ 1 if and only if one of the following holds: (1) X ⊆ P5 is the image of the Veronese embedding v2 : P2 → P5 , and L is a 2-secant line. (2) X ⊆ P4 is the isomorphic image of the Veronese surface v2 (P2 ) by the projection from a point of P5 \ Sec(v2 (P2 )), and L is a 3-secant line, where Sec(v2 (P2 )) denotes the secant variety of v2 (P2 ). (3) X is a rational scroll with a (d − e + 1)-secant line L in Theorem 3.4.

358

Atsushi Noma

Proof. We prove the “only if” part. By Theorem 2.4 and the classiﬁcation of varieties of sectional genus zero (see [3], (12.1); [7], (2.3)), X is isomorphic to v2 (P2 ) or a rational scroll. The latter case implies (3). The former case implies that X is v2 (P2 ) ⊆ P5 or its projection from a point since dim Sec(v2 (P2 )) = 4. Thus we have (1) or (2). Conversely “if” part for (1) and (3) is clear. From Lemma 4.2 below, we know every X in (2) has a 3-secant line, as required.

Lemma 4.2. Let X ⊆ PN (N = d − g + n − 1) be a nondegenerate, linearly normal, smooth, projective variety with h1 (OX ) = 0 of dimension n ≥ 2, degree d and sectional genus g. Assume that d ≥ 2g + 3, dim Sec(X) = N − 1 and d − g = 4 or 5. Then for any projection πv : PN \ {v} → PN −1 from v ∈ PN \ Sec(X), the image πv (X)(∼ = X) has a secant line L with l(πv (X) ∩ L) = 3. Proof. If H1 , . . . , Hn−1 are general hyperplanes through v, then C := X ∩H1 ∩· · ·∩ Hn−1 is a smooth curve in Pd−g := H1 ∩ · · · ∩ Hn−1 . By assumption d ≥ 2g + 3 and h1 (OX ) = 0, for any x1 , x2 , x3 ∈ C, we have dimx1 + x2 + x3 = 2 and x1 + x2 + x3 ∩ C = x1 + x2 + x3 . Thus S := {(x1 , x2 , x3 , w)|w ∈ x1 + x2 + x3 } (⊆ C 3 × Pd−g ) is a P2 -bundle over C 3 . By [10], (1.4), S is surjective onto Pd−g . Hence v lies on a curvilinear 3-secant 2-plane to X, which is mapped to a 3-secant

line to πv (X) as required. Theorem 4.3. Let X ⊆ Pn+e (n ≥ 2, e ≥ 3) be a nondegenerate n-dimensional smooth projective variety of degree d. Then X ⊆ Pn+e has a secant line L with l(X ∩ L) = d − e ≥ 2 if and only if one of the following holds: (1) X is a rational scroll with a (d − e)-secant line L in Corollary 3.5. (2) X ⊆ P4 is the isomorphic image of the Veronese surface v2 (P2 ) by the projection from a point of P5 \Sec(v2 (P2 )), and L is a secant line with l(X ∩L) = 2. (3) X is an elliptic scroll of dimension 2 with a (d − e)-secant line L in (3.6). (4) X is the isomorphic image of a linearly normal, nondegenerate Del Pezzo ˜ ⊆ PN ([3], §8) by the linear projection from a (d − e − 3)-plane Λ variety X ˜ = ∅, and L is the image of a curvilinear secant (d − e − 1)with Λ ∩ Sec(X) ˜ ∩ L) ˜ = d − e. ˜ in PN containing Λ with l(X plane L Proof. We prove the “only if” part. By Theorem 2.4, g(X) = 0 or 1. If g(X) = 0, we have (1) or (2). If g(X) = 1, by the classiﬁcation ([3], (12.3); [7], (2.6)), X is an elliptic scroll or a Del Pezzo variety. In the former case, by Theorem 3.6, we have (3). In the latter, by Lemma 4.4, we have (4). In each case, the “if” part is obvious.

Lemma 4.4. Let X ⊆ Pn+e be a nondegenerate n-dimensional smooth projective variety with h1 (OX ) = 0 of degree d and sectional genus g. Let φ : X → PN (N = d − g + n − 1) be the embedding by complete linear system |OX (1)|. Let Λ ⊆ PN be the (d − e − 2 − g)-dimensional linear space such that the projection

Multisecant lines to projective varieties

359

from Λ gives X ⊆ Pn+e . Let Z ⊆ X is a 0-dimensional curvilinear scheme of deg Z = d − e + k − δ =: t + 1 for some δ ≥ 0. Assume e − k ≥ δ + 1. Then L := Z ⊆ Pn+e is a secant k-plane to X with l(X ∩ L) = t + 1 if and only if ˜ := φ(Z) ⊆ PN is a secant t-plane to φ(X) with l(φ(X) ∩ L) ˜ = t + 1 such that L ˜ \ Λ). ˜ ˜ dim Λ ∩ L = d − e − 2 − δ and Λ, L ∩ φ(X) = φ(Z). In this case, L = πΛ (L Proof. The “if” part is clear, so we show the “only if” part. By Lemma 2.1, Z is a subscheme of a smooth curve C obtained by successive hyperplane sections. Set OC (1) = OX (1)|C. From h1 (OX ) = 0, we have φ(C) ⊆ P(H 0 (OC (1))) ⊆ P(H 0 (OX (1))) and this embedding of φ(C) is given by |OC (1)|. Since deg OC (1)(−Z) = e − k + δ ≥ 2δ + 1 ≥ 2g + 1, we have h1 (OC (1)(−Z)) = 0 and h1 (OC (1)(−Z − x)) = 0 for any x ∈ C. Hence ˜ \ Λ) by construction, ˜ = t and L ˜ ∩ φ(X) = L ˜ ∩ φ(C) = φ(Z). Since L = πΛ (L dim L we have the required condition.

References [1] Bertin, M. A., On the regularity of varieties having an extremal secant line, J. Reine Angew. Math. 545 (2002), 167–181 [2] Flenner, H., O’Carroll, L. and Vogel, W., Joins and intersections, Springer Monographs in Mathematics, Springer-Verlag (1999) [3] Fujita, T., Classiﬁcation theories of polarized varieties, London Mathematical Society Lecture Note Series 155 Cambridge University Press (1990) [4] Harris, J., Curves in projective space, Seminar on Higher Mathematics, Presses de l’universit´e de Montr´eal, Montreal, Que. (1982) [5] Harris, J., Algebraic Geometry, Graduate Texts in Mathematics 133 SpringerVerlag [6] Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics 52 Springer-Verlag [7] Ionescu, P., Embedded projective varieties of small invariants, Lecture Note in Math. 1056 (1984), 142–186 [8] Kwak, S., Smooth projective varieties with extremal or next to extremal secant curvilinear spaces, Trans. Amer. Math. Soc. 357 (2005), 3553-3566 [9] Zak, F. L., Tangents and secants of algebraic varieties, Translations of Mathematical Monographs 127 (1993) [10] Zak, F. L., Projective invariants of quadratic embeddings, Math. Ann. 313 (1999), 507–545 Atsushi Noma Department of Mathematics Faculty of Education and Human Sciences Yokohama National University Yokohama 240-8501 Japan Email: [email protected]

