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London Mathematical Society Lecture Note Series. 155
Classification Theories of Polarized Varieties Takao Fujita Professor, Department of Mathematics, Tokyo Institute of Technology
The right of1he
Vnioceiry (Cambridge Sc pool and eeg eli manner of boo St
net granted by Henry VIII in 1534. The Univeceiry has poled endpabuiehed condnaouely since 1584.
CAMBRIDGE UNIVERSITY PRESS Cambridge
New York Port Chester Melbourne Sydney
Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 I RP 40 West 20th Street, New York, NY 10011, USA 10, Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1990
First published 1990 Library of Congress cataloguing in publication data available British Library cataloguing in publication data available
ISBN 0 521 39202 0 Transferred to digital printing 2003
Contents
Introduction Chapter 0. Summary of Preliminaries §1. Relative viewpoint
1
§2. Singularities
6
§3. Intersection theories
P. Semipositive line bundles and vanishing theorems §5. Birational classification of algebraic varieties Chapter I.
§1. §2. §3. §4. §5.
9
14
19
d-genus and Apollonius Method
Characterizations of projective spaces Basic notions in the Apollonius method Iteration of the Apollonius method Existence of a ladder Classification of polarized varieties of d-genus zero
22 25 28 31
38
§6. Polarized varieties of d-genus one: First step
43
V. Results of Lefschetz type §8. Classification of Del Pezzo manifolds §9. Polarized varieties of d-genus one: remaining cases §10. Polarized manifolds of d-genus two
55
62 77
83
Sectional Genus and Adjoint Bundles §11. Semipositivity of adjoint bundles §12. Polarized manifolds of sectional genus s 1
107
§13. Classification of polarized manifolds of a fixed sectional genus: higher dimensional cases
108
§14. Classification of polarized surfaces of a fixed sectional genus
114
§15. Polarized manifolds of sectional genus two
122
Chapter II.
Chapter
I.
93
Classification Theories of Projective Varieties
§16. Castelnuovo bounds
139
§17. Varieties of small degrees §18. Adjunction theories
147
Related Topics and Generalizations §19. Singular and quasi- polarized varieties §20. Ample vector bundles
151
Chapter N.
164 171
§21. Computer-aided enumeration of ruled polarized surfaces of a fixed sectional genus
176
References
184
Subject Index
202
Introduction
By a polarized variety we mean a pair ing of a projective variety on it.
(V, L)
consist-
and an ample line bundle
V
L
We will classify such pairs and describe their
structure as precisely as possible.
Needless to say, algebraic varieties are the main In this book, however, we
object in algebraic geometry.
mainly consider the pair itself.
rather than the variety
(V, L)
There are several reasons of taking this viewpoint.
First of all, polarization (or the linear system defined by it) is very important for describing the structure Fn is of a variety. For example, the projective space
described by a homogeneous coordinate system, namely a linear parametrization of
But it is by no
H0(IPn, 0(1)).
means easy to recognize a projective space without being For beginners it takes some thought
given a polarization.
to see that a twisted cubic in
is isomorphic to
F3
and this is because the polarization priori.
V
IF1,
is not given a
0(1)
Another example is the space parametrizing linear
F2's contained in a smooth hyperquadric in actually isomorphic to
IF5.
This is
Is that obvious to you ?
i3.
There are polarizations which are not very ample but Fn For example, let f: V -+ are useful for this purpose. be a finite double covering. but not very ample.
To recognize
to considering the pair map defined by
f*10(1)1.
L = f 0(1)
Then
V
(V, L), since
via
f
f
is ample,
is equivalent
is the rational
In this case the graded algebra
viii
Introduction
G(V, L) = ®t20H0(V, tL) (V, L)
is a hypersurface in a weighted projective space.
Although mL G(V, mL)
has a very simple structure and
is very ample for
is not as simple as
m >> 0, the graded algebra
Another classical
G(V, L).
example is Weierstrafi's normal form
y2
=
4x3 - g2x - g3,
which exhibits an elliptic curve as a weighted hypersurface of degree six in the weighted projective space In general, by the ampleness of G(V, L)
by taking
Proj.
L, V
IP(3,
2, 1).
is recovered from
This provides us with an
algebraic approach to algebraic varieties.
Thus, the study
of polarized varieties is indeed an algebraic geometry (but of course geometric approaches are also useful).
Second, we note that most interesting examples of algebraic varieties carry natural polarizations.
In classi-
cal projective geometry the hyperplane section gives such polarizations. 0-divisors.
In case of Jacobian varieties we have the
On the other hand, abelian varieties can be
recognized by their periods.
The interplay of these two
approaches yielded the theory of 6-functions.
We should
also observe the importance of polarizations in Torelli type theorems for many types of varieties, e.g.
K3-surfaces.
In moduli theories you find many examples of moduli spaces which carry polarizations constructed in a natural way. In my opinion, God did not make abstract varieties, but polarized varieties.
Third, we notice the relationship with the theory of singularities.
The vertex of a cone over a polarized varie-
ty gives a typical example of a singularity.
Thus, the
category of polarized varieties can be viewed as a subcate-
Introduction
ix
gory of singularities.
Moreover there is an evident simi-
larity of notions and theorems between them.
It is not easy
to formulate this analogy in a logically strict way, but nevertheless it is useful for heuristic purposes to have Recently classification theory
this parallelism in mind.
and the theory of minimal models of algebraic varieties has made remarkable progress (cf. [KMM], [Mor5]), where the study of certain singularities (terminal, canonical, log-terminal etc.) plays an important role.
I believe that
these objects, namely, polarized varieties, singularities and algebraic varieties, should be and will be studied together more and more in the future.
OK, let us now agree that we will study polarized varieties here.
But how ?
We want to recognize each polarized variety what it is.
(V, L)
as
However, there are too many types of polarized
varieties and there is no almighty method for all of them. We must employ various tools according to the types of (V, L).
Therefore we need first to classify polarized
varieties, so that we can distinguish those objects for which a certain approach works well.