Cycle map on Hilbert schemes of nodal curves Ziv Ran∗

Abstract. We study the structure of the relative Hilbert scheme for a family of nodal (or smooth) curves via its natural cycle map to the relative symmetric product. We show that the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We discuss some applications and connections, notably with birational geometry and intersection theory on Hilbert schemes of smooth surfaces. 2000 Mathematics Subject Classiﬁcation: 14H10, 14C05

1. Introduction An object of central importance in classical algebraic geometry is a family of projective curves, given by a projective morphism π:X→B with smooth general ﬁbre. One wants to take B itself projective, which means one must allow some singular ﬁbres. We will assume that our singular ﬁbres are all nodal. Of course, by semistable reduction, etc., any family can be modiﬁed so as to have this property, without changing the general ﬁbre Xb = π −1 (b). Many questions of classical geometry involve point-conﬁgurations on ﬁbres Xb with b ∈ B variable. From a modern standpoint, this means they involve the relative Hilbert scheme [m]

XB = Hilbm (X/B). [m]

This motivates the interest in studying XB and setting it up as a tool for studying the geometry, e.g. enumerative geometry, of the family X/B. This paper is a step in this direction. Our focus will be on the cycle map (sometimes called the ’Hilb-toChow’ map) cm , which in this case takes values in the relative symmetric product (m) XB . (m) Our main result is that cm is the blowing-up of the discriminant locus in XB . This result will be proven in §1. As we shall see, the proof amounts to a fairly ∗ Research

supported in part by the NSA under grant MDA 904-02-1-0094

362

Ziv Ran (m)

complete study, locally over XB , of cm . We shall see in particular that cm is a small resolution of singularities; in fact in ’most’ cases the non-point ﬁbres of cm are chains of rational curves (with at most m − 1 components). In §2 we will consider [m] applications of the result of §1 to the further study of XB and cm . We will give [m] a formula for the canonical bundle of XB showing that cm ’looks like’ a ﬂipping contraction; in fact, c2 is none other than the Francia ﬂip and admits a natural 2:1 covering by the ﬂop associated to a 3-fold ODP . We will also give a simple [m] formula for the Euler number of XB . In §3 we will discuss the Chern classes of tautological bundles. These are bundles whose ﬁbre at a point representing a scheme z is H 0 (E ⊗ Oz ), where E is a ﬁxed vector bundle on X. This paper has substantial intersection with the Author’s papers [7, 8, 9] where some of the results are proven in greater detail. As to the relevance of this paper to the theme of ’projective varieties with unexpected properties’ I can only say that the close links- some exposed belowof the Hilbert scheme, a priori a purely algebraic object, to classical projective geometry were quite unexpected by me, though this is probably due only to my own ignorance.

Acknowledgments. A preliminary version of this work was presented at the Siena conference in June ’04. I would like to thank the conference’s organizers, especially Luca Chiantini, for their hard and successful work putting together this memorable and valuable mathematical event. I would also like to thank the participants of both the Siena conference and a subsequent one in Hsinshu, Taiwan, especially Rahul Pandharipande and Lih-chung Wang, for valuable input into §3.

2. The cycle map as blow-up Our main object of study is family of projective curves π:X→B whose ﬁbres Xb = π −1 (b) are smooth for b ∈ B general. We shall make the following Essential hypothesis: Xb is nodal for all b ∈ B. We shall also make the (nonessential, but convenient) hypothesis that X, B are smooth of dimension 2,1 respectively. Geometry of the family largely amounts to the study of families of subvarieties (more precisely subschemes) {Zb ⊂ Xb , b ∈ B}

Cycle map on Hilbert schemes of nodal curves

363

of some ﬁxed degree (length) m over B. The canonical parameter space for subschemes is the relative Hilbert scheme [m]

XB = Hilbm (X/B). [m]

So (ordinary)points z ∈ XB correspond 1-1 with pairs (b, Z) where b ∈ B and Z ⊂ Xb is a length-m subscheme. More generally, for any artin local C-algebra R and S = Spec(R), we have a bijection between diagrams S

f

[m]

→ XB

f0

↓ B

and Z

⊂ XS * ↓ S

→ X ↓ f0

→

B

with the right square cartesian and Z/S ﬂat of relative length m. [m] As usual in Algebraic Geometry, we study a complex object like XB by relating it (mapping it) to other (simpler ?) objects. One approach (not pursued here, [m] but see [7]) is to relate XB (albeit only by correspondence, not morphism) to [m−1] . This leads to studying ﬂag Hilbert schemes. These have a rich geometry; XB they are generally singular. We focus here on another approach, based on the cycle map [m]

(m)

cm : XB → Symm B (X) =: XB Z →

lengthp (Z)p.

p∈X

Clearly, cm is an iso oﬀ the locus of cycles whose support meets the critical or singular locus sing(π) = locus in X of singular points of ﬁbres of π. Main Theorem 2.1. cm is the blowing-up of the discriminant locus

(m) Dm = mi pi : ∃mi > 1 ⊂ XB Recall that if I is an ideal on scheme X, we have a surjection of graded algebras from the symmetric algebra on I to the Rees or blow-up algbera Sym• (I) →

∞ 6 0

Ij.

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Applying the Proj functor, we get a closed embedding (maybe strict) of schemes over X BI (X) ⊆ P(I) of the blow-up into the ’singular projective bundle’ P(I), whose ﬁbres over X are projective spaces of varying dimensions. Note that P(I) may be reducible, while BI (X) is always an integral scheme if X is. Concretely, these schemes may be described, locally over X, as follows: if f1 , . . . , fr generate I, take formal homogeneous coordinates T1 , . . . , Tr , then as subschemes of X × Pr−1 , BI (X) = Zeros(G(T1 , . . . , Tr ) : G(f1 , . . . , fr ) = 0, G homogeneous), P(I) = Zeros(G(T1 , . . . , Tr ) : G(f1 , . . . , fr ) = 0, G homogeneous linear). Thus, the inclusion BI (X) ⊆ P(I) is strict iﬀ I admits nonlinear syzygies; the case of the discriminant locus, to be studied below, will provide examples of such ideals. Remark We will see in the proof that [m] • XB is smooth (over C) of dimension m + dim B, • cm is a small map (in fact, if each Xb has at most ν nodes– usually, ν = 1 – then ﬁbres of cm have dimension at most min(ν, m/2)). (m) Clearly, Dm is a prime Weil divisor on XB , in fact (m−2)

Dm ∼bir X ×B XB because a general z ∈ D has the form

z = 2p1 + p2 + . . . + pm−1 . On the other hand, near cycles meeting sing(π), esp. ’maximally singular’ cycles z = mp, p ∈ sing(π), it’s not clear a priori what (or how many) deﬁning equations Dm has (the proof below will yield a posteriori a set of equations locally at maximally singular cycles). m Note that locally at maximally singular cycles, the relative Cartesian product XB is a complete intersection with equation x1 y1 = . . . = xm ym , with the projection (m) to B given by t = x1 y1 , while XB is a quotient of a complete intersection (x1 y1 = . . . = xm ym )/ symmetric group Sm . (m)

We will see that XB is not Q-factorial: in fact, Dm is not Q-Cartier; (m) Worse, XB is not even Q-Gorenstein: we shall see that it admits a small dis[m] crepant resolution XB . (m) Nonetheless, being quotient by a ﬁnite group and smooth in codimension 1, XB is normal and Cohen-Macaulay. The plan of proof is as follows. [m] (m) - Construct explicit (analytic) model of XB and cm , locally over XB ; in partic-

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m ular, conclude that XB is smooth and cm is small, so c−1 m (D ) is Cartier divisor. - The Universal property of blowing up now yields a factorization [m]

[m]

XB

cm

→

cm

(m)

BDm XB ↓ b (m) XB

Then we check locally (over the blowup) that cm is an iso. To start the proof, ﬁx an analytic neighborhood U of ﬁbre a node p, so the family is given in local analytic coordinates by xy = t. For the local study, the ﬁrst question is: what are ﬁbres of cm ? Now locally in the ´etale topology, all ﬁbres are (essentially) products of ﬁbres c−1 mi (mi pi ). So suﬃces to study c−1 m (mp), p ∈ sing(π). Then, 0 c−1 m (mp) = Hilbm (R)

where R is the formal power series ring R = C[[x, y]]/(xy). Here

Hilb0m

denotes the punctual Hilbert scheme.

Proposition 2.2. Hilb0m (R) is a chain of m − 1 smooth rational curves meeting m m Cm−1 : normally C1m ∪q2m . . . ∪qm−1 m

m

Qm−1

Q2

m

m

C1

Cm−1

m

C2

m

Qm

m

Q1

Fig. 1

where qim = (xm+1−i , y i ), m Cim \ {qim , qi+1 } = {Iim (a) = (axm−i + y i ) : a = 0} m NB lim Iim (a) = qim , lim Iim (a) = qi+1 . a→0

Proof. See [8].

a→∞

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Given this, the next question is: what does the full Hilbert scheme look like along Hilb0 , e.g. locally near qim ? Proposition 2.3. The universal ﬂat deformation of the ideal qim = (xm+1−i , y i ), i = 1, . . . , m, rel B, is (f, g) where 1 2 f = xm+1−i + fm−i (x) + vy i−1 + fi−2 (y), 1 2 g = y i + gi−1 (y) + uxm−i + gm−i−1 (x) 1 1 and gi−1 have no constant term, and the where each fba , gba has degree b, fm−i following relations, equivalent to ﬂatness, hold

yf = vg xg = uf.

Proof. See [8]. Concretely, the above relation mean 1 1 (x), gi−1 (y) are free parameters (no relations); - the coeﬃcients of fm−i - the relation uv = t holds; 2 2 , gm−i−1 are determined by the other data. - fi−2 A similar and simpler story holds at the principal ideals Iim (a). We conclude [m]

- XB

is smooth;

- its ﬁbre at t = 0, i.e. Hilbm (X0 ) has, along Hilb0m (R), (m+1) smooth components crossing normally, D0 , . . . , Dm .

m

Dm

m

D0

m

D2

m

D1

Fig. 2 In fact, if X0 = X0 ∪ X0 then Di ∼bir (X0 )m−i × (X0 )i . The next question is: how to glue together the various local deformations ?

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Construction Let C1 , . . . , Cm−1 be copies of P1 , with homogenous coordinates ui , vi on the i-th copy. Let C˜ ⊂ C1 × . . . × Cm−1 × B be the subscheme deﬁned by v1 u2 = tu1 v2 , . . . , vm−2 um−1 = tum−2 vm−1 . Fibre of C˜ over 0 ∈ B is C˜0 =

m−1

C˜i ,

i=1

where C˜i = [1, 0] × . . . × [1, 0] × Ci × [0, 1] × . . . × [0, 1]. In a neighborhood of C˜0 , C˜ is smooth and C˜0 is its unique singular ﬁbre over B. We may embed C˜ in Pm−1 × B via Zi = u1 · · · ui−1 vi · · · vm−1 , i = 1, . . . , m. These satisfy Zi Zj = tZi+1 Zj−1 , i < j − 1 so embed C˜ as a family of rational normal curves C˜t ⊂ Pm−1 , t = 0 specializing to a connected (m − 1)-chain of lines. Next consider A2m with coordinates a0 , . . . , am−1 , d0 , . . . , dm−1 ˜ ⊂ C˜ × A2m be deﬁned by Let H a0 u1 = tv1 , ... , am−1 um−1 = d1 vm−1 d0 vm−1 = tum−1 . ˜ over A2m are: a point (generically), or a chain of i ≤ m − 1 rational Fibres of H ˜ ×B U curves; all values i = 1, . . . , m − 1 occur. Consider the subscheme of Y = H deﬁned by a1 u1 = dm−1 v1 ,

F0 F1

:= :=

xm + am−1 xm−1 + . . . + a1 x + a0 u1 xm−1 + u1 am−1 xm−2 + . . . + u1 a2 x + u1 a1 +v1 y .. .

Fi

:=

Fm

:=

ui xm−i + ui am−1 xm−i−1 + . . . + ui ai+1 x + ui ai +vi dm−i+1 y + . . . + vi dm−1 y i−1 + vi y i .. . d0 + d1 y + . . . + dm−1 y m−1 + y m .

The following is proven in [9].