Thus, philosophically,
each approach should have its own classification theory. This is why I wrote
"Theories"
in the title of this book.
Here I will present two such theories.
The first one may be called Apollonius method. given a polarized variety ILI
LD
pair
such that
(D, LD)
is the restriction of (D, LD)
(V, L), we take a member
Namely, D
of
is also a polarized variety, where L
to
D.
We first study the
which is of lower dimension, and then proceed
Introduction
x
to
(V, L)
type.
by making use of various results of Lefschetz
Thus, we use induction on
dim V
for proofs.
This
approach has a long history in classical geometry (this idea can be traced back to Apollonius, so I would like to call this method 'Apollonius method').
However, since we do not
assume that
L
is very ample, there is not always such a
good member
D
of
in general. The theory of
ILI
which is defined by
4(V, L) = n + Ln - h0(V, L)
d-genus,
where
n = dim V, provides several sufficient conditions for the existence of such a member
D.
Moreover, d-genus turns out
to be a powerful invariant for characterizing d
(V, L)
if
Chapter I is devoted to this theory.
is small enough.
In Chapter U we present the theory of adjoint bundles K + mL, where
K
is the canonical bundle of
a positive integer.
defined over
Here
V
K + mL in
Z
is
V.
has a special
(V, L)
is not nef, i.e.
(K + mL)Z < 0
for
The description of such a structure
is very precise when n - m is always nef.
m
is assumed to be smooth and
theory on minimal models, we see that
some curve
and
By a polarized version of Mori-Kawamata's
C.
structure when
V
is small, while
K + (n + 1)L
Using this theory we can classify polarized
manifolds by their sectional genus, which can be defined by the formula
2g(V, L) - 2 = (K + (n - 1)L)Ln-l.
We remark
that adjoint bundles have long been studied (especially when n = 2, m = 1) by the Apollonius method. many beautiful results in this way.
assumption is weaker (L
Sommese obtained
In our case the
is just ample) and we need
different techniques, but the conclusions are very similar to classical ones.
Introduction
xi
In Chapter I we survey Ionescu's classification theory of projective varieties (= varieties embedded in projective spaces) and Sommese's theory on adjunction process (when
L
is ample and spanned), but technical details of the proofs are omitted.
In Chapter N we discuss several further developments and generalizations, and at the end we present a computeraided enumeration of ruled polarized surfaces of a fixed sectional genus.
In addition, numerous related topics are mentioned in various parts, which will be useful especially for advanced readers who want to know about the present state of investigations.
There are many interesting and important theories which
we do not treat in this book - e.g. moduli theories amongst others.
However, I hope that our studies will provide a
good starting point for moduli theories in many cases.
The reader is assumed to have some knowledge of algebraic geometry, as can be found in Hartshorne's book [Ha4] for example.
For the sake of convenience, in Chapter 0,
I
give a brief summary of matters which are often used freely in this book.
For their proofs the reader should consult
appropriate references.
This book is not quite self-
contained in the sense that we usually present a result with an outline of proof of it, and refer to other papers for technical details.
I hope that this rather helps the reader
to have a good idea of what is most important.
I would be
happy if young people can enjoy the experience of applying modern general theories in concrete problems.
xii
Introduction
Acknowledgements
I would like to express my hearty thanks to all persons who have contributed to the preparation of this book.
I am
especially grateful
to Professors K. Kodaira, S. Iitaka and K. Ueno who introduced me to algebraic geometry, an area with numerous interesting problems;
to members of the Department of Mathematics at Komaba,
University of Tokyo, where I was able to make most of the studies in this book, enjoying excellent facilities with spiritual support;
to Professors L. Badescu, M. Beltrametti, R. Hartshorne, E. Horikawa, P. Ionescu, Y. Kawamata, A. Lanteri, R. Lazarsfeld, M. Miyanishi, Y. Miyaoka, S. Mori, H. Popp,
J. Roberts, F. Sakai, M. Schneider, T. Shioda, A. J. Sommese and many other mathematicians for comments, advice,
encouragement, support and/or correspondence with many interesting preprints and offprints;
to Professor Oshima, whose excellent word processor has made it possible for me to prepare the manuscript;
to the editor and the staff of the Cambridge University Press, who gave me the opportunity of publishing this book and gave numerous valuable comments on the first version of the manuscript; and to all persons who are willing to read this book. TO_
k 0- 0
Fj
ta,
Department of Mathematics Tokyo Institute of Technology Oh-okayama, Meguro, Tokyo 152 Japan
Introduction
x ii i
Notation, Convention and Terminology Basically we employ the customary notation in algebraic geometry as in [Ha4].
In most cases we work in the category of algebraic spaces defined over a fixed algebraically closed field A
They are assumed to be proper and of finite type over
By a variety we mean
unless specifically stated otherwise. an irreducible reduced space. variety.
A.
Manifold means a non-singular
Point means a .R-rational point, while scheme-point
means a scheme theoretical point.
Vector bundles are often identified with the locally free sheaves of their sections, and these words are used interchangeably.
Line bundles are identified with linear
equivalence classes of Cartier divisors, and their tensor products are denoted additively, while we use multiplicative notation for intersection products in Chow rings. The pull-back of a line bundle morphism
less.
L
via a or
LY, but
when confusion is impossible or harm-
This convention applies for pull-backs of other
objects too.
For example, if
Cartier divisor on of
is denoted usually by f L
f: Y -+ X
sometimes just by
on X
L
X, by DY
f
is birational and
D
is a
we mean the total transform
D (we never mean the strict transform). To avoid possible confusion, the canonical bundle of a
manifold M ary notation
is denoted by KM.
or
K(M)
Similarly
©(M)
normal bundle of a submanifold XCcM'
and
M
tangent and cotangent bundles of C
KM, unlike the customQ(M)
denote the
respectively.
But the
in M is denoted by
xiv
Introduction
Now we list notation used very often. Sing(X): the set of singular points of a space
X.
Supp(Y): the support of a subspace (or subscheme) Y
of
X.
OX: the structure sheaf (or the trivial line bundle) of
X.
hl(X, T): the dimension of the F
coherent sheaf
on
Y[L] (or Y(L)): = Y ®0 L
i-th cohomology group of a
X.
for a line bundle
L.
ILI: the complete linear system associated with
L.
[d]: the line bundle associated with a linear system
A.
BsA: the set-theoretic intersection of all the members of PA: the rational map defined by
A.
(Z): the Chow homology class of an algebraic cycle 8V: the dual bundle of a vector bundle Sk8: the
k-th symmetric product of pr-l-bundle
IP(8), 1X(8): the
p
A point
over
Ny
subspace
x
on
space
Ex/Ny.
Ha, H. Fa, 1P
:
,
X.
8.
of codimension one in
p E P(8)
at
over
Z.
8, r = rank 8.
corresponds to a linear
H(8): the tautological line bundle on H(8)p
8
associated with
X
A
EX = Ar
IP(8).
The fiber
is identified with the quotient
O[tH(8)]
is often denoted by
0(t).
the (pull-backs of) 0(l)'s of projective spaces indicated by the same Greek letters.
Chapter 0. Summary of Preliminaries
The matters in this chapter are well known among We review
experts and will be freely used in this book.
them for the convenience of the reader and hope that it serves as a sort of 'dictionary'. The reader is advised to skip this chapter at first, and then to refer to it if necessary. A paragraph (x.y) will be referred to as (O.x.y) in later chapters, but as (x.y) within this chapter. §1. Relative viewpoint
Many notions and theorems in algebraic geometry can be formulated in a relative situation. Formally this is a very simple (apparently trivial) process, but it provides a The 'Bible' of powerful approach for many applications. this philosophy is [EGA].
X
(1.1) Let
be an algebraic space defined over an
algebraically closed field any sheaf
d
For
char(t) = p 2 0.
OX-algebras of finite type, we can define
of
the relative spectrum
morphism
with
St
a: Z -a X
Z = 9'ftecX(d)
such that
with a natural affine
n*OZ = d (cf. [EGA;II,Sl] or
[Ha4;II, Ex. 5.17]).
(1.1.1) Example (Stein Factorization). be a proper morphism of spaces. coherent sheaf of
morphism for
0: Y - Z
f = n
-
0
0
.
'
is connected.
and
(1.1.2) Remark.
are varieties and if
X
is a
is a finite
O*OY = OZ.
Hence
Such a factorization
is determined uniquely by
Stein factorization of
= f*OY
f: Y -b X
Moreover there is a morphism
Z = SafecX(d).
f = n
.
r: Z
OX-algebras, so
such that
every fiber of
Then
Let
f
and is called the
f.
If
Y
f
is surjective, both
is normal, then
Z
X
and
is just the
Y
0: Summary of Preliminaries
2
X
normalization of (1.2) Let
in the rational function field of be a sheaf of graded
J = ®d20 91d
algebras of finite type over a space
OX-
Then we obtain a
X.
with a projective morphism
S = 3to/X(9)
space
Y.
n: S -+ X Here we
in a natural way (see [EGA;II,S3] for details).
give a few important examples. (1.3) Scroll of a vector bundle. Let let
be the symmetric algebra of
9'
a 1Pr-l-bundle over
on
n*XOt
and the pair
quotient bundle natural
of
H(8)
of the total space of the dual
Thus
corresponds to a quotient space
of dimension one.
n E -+ H(8)
is the
E, minus the zero section, modulo the
y E F(6)
En(y)
8.
P(8)
E,
Gm-action via the scalar multiplication.
each point
X
and is denoted by
IP(8)
for some vector bundle
(EV - O(X))/Gm
EV
is
This is
is called the scroll of
(P(8), H(8))
8 = OX[E]
If
and
or
1PX(8)
The corresponding line bundle is denoted by
0(1).
*
X
9hojX(9J)
- St8 for any t e Z.
called the tautological sheaf on
of
Then
on
Moreover there is an invertible sheaf
such that
IP(8)
S.
X, and will be denoted by
from now on.
IP(8)
r
be a locally free sheaf of rank
6
There is a surjection
of vector bundles on
identified with H(8) y.
naturally isomorphic to
Hy
IP(8), and
Hy
is
The kernel of this homomorphism is QIP(8)/X
the relative cotangent bundle of
0 H(8), where n.
S1F(8)/X
is
In particular we have
the following canonical bundle formula: K1P(8)
=
*(KX + det 8) - r H(8).
Remark. The word 'scroll' is used in different meanings
0.1: Relative viewpoint
3
in several papers.
(1.4) Fact (Relative rational map).
E be a vector bundle on X
a morphism,
line bundle on
f E --i L.
f= n
. p
f: Y -* X be
Let
and let
be a
L
Suppose that there is a surjection
Y.
Then there is a morphism p: Y - IP(E) * and p H(E) = L.
such that
For a proof, see [EGA;II,(4.2.3)] or [Ha4;II,7.12].
L
is said to be relatively very ample with respect to
f
(or
f-very-ample for short) if there is a vector bundle
E
as
above such that f-ample if
is a closed embedding.
p
mL
f-very-ample for some
is
is said to be
L
m > 0.
(1.5) Blowing-up (cf. [EGA;II,S8] or [Ha4;p.163]). Let
subscheme of
in
i
be a coherent sheaf of
i C
OX
X' = 9 oi.X(g)
of a variety and set
Let
X.
OX-ideals defining a 3d
Then
30 = OX.
9' = ED d20'gd, where
is called the blowing-up of X
is called the center of this blowing-up.
d-th power
be the
along
X
a: X'
C.
C
is a
birational morphism with the following properties. (1.5.1) The inverse image ideal the image of the natural homomorphism
3' on n*3
X', which is
-' OX
principal ideal defining a Cartier divisor E E
is called the exceptional divisor of
(1.5.2) n(E) = C
and
on
x'.
a.
X' - E = X - C.