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˜ is smooth and irreducible. Theorem 2.4. (i) H ˜ ×B U that (ii) The ideal sheaf I generated by F0 , . . . , Fm deﬁnes a subscheme of H ˜ is ﬂat of length m over H. (iii)The classifying map ˜ → Hilbm (U/B) Φ = ΦI : H is an isomorphism. ˜ is covered by open sets The proof shows furthermore that H Ui = {Zi = 0}, i = 1, . . . , m. U2 Um−1

U3 Um U1

Fig. 3 On Ui , we have Fj = uj xi−j−1 Fi−1 , j < i − 1 Fj = vj y j−i Fi , j > i hence Fi−1 , Fi generate I on Ui (they yield the f, g in the universal deformation of Proposition 2 above). x are the elementary symmetric functions in the roots of Also, ai = (−1)i σm−i y ˜ → A2m factors through the F0 , and ditto for di , σm−i , Fm . So the projection H B cycle map ˜ H c↓ (m) XB

* σ → A2m B

x y σ = (σ1x , . . . , σm , σ1y , . . . , σm )

(one can show σ is embedding).To prove the Main Theorem, we must show: c is the blow-up of Dm . It is convenient to pass to an ’ordered’ model, deﬁned by the following Cartesian diagram: m

XB ↓ m XB

→

[m]

XB ↓ (m) → XB

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369

In this diagram, the right vertical arrow is the cycle map, the bottom horizontal arrow is the natural map between the Cartesian and symmetric products, and the other arrows are deﬁned by the ﬁbre product construction. Recall the description of the blowup of an ideal I as subscheme of P(I). Let us rewrite the deﬁning local [m] equations for XB in terms of the homogeneous coordinates Zi on Pm−1 : they are linear : y Zi = tm−j−i σjx Zi+1 , i = 1, . . . , m − 1, j = 0, . . . , m − 1; σm−j x Zi = tm−j−i σjy Zi−1 , i = 2, . . . , m, j = 0, . . . , m − 1. σm−j

quadratic: Zi Zj = tZi+1 Zj−1 , i < j − 1. Our task at this point is to ’reverse engineer’ an ideal whose generators G1 , . . . , Gm satisfy (precisely) these relations. Actually, the choice of G1 determines G2 , . . . , Gm via the linear relations, though a priori, G2 , . . . , Gm are only rational functions. Now recall that Z1 generates O(1) over the open U1 which meets the special ﬁbre t = 0 in the locus of m-tuples entirely on x-axis. On that locus, an equation for the discriminant is given by the Van der Monde determinant: vxm = det(Vxm ), ⎡

1 x1 .. .

⎢ ⎢ Vxm = ⎢ ⎣

xm−1 1

... ...

1 xm .. .

⎤ ⎥ ⎥ ⎥. ⎦

. . . xm−1 m

Thus motivated, set G1 = vxm . This forces Gi =

y i−1 ) (σm vxm (i−1)(m−i/2) t

=

y i−1 ) (σm G1 , (i−1)(m−i/2) t

i = 2, . . . , m.

If the construction is to make sense, these better be regular. In fact, Gi = ± det(Vim ), ⎡

Vim

1 x1 .. .

... ...

1 xm .. .

y1i−1

...

i−1 ym

⎢ ⎢ ⎢ ⎢ ⎢ m−i =⎢ ⎢x1 ⎢ y1 ⎢ ⎢ . ⎣ ..

⎤

⎥ ⎥ ⎥ ⎥ ⎥ m−i ⎥ . . . xm ⎥ . . . ym ⎥ ⎥ .. ⎥ . ⎦

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(we call this the ’Mixed’ Van der Monde matrix). The Gi satisfy same relations as the Zi , so we can map isomorphically O(1) → J = Ideal(G1 , . . . , Gm ) Zi → Gi . Then J is an invertible ideal deﬁning a Cartier divisor Γ. The Main Theorem’s assertion that c is the blowup of Dm means J = c∗ (IDm ), i.e. Γ = c∗ (Dm ). Containment ⊇ of schemes is clear. Equality is clear due to the special ﬁbre t = 0. Now this special ﬁbre is sum of components ΘI = Zeros(xi , i ∈ I, yi , i ∈ I), I ⊆ {1, . . . , m}. Set Θi =

ΘI .

|I|=i

Note that the open set Ui meets only Θi , Θi−1 . One can check that the vanishing order of Gi on any ΘI , |I| = k, is ordΘI (Gi ) = =

(k − i)2 + (k − i) k = i, i − 1.

0 if

So Zeros(Gi ) = c∗ (Dm ) on Ui , i.e. Γ|Ui = c∗ (Dm )|Ui , ∀i ∴

Γ = c∗ (Dm ).

This concludes the proof of the Main Theorem. One point of interest is the interpretation of mixed Van der Monde matrices, whose determinants played a large role in the proof : The universal subscheme m

Ξ = Zeros(I) ⊂ XB

×X

contains sections m

Ψi = graph(pi : XB

→ X).

The universal quotient Qm = pX m ∗ (O/I) B

maps to OX (m) via restriction on Ψi . Assembling together, get map B

V : Qm → mOX m . B

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Then Vim = is just the matrix of V with respect to the basis 1, x, . . . , xm−i , y, . . . , y i−1 of Qm on Ui . A somewhat mysterious point that comes up in the above proof is: as the Zi are interpreted as the equations of the discriminant, what, if any, is the interpretation of ui , vi ?

3. Applications Canonical bundle. [m]

A ﬁrst application is a formula for the canonical bundle of XB . For any class α on X, denote α[m] = q∗ p∗ (α) [m]

[m]

where Ω ⊂ XB ×B X is universal subscheme and p : Ω → X, q : Ω → XB are natural maps. Here q∗ denotes the cohomological direct image, sometimes called the norm or denoted q! , not the sheaf-theoretic direct image. Another way to construct α[m] is as follows. First note the natural isomorphism over Q H ∗ (X (m) ) Symm (H ∗ (X)). This yields a class αm ∈ H ∗ (X (m) ), and α[m] is the image of the latter via the composite c∗

m XB . H ∗ (X (m) ) → H ∗ (XB ) →

(m)

[m]

(3.1)

Also set OX [m] (1) = O(−Γ) B

(the canonical O(1) as blowup, via Proj).

Corollary 3.1. KX [m] /B = (KX/B )[m] ⊗ OX [m] (1) B

B

Proof. It suﬃces to note that both sides agree oﬀ the exceptional locus of cm .

In particular, KX [m] /B .Cim = +1, so cm ’looks like’ a ﬂipping contraction. The B following example partially conﬁrms this.

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Example: m = 2. We have a diagram 2

XB c2 ↓ 2 XB

→

[2]

XB ↓ c2 (2) → XB

2 with horizontal maps of degree 2. Local equations for XB are:

x1 y1 = x2 y2 = t (so this is a 3-fold ODP); 2 2 for XB in XB × P2 : x1 u = y2 v x2 u = y1 v so c2 is a small resolution of the ODP, known as a ﬂopping contraction; it can be ﬂopped to yield X ∗∗ smooth that is the source of the ’opposite’ ﬂopping contraction. (2) Local equations for XB are: σ2y σ1x

= tσ1y

σ2x σ1y σ2x σ2y

= tσ1x = t2 . [2]

Viewed in A5 , this is a cone over a cubic scroll in P4 . XB is small resolution of the cone, with exceptional locus C12 = P1 . A well-known procedure, due to Francia, [2] yields a ﬂip, called Francia’s ﬂip, of c2 : blow up C12 in XB (which is the same as blowing up the vertex of the cone); the exceptional divisor is a scroll of type F1 ; then blow up the negative curve of F1 to get a new exceptional surface of type F0 ; then blow down F0 in the other direction to C ∗ = P1 so the F1 becomes a P2 ; then ﬁnally blow down P2 to a (singular) point on a new 3-fold X ∗ , which is 2:1 covered by X ∗∗ . This situation is intriguing in view of recent work of Bridgeland [2] and Abramovich and Chen [1] which shows that the ﬂop X ∗∗ and the ﬂip X ∗ can be interpreted 2 [2] as moduli spaces of certain ’1-point perverse sheaves’ on XB and XB , respectively. This raises the question of ﬁnding a natural interpretation of X ∗ , X ∗∗ and their higher-order analogues, if they exist, in terms of our family of curves X/B.