(1.5.3) For any morphism
f: Y --' X
*
image of the homomorphism f 3 --' OY is a unique morphism
is a
g: Y -. X'
such that the
is invertible, there
with f = n
.
g.
This is called the universal property of the blow-up. (1.5.4) Let
Z
be a closed subvariety of X not in
C
0: Summary of Preliminaries
4
and let
be the image of
3Z
blowing-up
of
Z'
subvariety of
i
via
OX --+ OZ.
with respect to
Z
is a closed
3Z
X'.
is called the strict (or proper) transform of
Z'
of the blowing-up
ifold of
X, then
is smooth along
C
(1.6) Fact (Blowing-down).
Cartier divisor on
Xb
I
is a subman-
r = codim C.
E be an effective X
is smooth along E
is isomorphic to the scroll of a
(E, [-E] E)
vector bundle
C
The relative canonical
Let
Suppose that
X.
and
is isomorphic to the scroll
(E, OE[-E])
of the conormal bundle of C in X. KX'/X is [(r - 1)E], where bundle
and that
Z
X' -i X.
X
(1.5.5) If
space
Then the
over a manifold
containing
the blowing-up of Xb
C
Then there exists a
C.
X
as a submanifold such that
along
C, where
E
is
is identified
with the exceptional divisor.
For a proof, see [Ar] (or [Moi2], [Nkn] if
A = C).
This criterion is not true in the category of schemes.
(1.7) The relative version of the sheaf cohomology is the higher direct image.
For any continuous map spaces and for all
f: X -i Y
of topological
i s 0, we have natural functors
between the categories of sheaves of abelian groups which have the following properties.
Rif* : Xb (X) - l1b (Y)
(1.7.0) R0f* = f*. (1.7.1) For any exact sequence in
21b(X)
and for all
homomorphisms
0
8 -> S
%
0
i 1 0, there are naturally defined
8i: Rif*` - Ri+1f*d
such that
0.1: Relative viewpoint
0 -+ f*8 -, f*T - f*`
5
Rlf*8
-* ... -,
8
is exact in
-+ R1f*F -p R1f*C4 -ii Ri+lf*8 (1.7.2) If
is a point, then
Y
Ri-1 f* 8i-+ R1f*t8
Rif*.T ti Hi(X, S).
(1.8) Fact (Leray Spectral Sequence). and
g: Y
Rp+q(g.f)*3.
converging to
This means that there are objects
p, q
1 s r g -
and
drq: Er'q -
r
p+r,q-r+l
in
for
Zfb(Z)
such that and
Ker(dpq)/Im(dp-r,q+r-1)
r
Ep'q
together with homomorphisms
1) dp'q . dp+r,q-r+l = 0
Ep,q = r+l
Then
Y e Z(b(X).
E2'q = Rpg*(RqfY)
there is a spectral sequence with
all
f: X -+ Y
Let
be continuous maps and let
Z
Xb(Y).
r
p, q, r e Z,
for any
2) EP ,q -- Rpg*(Rqf*`3),
3) for each that
dp'q = 0,
p, q
there is an integer
dp-r,q+r-1
= 0
and
r(p, q)
Ep'q -- Ep'q
such
for any
r a r(p, q), and 4) there is a descending filtration of that
and
FO,k o Fl,k D
... D Ft,k = 0
Ep'q = EP'q
for all
p
If
f: X --+ Y
algebraic spaces and if Rif*.T
>> 0
T
at a point
Er
terms.
is a proper morphism of
is a coherent
is coherent on Y for every
(R1f*Y)y = 0
£
q, we say that the
and
spectral sequence degenerates at the (1.9) Fact.
for
EP,q
Fp,p+qIFp+l,p+q When
such
y
on
Y
i a 0.
if
For a proof, see [EGA;X,(4.2.2)].
OX-module, then
Moreover
dim f(y) < i.
0: Summary of Preliminaries
6
f: X -> Y
(1.10) Fact (Serre's Vanishing Theorem). Let
Then a line
be a proper morphism of algebraic spaces.
bundle
on x
L
f-ample if and only if, for every
is
coherent sheaf X
on
X, there is an integer for any
Rif*(S ® [tL]) = 0
that
Let
(1.11) Fact.
f: X - Y be a proper morphism as
Y be a coherent sheaf on
above and let for every point
x
on
i > 0.
and
t i N(3)
such
N(3)
X, the stalk
0Y f x -module via the mapping
0Y
gx
Suppose that,
X.
is a flat
f(x)
Then
0X x'
x(Xy, Y ) _ 2i(-1)1hi(Xy, Yy) is a locally constant y function in y e Y, where Xy = f 1(y) and 3 is the
restriction of Y
to
Xy.
Moreover
upper-semicontinuous function in is locally constant at vector bundle
E
hi(Xy, Sy)
for each
y
y E Y, then
I.
Rif*Y = 0[E]
over a neighborhood of
identified naturally with
is an If this
for some Ey
y, and
is
Hi(Xy, 3y ).
For a proof, see [EGA;1Q,S7] or [Ha4;1Q,12.9]. (1.12) We have the following criterion for flatness.
f: X -i Y
Fact (cf. [EGA;1V,(6.l.5)]). Let
morphism of algebraic varieties.
and X
is locally Cohen Macaulay.
Suppose that Then
f
Y
is smooth
is flat if and
dim f 1(y) = dim X - dim Y for every
only if
be a proper
y E Y.
§2. Singularities (2.1) Fact (Serre Duality).
dim M = n Exti(Y, WM) ,n-i(M,
Y)
and let
M be a manifold with
Let
wM = Dn be its canonical sheaf.
Then
is naturally isomorphic to the dual space of for every coherent sheaf
S
on
M.
Here we give a generalization of this fact on singular
0.2: Singularities
7
spaces for the convenience of later use.
For a more
complete treatment, see e.g. [Ha2].
X
(2.2) Let
and P
dim P = N (both X
with
may be non-complete).
Suppose that any irreducible component of dimension 0) S)X
n.