Euler number. As an application of our study of cm , we can compute (topological) Euler number [m] e(XB ) = cm+1 (TX [m] ), at least for case of ≤ 1 node in any ﬁbre: B

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Corollary 3.2. If X/B has σ singular ﬁbres and each has precisely 1 node, then [m] the topological Euler number of XB is given by m − 2g + 2 [m] m 2g − 2 e(XB ) = (−1) (2 − 2g(B)) + σ . (3.2) m m−1 Proof. Let (Xi , pi , Xi,0 = Xi \ pi ), i = 1, . . . , σ be the singular ﬁbres with their respective unique singular point and smooth part, and X0 = X \ (X1 ∪ . . . ∪ Xσ ), B0 = π(X0 ). Then

(m) XB

admits a (locally closed) stratiﬁcation with big stratum (m)

(X0 )B0 and other strata

Σi,j = ipj + (Xj,0 )(m−i) , i = 0, . . . , m, j = 1, . . . , σ. The ﬁbre of cm over each of these strata is, respectively, a point over the big stratum, and over the Σi,j , a point for i = 0, 1, a chain of (i − 1) P1 ’s for i = 2, . . . , m. Since the Euler number is multiplicative in ﬁbrations and additive over strata, we get

[m] (m) e((Xj,0 )(m) ) e(XB ) = e((X0 )B0 ) +

ie((Xj,0 )(m−i) ). + i>0

Now MacDonald’s formula [5] says that for any X, the Euler number of its mth symmetric product is given by (m) m −e(X) e(X ) = (−1) . m (m)

Plugging this into the above and using multiplicativity for the ﬁbration (X0 )B0 over B0 yields m−1

2g − 2 2g − 2 [m] (−1)k (m − k) . (3.3) e(XB ) = (−1)m (2 − 2g(B)) + σ k m k=0

Now, as pointed out by L.C. Wang, (3.2) follows from (3.3) by the elementary formula b

a a−1 (−1)k = (−1)b k b k=0

which in turn is an easy consequence a−1 a−1of Pascal’s relation a = + k k k−1 .

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Remark 3.3. Suppose our family X/B is a blowup β:X→Y of a smooth P1 bundle; equivalently, each singular ﬁbre of X/B has consists of two [m] P1 components. Then there is another way to construct XB and obtain formula (3.3) above, as follows. Note that the natural map [m]

η : YB

(m)

= YB

→B

is a Pm -bundle. Blow up a Pm−1 in each ﬁbre of η over a singular value of π, giving rise to exceptional divisors E1,i , i = 1, . . . , σ; then blow up a Pm−2 in general position in each exceptional divisor E1,i , giving rise to new exceptional divisors E2,i , etc. Finally, blow up general point on each exceptional divisor Em−1,i . This [m] yields XB . In these blowups, the change in Euler number is easy to analyze, yielding (3.2).

Further developments (under construction). We mention some natural questions and possible extensions. • What is the total Chern class c(TX [m] )? B

[m]

• Develop intersection calculus for diagonal loci of all codimensions in XB , i.e. degeneracy loci m Γm r = rk(Vi ) ≤ m + 1 − r

(locus where r points come together) More generally, loci Γm (m.) , m1 + . . . + mk = m,

& % mi p i . Γm (m.) = z : cm (z) = In particular, the small diagonal Γm (m) = locus of length-m schemes supported at 1 point which coincides with the blowup of X, locally at each ﬁbre node, in a punctual subscheme of type ( m) m−i i m x 2 , . . . , x( 2 ) y (2) , . . . , y ( 2 ) . A potential application of this calculus is is to enumerative geometry (multiple points, multisecant spaces, special divisors on stable curves. . . ) A Sample corollary which however can also be derived by other means) is the following relative triple point formula: for a map f : X → P2 , the number of relative triple points is > = (d − 2)(d − 4) = d> + g − 1 L2 + 3 − ωL + 2ω 2 − 4σ N3,X (f ) = 2 2

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375

where L = f ∗ O(1), L2 = deg(f ), d = deg(ﬁbre), g =genus(ﬁbre). See [9] for some progress on this. • If X/B is of compact type (assume for simplicity there exists a section), we have an Abel-Jacobi morphism to the Jacobian: [m]

XB → J(X/B). Fibres give a notion of ’generalized linear system’ on reducible ﬁbres. How is this related to other approaches to such notions in the literature ?

4. Chern classes of tautological bundles In [7] we gave a simple formula for the Chern classes of the tautological bundles λm (L), where L is a vector bundle on X. Here X need not be a surface; we just need a family of nodal curves X/B. More precisely, we gave in [7] a formula for the pullback of λm (L) on the (full) ﬂag relative Hilbert scheme, denoted W m (X/B). The formula is simple and involves only divisor classes plus classes coming from X, but has the disadvantage that these classes, unlike λm (L) itself, do not descend to [m] the Hilbert scheme XB . Though it is, broadly speaking, obvious that a formula [m] on XB can be derived from the one on W m (X/B), it is still of some interest, in view of possible applications, to work this out. It turns out that for X a surface, a formula for the Chern classes of tautological bundles was already derived, in the context of the (absolute) Hilbert scheme X [m] , by Lehn [4], using the Fock space formalism introduced earlier by Nakajima [6, 3]. Since our tautological bundles λm (L) are pullbacks of the analogous bundles on X [m] via the natural inclusion [m]

XB ⊂ X [m] , [m]

Lehn’s formula yields an analogous one on XB . Our purpose here, then, is to [m] verify that when X is a surface, the push-down from W m (X/B) to XB of the formula of [7] coincides with the restriction of Lehn’s formula, at least when L is a line bundle. Thus, we have compatibility in the natural diagram W m (X/B) wm ↓ [m] XB → X [m] We begin with some formalism. First, we have the operation of exterior mul[m] tiplication of cohomology classes on various XB , deﬁned as follows. Let [m]

[n]

[m+n]

Zm,n ⊂ XB ×B XB ×B XB

be the closure of the locus

? (zm , zn , zm zn ) : zm ∩ zn = ∅ ,

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and let [m]

[n]

[m+n]

p : Zm,n → XB ×B XB , q : Zm,n → XB

[m]

[n]

be the projections, both generically ﬁnite. For α ∈ H r (XB ), β ∈ H s (XB ), identifying homology and cohomology, set α β = q∗ p∗ (α × β) ∈ H r+s (XB

[m+n]

).