P
be a closed subspac6 of a manifold
We define
X
is of the same
qX = dxtp+q(OX, c)p).
is an invariant of X
Then
and is independent of the
choice of an embedding X (; P. unless
1) qX = 0
0 s -q s n.
and
2) dim Supp(EbXq) s q
3) When X
Supp(cXn) = X.
OXn = wX
is smooth,
and
OXq = 0
for
q # n.
The proofs are easy (see e.g. [F8]). (2.3) A subvariety
X
component of
if
of X
Z
is called an embedded
dim Z < dim X
some coherent subsheaf
,N
of
and if
for
Z = Supp(.N)
Here, as usual, Supp
OX.
denotes the set-theoretical support. Fact (cf. [F8;(1.11)]).
an embedded component of X (2.4) Definition.
on X
x
point
any component tion means
Z
if
of
X
q = dim Z.
Set
x g Supp(gxq)
for
0 s q < n
for any
with x e Z. q < k.
X
and
This condi-
is said to be
k-Macaulay if it is so at every point on
locally
is
k-Macaulay at a
is said to be
Supp(qXq)
Z
Z c Supp(cXq).
if and only if
dim Z s q - k
Then
X.
Then we have the following facts. 1) X
is locally 1-Macaulay if and only if X has no
embedded component. 2) Let
D
such that
be a subscheme of x JD = Ker(OX -+ OD)
of pure dimension n - 1 is invertible.
Then
D
is
0: Summary of Preliminaries
8
x e D
(k - l)-Macaulay at 3) Let D
D
Then
be as above.
k-Macaulay at
is
If X X
k-Macaulay at
is
x.
k-Macaulay at
is
x
if
x.
These are easily proved.
From these facts we deduce
further the following. (2.5) Fact.
X
depth(OX'r) 2 Min(k, height p)
with x E
x
k-Macaulay at
is
if and only if
for any scheme-point
p
C X.
This is called usually 'Serre's condition Sk'. 0X
Thus,
is a Cohen Macaulay local ring if and only if
x
x, namely
n-Macaulay at
(9xq)X = 0
for
q # n.
X
is
9 xn
is
the dualizing sheaf in this case. Let
(2.6) Fact.
assume that some
X
k < n.
k-Macaulay on
be a linear system on X
A
k-Macaulay on an open set
is
Then any general member
of
D
and
U c X d
for
is
D n U n (X - BsA).
This follows from the observation in (2.4). (2.7) Fact.
Let
X be a locally
k-Macaulay variety
and let H be an ample line bundle on locally free sheaf and
£
8
on
X.
Then, for any
X, H1(X, 8[-tH]) = 0
i < k
for
>> 0.
For the proof, use the spectral sequence with Ep,q = Hp(X, 9X ® a")
H-p-q(X, S)v,
converging to
we obtain by using (2.1).
A variety
(2.8) Fact (Serre's criterion).
normal if and only if it is locally singular locus is of codimension
V
is
2-Macaulay and its
> 1.
which
9
0.3: Intersection theories
For a proof, see e.g. [EGA;W,§5.8].
As a corollary we
obtain the following. (2.9) Fact (cf. [Sei]).
V such that
a normal variety that
Let
char(t) = 0
or that
A
A be a linear system on Suppose further
BsA = 0.
Then any
is very ample.
connected component of any general member of
A
is normal.
(2.10) Finally we recall a result in Hironaka's desingularization theory [Hirn]. Definition.
n: X' - X be a blowing-up of X
Let
This is said to be admissible if the center
in (1.5).
is smooth and
X
is normally flat along
as
C
C.
We omit the precise definition of normal flatness.
Roughly it means that the algebraic nature of the singularities of
X
relative to Let
(2.11) Fact.
closed subset of
V.
C
varies continuously along
char(l) = 0.
is a finite sequence V' = Vk - Vk_l of admissible blowing-ups such that inverse image of
-
V'
Then there
-+ V1 - VO - V is smooth and the
V' is a divisor whose support has
in
S
S be a
V be a variety and let
Assume that
C.
only normal crossing singularities.
§3. Intersection theories (3.1) Let A.
19
be the category of algebraic spaces over
We use a theory which has the following features:
a) There is a contravariant functor
A'(*)
category of graded rings with unit.
Thus, for any morphism
from
18
to the
*
f: X --i Y
in
A'(Y) -p A'(X)
19, we have a ring homomorphism
such that
The multiplication in
f
= A'(f):
f*(Ad(Y)) c Ad(X), f*lY = 1X.
A'(*)
is called the cup product.
0: Summary of Preliminaries
10
b) There is a covariant functor
A.(*)
has a natural structure of graded A'(X)
action of
on A.(X)
A.(f): A.(X) --> A.(Y)
is a homomorphism of
where
A.(X)
d) Ad(X) = Ad(X) = 0 Y
unless
in
T, the map
via the map
f*
def
A'(Y)-modules, A'(f).
0 s d s dim X.
is a point, there is an isomorphism
we have
The
is called the cap product.
f: X - Y
acts on
X,
A'(X)-module.
c) For every morphism
A'(Y)
to the
Moreover, for each space
category of abelian groups. A.(X)
T
from
Moreover, if
deg: A0(Y) z Z.
degX = deg > AO(f): AO(X) -p Z, where
So
f: X -+ Y
is the trivial map.
e) For each complete variety natural element
f: V -- W
V
morphism with
in
An(V).
n = dim V, there is a
Moreover, for any morphism
of varieties of the same dimension
f*(V) = d(W) f) If
(V}
V with
for the mapping degree
d
of
is a smooth complete variety with
8: Ad(V) - An_d(V)
(V)
is bijective for any
is said to be Poincare dual of
the inclusion
n = dim V, the
a e Ad(V)
d.
a
if
and
8(a)
A subvariety
t(Z) = b(a)
for
c: Z g V.
(3.2) Example.
H2d(Xan; Z)
f.
defined by the cap product
are said to be Poincarb duals of each other. Z
n, we have
and
When
.
= C, by setting
Ad(X)
=
Ad(X) = H2d(Xan; Z), we obtain a theory
with the above property.
(3.3) There exists a theory as in (3.1) such that Ad(X)
X
is the group of algebraic cycles of dimension
modulo rational equivalence (see [Ful]).
A'(X)
is called the Chow ring of
X.
d
In this case
on
0.3: Intersection theories
11
(3.4) Given a theory as in (3.1), we have a theory of Chern classes of vector bundles as follows. E
For every vector bundle
in A'(X)
c(E)
X, an element
on a space
is defined in a natural way so that the
following 'axioms' are satisfied. a) If
c.(E) = 0
and
for
Chern class while
c3(E) a A1(X), then
with
c(E) = 21 ci (E)
j > rank(E).
is called the total
c(E)
is called the
ci (E)
c0 = IX
j-th Chern class.
b) If 0 --+ E - F - G - 0 is an exact sequence of vector bundles on X, then c(F) = c(E)c(G). for any morphism
c) c(f E) = A'(f)(c(E))
d) For any Cartier divisor 2 µiZi
on a variety
D
inclusion map
(Zi)
A.(ti)(Zi)
stands for Then
ti: Zi c, V.
cI([D]), where
D =
is the Poincarg dual
(D)
cycle in X bundles on
Then
Z = 2 µiZi
Let
of dimension X.
For example
for the dual bundle
(3.5) Definition.
d
EV.
be an algebraic
L1, -1 Ld
and let
degX((cl(L1) A
(Z) = 2 µi(ci)*(Zi)
D.
a) - d) we can deduce well known
formulas for Chern classes, as in [Hirz]. (-1)ici (E)
for the
is the line bundle defined by
[D]
From these properties
where
V, let
be the prime decomposition as a Weil divisor and set
(D) = 2 µi(Zi), where
of
f: Y --p X.
cup and cap products respectively.
the intersection number of
L1,
be denoted by
If
a Z,
A
as above, and
A
and
denote
This integer is called
Ld with X
be line
Z, and will
is a variety and
we often write simply L1 Ln, where
n = dim X.
Z = X,
0: Summary of Preliminaries
12
We have If
(3.4.d).
for any Z
is a curve, then
of the line bundle
on
Lz
(3.6) Definition.
total Segre class s(E)c(EV) = 1X s1(E) = cl(E), (3.7) Fact.
bundle
E
of rank
r
n-j
(V)
for any
E
on
X, the
is defined by the formula
By easy computation we get
S2(E) = C1(E)2 - c2(E),
be the natural map. R
Z.
A'(X).
Let
by
ILnI
is the usual degree
L(Z)
For any vector bundle
s(E)
in
D E
(P, H)
be the scroll of a vector
on a variety
V
Therefore, for any C1(H)J+r-1)(P)
a n si(E)(V) = (x*a n
x: P - V
and let
x*(C1(H)J+r-1(PJ)
Then
j s 0.
and so on.
= si(E){V} E a E An-j(V), a Z.
Using this formula and (1.5.5), we can calculate
various intersection numbers on the blowing-up
X
of
X'
as in (1.5).
(3.8) We have the following theory of Todd classes for any reasonable theory having the properties in (3.1). For each coherent sheaf element
Y
on a complete space
r(S) = 21 r (Y) E A.(X)OQ
is defined in a natural
way so that the following 'axioms' are satisfied. called the total Todd class of
X, an
T(Y)
Y.
a) If 0 -i T -- % --a X --p 0 is an exact sequence, then T ('S) b)
= T (Y) + T(X).
idim
c) If
for any complete variety
V(0V) _ (V)
f: X - Y
is a proper morphism, then in
V.
f*r(S) _
A.(Y)@ = A.(Y) 0 Q.
(3.9) The following 'theorems' follow from the above
is
0.3: Intersection theories
13
'axioms' in (3.8).
1) degX(i0(f)) = X(Y) der
2jz0(-1)Jhj(3).
2) r(3 0 L) =
for any line bundle
More generally, we have vector bundle
E
for any
:(3' 0 E) =
and its Chern character
a A'(X)@,
See [Hirz] for its precise definition.
r = rank(E).
3) If M is.a smooth complete variety, then expressed explicitly by the Chern classes tangent bundle of
Note that
ch(E).
ch(E) = rX + c1(E) + (cl(E) 2 - 2c2(E))/2 +
where
L.
M.
i(OM)
ci
is
of the
For example, if n = dim M, then
sn = (M),
c(M)/2,
to-1
sn-2 = (c12 +
(Noether's formula),
to-3 =
and so on.
See [Hirz] for a general formula for
t1(0M).
4) Combining 1), 2) and 3), we obtain a formula, which is called the Riemann-Roch-Hirzebruch formula, expressing X(OM[E])
of a vector bundle
E
on a manifold M
in terms
of Chern classes.
(3.10) The above theory was established by Hirzebruch
Grothendieck generalized the theory to
in the case (3.2).
an abstract context. (3.11) Let
L
results in (3.9),
polynomial in (d/n!)tn
for
t
be a line bundle on a variety
V. By the
X(V, tL) = 2J>0(-1)ihJ(V, tL)
is a
d = Ln{V).
is a polynomial in the Ln.
n = dim V, and its main term is
of degree
More generally,
ti's
X(V,
for any line bundles
L1,
This fact can be proved by an elementary argument
0: Summary of Preliminaries
14
without using the intersection theory as in this section. Hence we can define the intersection number
by
using the coefficients of the above polynomial and deduce various properties of it (cf. [Kl]).
§4. Semipositive line bundles and vanishing theorems (4.1) Definition.
An element of
a Q-bundle on
X.
is denoted by
[L]Q, or often just by
For a line bundle
Pic(X) 0 Q
is called
L, the Q-bundle L
L ® 1
when confusion is
impossible or harmless.
Div(X) 0 Q
Similarly, an element of
Q-divisor (or IQ-Cartier divisor), where
group of Cartier divisors on Q-divisor
D
is denoted by
The
X.
[D]@,
is called a Div(X)
is the
Q-bundle defined by a or just by
[D]
D.