This operation is obviously associative and commutative on even (in particular, algebraic) classes. In particular, taking β = 1, we get a natural way of mapping [m] [m+n] H r (XB ) to H r (XB ) for each n ≥ 0. Next, consider the small diagonal im

[m]

Γm (m) → XB . The restriction of the cycle map yields a birational morphism βm : Γ m (m) → X. For any α ∈ H r (X), we set ∗ (α)) ∈ H r+2m−2 (XB ), qm [α] = im∗ (βm [m]

Via multiplication, qm [α] may be viewed as with an operator on ∞ 6

[n]

H s (XB )

s,n=0

which has operator bidegree (r + 2m − 2, m). This is known as Nakajima’s creation operator (cf. [6, 3]). Lehn’s formula is as follows, Theorem 4.1. (Lehn [4]) For a line bundle L, the total Chern class of λm (L) is the part in bidegrees (∗, m) of exp

∞ > = (−1)n−1 qn [c(L)] . n n=1

Now our formula is the following: Theorem 4.2. For a line bundle L, we have c(λm (L)) =

(i1 − 1)! . . . (ik − 1)! qi1 [c(L)] . . . qik [c(L)]. (−1)|I|−k |I|!(m − |I|)! I = (1 ≤ i < . . . < i ) 1 |I| ≤ m

k

It is elementary to derive Theorem 4.2 from Lehn’s theorem (whose proof is rather long). Our purpose here, however, is to derive Theorem 2 from a result in

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377

[7], as follows. Let [m]

wm : W = W m (X/B) → XB

be the natural morphism from the ﬂag Hilbert scheme to the ordinary one, let pi : W → X be the ith projection, mapping a ﬁltered scheme z1 < . . . < zm to the support of zi /zi−1 , and let Δij ⊂ W, i < j denote the (reduced) locus where the pi and pj coincide; also set, for any class c ∈ H ∗ (X), ci = p∗i (c). It is shown in [7] that each sum

j−1

Δij is a Cartier divisor (even though W is in

i=1

general singular and each summand individually is not Cartier). It is also shown there that the following result holds (for a line bundle L): c(w∗ λm (L)) =

j−1 m = > 1

Δij 1 + Lj − j=1

(4.1)

i=1

Deriving Theorem 4.2 from (4.1) is a matter of expanding the product as a sum of monomials, applying w∗ and dividing by m! = deg(w). In doing so, it is useful to observe the following. Let’s call a connected monomial on an index set I one which, after a permutation, can be written in the form qI [c] = ci1 Δi1 i2 Δi2 i3 . . . Δik−1 ik , I = (i1 < . . . < ik ) where c is either 1 or [L]. The intersection implicit in the above product is transverse, hence well-deﬁned even though the divisors are not Cartier. It is easy to see by induction that there are (k − 1)! unordered monomials in the expansion of (4.1) yielding the same qI [c]. Moreover it is clear that w∗ (qI [c]) = q|I| [c]. Now we note that each monomial appearing in the expansion of (4.1) may be decomposed uniquely as a product of connected monomials on pairwise disjoint index sets (its ’connected components’), yielding a term k

(|Ij |−1)

qI1 [c1 ] . . . qIk [ck ], 0 0 each cj ∈ {1, [L]} which, for ﬁxed I = I1 . . . Ik , appears (|I1 | − 1)!...(|Ik | − 1)! times. Applying w∗ , we get, for each choice of I ⊆ {1, . . . , m} and cj , a term in w∗ applied to (4.1): (−1)

j=1

(i1 − 1)! . . . (ik − 1)!(−1)i−k qi1 [c1 ] . . . qik [ck ],

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ij = |Ij |, i =

ij . Then multiplying by

j

m i

for the choice of subset I with |I| = i,

and dividing by m! yields the result.

References [1] D. Abramovich, J.C. Chen, Flops, ﬂips and perverse point sheaves on threefold stacks (math.AG/0304354). [2] T. Bridgeland, Flops and derived categories, Invent. Math. 147 (2002), 613-632 (math.AG/0009053). [3] G. Ellingsrud, L. G¨ ottsche, Hilbert schemes of points on surfaces and Heisenberg algebras, ICTP lectures, 1999. [4] M. Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136 (1999), pp. 157-207. [5] I.G. Macdonald, The Poincar´e polynomial Proc. Camb. Phil. Soc. 58 (1962), 563-568.

of

a

symmetric

product,

[6] H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. Math. 145 (1997), 379-388. [7] Z. Ran, Geometry on nodal curves (math.AG/0210209; to appear in Compositio Math. (2005)). [8]

,, A note on Hilbert schemes of nodal curves (math.AG/0410037; to appear in J. Algebra).

[9]

,, Geometry on nodal curves II: cycle map and intersection calculus (preprint available at http://math.ucr.edu/˜ziv/papers/cyclemap.pdf or at arXiv.org/math.AG/0410120).

Ziv Ran Mathematics Department UC Riverside, CA 92521, USA Email: [email protected]

Schedule of the conference TUESDAY JUNE 8 ARRIVAL and REGISTRATION of the PARTICIPANTS WEDNESDAY JUNE 9 9.30-10.20 10.30-11.20

EIN: Log canonical threshold and birational geometry CAPORASO: Completing roots of line bundles within the Picard scheme, and applications to projective geometry

coﬀee break 12.10-13.00 14.40-15.30

ARRONDO: Characterization of Veronese varieties via projections in Grassmannians MANIVEL: Trisecant planes to Severi varieties

coﬀee break 16.00-16.20 16.30-16.50 17.00-17.20 17.30-17.50 18.00-18.20

CALABRI: On the degeneration of projective surfaces DE FERNEX: On the birational rigidity of Fano projective hypersurfaces BERTIN: On trisecant lines to White surfaces ˜ MUNOZ: Varieties with small discriminant variety NOMA: Multisecant lines to projective varieties

THURSDAY JUNE 10 9.30-10.20 10.30-11.20

IONESCU: Birational Geometry of Rationally Connected Manifolds via Quasi-Lines STILLMAN: Bayesian networks and projective varieties

coﬀee break 12.10-13.00 14.40-15.30

CATANESE: Pencils of low genus and classiﬁcation of surfaces with low invariants POPESCU: The projective geometry of algebraic sets of 1”minimal degree”

coﬀee break 16.00-16.20 16.30-16.50 17.00-17.20 17.30-17.50 20.00

IARROBINO: Pencils of forms and level algebras DE POI: Congruences of lines and hyperbolic systems of conservation laws DI ROCCO: Dual defect of toric varieties and defect polytopes UGAGLIA: On a class of special linear systems on P3 SOCIAL DINNER