The notion of algebraic Q-cycles is defined similarly. (4.2) For any
@-bundle
L
and a curve
define the intersection number LC e Q
in
C
X, we
in a natural way.
L
is said to be numerically semipositive (or nef, for short) if
for any curve
LC 2 0
C
in
X.
'nef' is NEVER
Thus,
an abbreviation of 'numerically equivalent to an effective Q-divisor'.
positive
L
m E Q
is said to be ample if
and an ample line bundle
is said to be semiample if some line bundle line bundle BsjmFj = 0
F
L = m[H]@
H
L = m[H] Q
such that
Similarly, L
H.
for some
BsIHI _ 0.
for some
m > 0
and
In particular, a
is semiample as a c-bundle if and only if
for some positive integer
m.
The following results are well known for line bundles,
and they are obviously true for any Q-bundle
L.
(4.3) Theorem (Nakai's criterion, cf. [Nk], [Ha3]).
L
0.4: Semipositive line bundles and vanishing theorems
Ldim W{W} > 0
is ample if and only if
W of X with
15
for every subvariety
dim W > 0.
(4.4) Theorem (Kleiman's criterion, cf. [K1], [Ha3]). L
is nef if and only if
subvariety W of
X.
(4.5) Remark.
A + L
If
A
L
is a limit of ample
is nef, then
L
Hence any nef
Q-bundles:
This is a useful observation.
L = lima-++0(L + mA).
(4.6) Index Theorem.
nef
is ample and
is ample by the results above.
Q-bundle
for every
Ldim W{W) 2 0
v-bundles on a variety yij
711722 s 7122, where
Lit L2, Al, ***1 An-2
Let
with
V
be
Then
dim V = n.
is the intersection number
LILjA1...An-2(V). Outline of proof.
Ak's and
assume that the
may assume that the members of
By the above limit method, we may Li's
are all ample.
Moreover we
Taking general
Ak's are very ample.
we can reduce the problem to the case
IAkI
n = 2, which is the classical index theorem on surfaces. (4.7) The following simple lemma is useful in various situations. Lemma.
Let
Lit Hi (i - 1,
on a variety X Li - Hi = [Ei]
,
n) be nef '-bundles
with n = dim X. Suppose that, for each for some effective
Q-divisor E.
i,
Then
Ll ... Ln a H1 ... Hn, Proof.
Then
Let
Ei = 2 PaWa
L1 ... LiHi+l ... Hn - L1 ... Li-1Hi ... Hn =
L1 ... Li-lHi+1 ... Hn{Ei) = a 0
be the prime decomposition.
by (4.4).
So
L1...Ln 2
pa L1 ... Li-1Hi+1 ... Hn(Wa) 2
...
a H1...Hn.
0: Summary of Preliminaries
16
(4.8) Definition. L = m[F] Q
A Q-bundle
for some
m > 0
is said to be big if
L
F
and some line bundle
that the rational map defined by
JF)
its image (but we do not require
BsIFI = 0).
such
is birational onto Of course,
ample Q-bundles are nef and big, but the converse is not true in general.
We have the following criterion for the
bigness.
A Q-bundle
(4.9) Theorem (Kodaira's Lemma).
projective variety is big if and only if L - E for some effective Q-divisor
L
on a
is ample
E.
.For a proof, see [F8;(2.8)] or [KaMM;0-3-3]. (4.10) Theorem.
of dimension
n.
Let
Then
L
be a nef @-bundle on a variety
is big if and only if
L
Ln > 0.
For a proof, see e.g. [Fl5;(6.5)]. (4.11) We recall vanishing theorems of Kodaira type. Theorem ([Kodi]).
Let
A be an ample line bundle on a
compact complex manifold M whose canonical bundle is then
for
H1(M, K + A) = 0
K.
i > 0.
This famous result has been generalized by several authors.
Here we present one such generalization.
(4.12) Theorem (cf. [Kaw2], [Vi2]).
bundle on a smooth variety n = dim M,
M, defined over
whose canonical bundle is
there is an effective
Q-divisor D
decomposition D = 7 piDi i) 0 s µi < 1
for each
ii) Supp(D) = U Di
K.
A be a line
Let
C
and with
Suppose that
on M with prime
such that
i,
has only normal crossing singularities,
iii) A - D is nef and big.
0.4: Semipositive line bundles and vanishing theorems
Then
Hi(M, K + A) = 0 Remark.
for any
17
i > 0.
There are generalizations of the above result
in which the pair
is allowed to have certain mild
(M, D)
singularities (cf. [KMM] = [KaMM]). (4.13) Corollary.
A be a nef and big line bundle
Let
on a manifold M defined over
Then
C.
H'(M, -A) = 0
for
i < dim M. (4.14) Corollary.
Let
V
normal projective variety
be a nef line bundle on a
L
over
for some ample line bundle H on
C V.
such that Then
Outline of proof. We may assume that H
L2Hn-2 > 0
H1(V, -L) = 0. is very ample.
Using (2.7) and (2.9) we reduce the problem to the case n = 2, where (4.13) applies by the technique in [Muml]. (4.15) The vanishing theorem (4.12) is the most important technical tool in Kawamata's theory on minimal models (cf. [KMM]).
The following result, which originated from an
idea in [Rd2], is very useful for our study. Fibration Theorem.
and suppose that morphism
L = K + A
0: M --+ W
on W such that
M, K, A, D
Let
is nef.
be as in (4.12)
Then there is a
together with an ample line bundle
0 H = L
and
H
0*0M = OW.
This is just a reformulation of the Base Point Free Theorem (see e.g. [KMM;3-1-1]).
In the usual formulation,
it is asserted that
for every sufficiently
large integer
£
BsJLLl = 0
under the assumption of the theorem.
Obviously this follows from the existence of as above.
0, W
and H
The proof of the converse is as follows.
Take p > 0
such that
BsJpLJ = 0.
Let
M -* X c ]PN
0: Summary of Preliminaries
18
be the morphism defined by
Stein factorization of it and let 0X(1)
to
Then
W.