380

Schedule of the conference

FRIDAY JUNE 11 9.30-10.20 10.30-11.20

RUSSO: Varieties with quadratic entry locus MELLA: Singularities of linear systems and the Waring problem

coﬀee break 12.10-13.00 14.40-15.30

MIRANDA: Double Point Interpolation on Toric Surfaces ZAK: Towards elementary geometry of Grassmann varieties

coﬀee break 16.00-16.20 16.30-16.50 17.00-17.20 17.30-17.50 18.00-18.20

LAFACE: Linear systems of P3 through at most eight general points FONTANARI: Interpolation on higher dimensional projective varieties ZAVAREH: Lines on Fano hypersurfaces CARLINI: Varieties of binary forms FLAMINI: Brill-Noether theory for singular curves on a K3 surface

SATURDAY JUNE 12 9.30-10.20 10.30-11.20

GIMIGLIANO: Higher secant varieties, osculating varieties and 0-dimensional schemes RAN: Cycle maps on Hilbert schemes of nodal curves

coﬀee break 12.10-13.00

VERRA: Remarks on the uniruledness of Mg for low g

14.30-17.30

TOUR of the TOWN of SIENA

SUNDAY JUNE 13 DEPARTURE of the PARTICIPANTS

List of participants Abrescia Silvia Alzati Alberto Arrondo Enrique Artebani Michela Ballico Edoardo Beheshti Zavareh Roya Bernardi Alessandra

Universit` a di Bologna [email protected] Universit` a di Milano [email protected] Universidad Complutense de Madrid enrique [email protected] Universit` a di Pavia [email protected] Universit` a di Trento [email protected] MaxPlanck Institut [email protected] Universit` a di Milano [email protected]

Bertin Marie-Amelie Bocci Cristiano Brambilla Chiara Brivio Sonia Calabri Alberto Caporaso Lucia Carlini Enrico Casagrande Cinzia Catalisano M. Virginia Catanese Fabrizio Chikashi Miyazaki Chiantini Luca Ciliberto Ciro

Italy Italy Spain Italy Italy Germany Italy France

[email protected] Universit` a di Milano [email protected] Universit` a di Firenze [email protected] Universit` a di Pavia [email protected] Universit` a di Bologna [email protected] Universit` a di Roma 3 [email protected] Universit` a di Pavia [email protected] Universit`a di Pisa [email protected] Universit` a di Genova [email protected] Universit¨ at Bayreuth [email protected] University of the Ryukyus [email protected] Universit` a di Siena [email protected] Unviersit`a di Roma 2 [email protected]

Italy Italy Italy Italy Italy Italy Italy Italy Germany Japan Italy Italy

382 Cioﬃ Francesca Cobo Pablos Soﬁa Cools Filip Costa Laura De Fernex Tommaso De Poi Pietro Di Gennaro Roberta Di Rocco Sandra Dragotto Alessandra Ein Lawrence

List of participants

Universit`a di Napoli [email protected] Universidad Complutense Madrid [email protected] Universitat Leuwen [email protected] Universitat de Barcelona [email protected] University of Michigan [email protected] Universit¨at Bayreuth [email protected] Universit` a di Napoli [email protected] KTH [email protected] Universit` a di Trieste [email protected] University of Illinois at Chicago [email protected]

Fabre Bruno Faenzi Daniele Fania Maria Lucia Flamini Flaminio Fontanari Claudio Franciosi Marco Geramita Antony V.

Ghiloni Riccardo Gimigliano Alessandro Giuﬀrida Salvatore Gonzalez Pasqual Sonia

Italy Spain Belgium Spain USA Germany Italy Sweden Italy USA France

[email protected] Universti`a di Firenze [email protected] Universit` a de L’Aquila [email protected] Universit` a de L’Aquila [email protected] Universit` a di Trento [email protected] Universit`a di Pisa [email protected] Queens’ University [email protected] Universit` a di Genova [email protected] Universit` a di Trento [email protected] Universit` a di Bologna [email protected] Universit` a di Catania [email protected] Universidad Complutense de Madrid [email protected]

Italy Italy Italy Italy Italy Canada Italy Italy Italy Italy Spain

383

List of participants

Granha Otero Beatriz Guardo Elena Guida Margherita Harbourne Brian Holweck Frederick Iarrobino Antony Ilardi Giovanna Ionescu Paltin Keilen Thomas Kennedy Gary Kleppe Jan Oddvar Kloosterman Remke Kwak Sijong Laface Antonio Lanteri Antonio Mir´ o–Roig Rosa Maria Madonna Carlo Mallavibarrena Raquel Manivel Laurent Mella Massimiliano Mezzetti Emilia Miranda Rick

Unversidad de Salamanca [email protected] Universit` a di Catania [email protected] Universit` a di Napoli [email protected] University of Nebraska [email protected] Universit`e P. Sabatier, Toulouse [email protected] Northeastern University [email protected] Universit` a di Napoli [email protected] University of Bucharest [email protected] Universit¨at Kaiserslautern [email protected] Ohio State University [email protected] Oslo University [email protected] University of Groningen [email protected] KAIST [email protected] Universit` a di Milano [email protected] Universit`a di Milano [email protected] Universitat de Barcelona [email protected] Universit` a di Roma 1 [email protected] Universidad Complutense de Madrid [email protected] Institut Fourier, Grenoble [email protected] Universit` a di Ferrara [email protected] Universit`a di Trieste [email protected] Colorado State University [email protected]

Spain Italy Italy USA France USA Italy Romania Germany USA Norway Nederland Korea Italy Italy Spain Italy Spain France Italy Italy USA

384 Mu˜ noz Roberto Noma Atsushi Orecchia Ferruccio Ottaviani Giorgio Paoletti Raﬀaella Park Euisung Park Seong-Suk Pauer Franz Picco Botta Luciana Popescu Sorin Ran Ziv Ranestad Kristian Reilly Norman Roth Michael Russo Francesco Sabadini Irene Swinnerton-Dyer Peter Supino Paola Stillman Mike Tironi Andrea Luigi Tommasi Orsola Tortora Alfonso

List of participants

University Rey J. Carlos [email protected] Yokohama National University [email protected] Universit` a di Napoli [email protected] Universit` a di Firenze [email protected] Universit`a di Firenze [email protected] KAIST [email protected] ICTP [email protected] Universitat Innsbruck [email protected] Universit`a di Torino picco [email protected] New York State University at Stony Brook [email protected] University of California at Riverside [email protected] Oslo University [email protected] Simon Fraser University [email protected] Queens University [email protected] Universidade Federal de Pernambuco [email protected] Politecnico di Milano [email protected] Cambridge University [email protected] Universit` a delle Marche [email protected] Cornell University [email protected] Universit` a di Milano [email protected] Kath. Univ. Nijmegen [email protected] Universit` a di Milano [email protected]

Spain Japan Italy Italy Italy Korea Italy Austria Italy USA USA Norway Canada Canada Brazil Italy UK Italy USA Italy Nederland Italy

385

List of participants

Turrini Cristina Ugaglia Luca Van Tuyl Adam Verra Alessandro Yang Stephanie Zak Fyodor

Universit` a di Milano [email protected] Universit` a di Milano [email protected] Lakehead University [email protected] Universit` a di Roma 3 [email protected] Harvard University [email protected] Independent University of Moscow [email protected]

Italy Italy Canada Italy USA Russia

List of contributors

Enrique Arrondo Departamento de Algebra, Facultad de Ciencias Matem´ aticas, Universidad Complutense de Madrid 28040 Madrid, Spain Email: enrique [email protected] Edoardo Ballico Dipartimento di Matematica Universit` a degli Studi di Trento Via Sommarive 14, 38050 Povo, Trento, Italy Email: [email protected] Mauro C. Beltrametti Dipartimento di Matematica Universit` a di Genova Via Dodecaneso 35, I-16146 Genova, Italy Email: [email protected] Cristiano Bocci Dipartimento di Matematica “F. Enriques” Universit` a di Milano Via Saldini 50, 20133 Milano, Italy Email: [email protected] Andrea Bruno Dipartimento di Matematica Universit` a di Roma III Largo Murialdo, 00146 Roma, Italy Email: [email protected] Enrico Carlini Dipartimento di Matematica Universit` a di Pavia Via Ferrata 1, 27100 Italy Email: [email protected] Maria Virginia Catalisano Dipartimento di Ingegneria della Produzione e Modelli matematici Universit` a di Genova Piazzale Kennedy, 16129 Genova Italy Email: [email protected]

388

List of contributors

Fabrizio Catanese Lehrstuhl Mathematik VIII Universit¨ at Bayreuth, NWII D-95440 Bayreuth, Germany Email: [email protected] Luca Chiantini Dipartimento di Scienze Matematiche e Informatiche Universit` a di Siena Pian dei Mantellini 44, 53100 Siena, Italy Email: [email protected] Ciro Ciliberto Dipartimento di Matematica Universit` a di Roma Tor Vergata Via della Ricerca Scientiﬁca, 00133 Roma, Italia Email: [email protected] Tommaso De Fernex Department of Mathematics University of Michigan East Hall, 525, East University Avenue, Ann Arbor, MI 48109-1109, USA Email: [email protected] Pietro De Poi Mathematisches Institut Universit¨ at Bayreuth Lehrstuhl VIII, Universit¨ atsstraße 30, D-95447 Bayreuth, Germany Email: [email protected] Cindy De Volder Department of Pure Mathematics and Computeralgebra Ghent University Galglaan 2, B-9000 Ghent, Belgium Email: [email protected] Nicholas Eriksson Department of Mathematics University of California at Berkeley Berkeley, CA 94720-3840, USA Email: [email protected] Maria Lucia Fania Dipartimento di Matematica Universit` a degli Studi di L’Aquila Via Vetoio, loc. Coppito, 67100 L’Aquila, Italy Email: [email protected]

List of nontributors Claudio Fontanari Dipartimento di Matematica Universit` a degli Studi di Trento Via Sommarive 14, 38050 Povo, Trento, Italy Email: [email protected] Anthony V. Geramita Department of Mathematics and Statistics Queens’ University Kingston, Canada Dip. di Matematica Universit` a di Genova Via Dodecaneso 35, I-16146 Genova, Italy Email: [email protected]; [email protected] Alessandro Gimigliano Dipartimento di Matematica and C.I.R.A.M. Universit` a di Bologna Via Saragozza 8, I-40123 Bologna, Italy Email: [email protected] Elena Guardo Dipartimento di Matematica e Informatica Universit` a di Catania Viale A. Doria 6, 95100 Catania, Italy Email: [email protected] Brian Harbourne Department of Mathematics University of Nebraska Lincoln, NE 68588-0323 USA Email: [email protected] Anthony Iarrobino Department of Mathematics Northeastern University Boston, MA 02115, USA Email: [email protected] Atanas Iliev Institut of Mathematics Bulgarian Academy of Sciences Acad. G. Bonchev street 8, 1113 Soﬁa, Bulgaria Email: [email protected]

389

390

List of contributors

Paltin Ionescu Department of Mathematics University of Bucharest Academiei Str. 14, RO–70109 Bucharest, Romania Email: [email protected] Antonio Laface Dipartimento di Matematica “F. Enriques” Universit` a degli Studi di Milano Via Saldini 50, 20100 Milano, Italy Email: [email protected] Antonio Lanteri Dipartimento di Matematica “F. Enriques” Universit` a degli Studi di Milano Via Saldini 50, 20100 Milano, Italy Email: [email protected] Laurent Manivel Institut Fourier, UMR 5582 (UJF-CNRS) BP 74, 38402 St. Martin d’H`e res Cedex, France Email: [email protected] Emilia Mezzetti Dipartimento di Matematica e Informatica Universit` a degli Studˆıdi Trieste Via Valerio 12/b, I-34127 Trieste, Italy Email: [email protected] Rick Miranda Department of Mathematics Colorado State University Campus Delivery - 1801, Fort Collins, CO 80523-180, USA Email: [email protected] Roberto Mu˜ noz Departamento de Matem´ aticas y F´ısica Aplicadas y Ciencias de la Naturaleza Universidad Rey Juan Carlos C. Tulip´ an, 28933-M´ ostoles, Madrid, Spain Email: [email protected] Atsushi Noma Department of Mathematics, Faculty of Education and Human Sciences Yokohama National University Yokohama 240-8501 Japan Email: [email protected]

List of nontributors Raﬀaella Paoletti Dipartimento di Matematica e Applicazioni per l’Architettura Universit` a di Firenze Piazza Ghiberti 23, 50123 Firenze, Italy Email: raﬀ[email protected]ﬁ.it Ziv Ran Mathematics Department University of California at Riverside Riverside, CA 92521, USA Email: [email protected] Kristian Ranestad Department of Mathematics University of Oslo PB 1053 Blindern, 0316 Oslo, Norway Email: [email protected] Bernd Sturmfels Department of Mathematics University of California at Berkeley Berkeley, CA 94720-3840, USA Email: [email protected] Seth Sullivant Department of Mathematics University of California at Berkeley Berkeley, CA 94720-3840, USA Email: [email protected] Luca Ugaglia Via Petrarca 24 14100 Asti, Italy Email: [email protected] Adam Van Tuyl Department of Mathematical Sciences Lakehead University Thunder Bay, ON, P7B 5E1, Canada Email: [email protected] Alessandro Verra Dipartimento di Matematica Universit` a di Roma III Largo Murialdo, 00146 Roma, Italy Email: [email protected]

391

392

List of contributors

Bronislaw Wajnryb Department of Mathematics Technion 32000 Haifa, Israel Email: [email protected]