Every fiber
P
q and
defined by
p
are coprime.
Then
JqLj.
q
through W since W
is normal.
*
pH
(pH) = pL =
Thus we have
Hence
*
LF
Hence H
and H
0, W
factors
p
since p and
is numerically equivalent to
P.
is
comes from
qL
L = 0 H for some H E Pic(W)
are coprime.
BsjgLI = 0
is a point since
From this we infer that
So
0 P = pL.
such that
numerically trivial.
Pic(W).
is finite.
p: M -+ Y be the mapping
Let
p(F)
be the
be the restriction of
is connected and
0: M ---+ W
Now, take another positive integer and
X
is ample since W -+ X
P
F of
M -+ W
IpLI, let
is ample since
P
q
since is ample.
P
as desired.
The above argument is found in the proof of the Contraction Theorem (cf. [KMM;3-2-1]).
Combining this
theorem and the Cone Theorem (cf. [Mor3;(1.4)], [KMM;4-2-1]) we obtain the following. (4.16) Theorem.
suppose that
morphism
L = K + A
0: M -- W
1) 0*0M = 0W
and
2) for any curve Z
is not nef.
and a curve
from
z
in
F e Pic(M),
Pic(W), and Remark.
The ray
R
Then there is a
on M
such that
LR < 0, m,
is a point if and only if
O(Z)
is numerically equivalent to
3) for any
be as in (4.12) and
M, K, A, D
Let
mR
m,
if and only if
FR = 0
FR > 0
for some
if and only if
F
F
comes O-ample.
is
Such a fibration is not unique in general.
(mR)m>O
generated by
R
in the group of numerical
equivalence classes of 1-cycles with real coefficients is called the extremal ray represented by
R, and
0
is called
19
0.5: Birational classification of algebraic varieties
the contraction morphism of this exteremal ray.
R
is
called an extremal curve belonging to this ray.
§5. Birational classification of algebraic varieties (5.1) Definition (cf. [Iii], [1i2], [U], [F8]).
be a line bundle on a variety if
m > 0, and
for any
ImLI = 0
otherwise, where
PImLI
x = Maxm>0(dim PImLI(v))
is the rational map defined by ImLI.
is sometimes denoted just by
x(L, V)
KV
smooth and if
x(KV, V)
V.
x(L, V) k 0, we have
If
is
and is denoted by
V
This is a birational invariant of (5.2) Remark.
V
If
x(L).
is its canonical bundle, then
is called the Kodaira dimension of x(V).
L
x = x(L, V) = --
We set
V.
Let
x(L, V) =
tr.deg(®tz0H0(V, tL)) - 1 = limsupt,_(log(h0(V, tL))/log t). (5.3) Fact (Iitaka's Fibration Theorem).
line bundle on a normal variety there is a birational morphism 0:
-> W such that
with
V
such a fibration
0
F
Then
and a morphism
dim W = x(L, V)
for any generic fiber
x(LF, F) = 0
be a
L
x(L, V) k 0.
a: 1`---> V
0*0V = OW,
Let
of
0.
and
Moreover
is determined uniquely up to a
birational equivalence. 0
is called the Iitaka fibration of
smooth and
L
L.
When
is
V
is its canonical bundle, it is called the
canonical fibration of
V.
(5.4) litaka's classification Program.
Given a smooth variety n = dim M.
If
fibration by (5.3).
M, we have
0 < x < n, M
x(M)
0,
1,
admits a non-trivial
We hope that the investigation of such
manifolds can be reduced to the study of lower dimensional
20
0: Summary of Preliminaries
cases.
Hence we will study the three cases
x =
0, n.
When n = 1, this corresponds to the classification of Riemann surfaces by their universal coverings. If
x < 0, Mori-Kawamata theory will be useful.
If
x = 0, the Albanese mapping gives more precise information on the structure of
M (cf. [Kawl]).
The case
x = n
is
the most difficult to study.
(5.5) Classification of smooth algebraic surfaces [Beal], [BaPV], [112], [Zr], [Sha], [Mum2], [B0M],
...
(cf. ).
In the birational classification theory of surfaces, we
may consider only relatively minimal surfaces, i.e. surfaces having no (-1)-curve.
For such a surface
K
1) The canonical bundle
S
we have:
is nef if and only if
x(S) k 0.
2) If
x(S) < 0, then
over a curve.
3) If BsjmKI = 0 4) If in
In particular
S
x(S) 2 0, then K for some x(S) = 0,
Pic(S), then
If K $ 0
S = F2
in
S
or
S
is a
P1-bundle
is ruled.
is semiample, which means
m > 0.
K
If K = 0
is numerically trivial.
is an abelian variety or a K3-surface.
Pic(S), then
S
hyperelliptic surface, unless
is an Enriques surface or a char(St) = 2
or
3.
There are several ways of defining the notion of a K3-surface.
Roughly speaking, it is a surface having the
same numerical invariants as smooth quartics in
An Enriques surface (when the form
where
char(.) # 2) is a surface of
is a K3-surface and
point free involution of .
P3.
t
is a fixed
0.5: Birational classification of algebraic varieties
A hyperelliptic surface (when surface of the form and
char(&) # 2, 3) is a is an abelian surface
2S
is a cyclic group acting on
points.
Sf/, where
21
By the Albanese mapping, S
without fixed
has the structure of an
etale fiber bundle over an elliptic curve where every fiber is isomorphic to a fixed elliptic curve.
example in [Beal]) S 5) If
Sometimes (for
is called bi-elliptic surface.
x(S) = 1, S
is an elliptic surface (or possibly
a quasi-elliptic surface when
char(SF) = 2, 3).
The ellip-
tic fibration is unique and is the canonical fibration. 6) There are many types of
S
with
x(S) = 2.
They
are called surfaces of general type.
(5.6) We have many results which can be viewed as 3-dimensional versions of those in (5.5). [Vil], [W], [Mor4], [Miy2],
See [U], [Ii2],
.
When n k 4, it is still dificult to get satisfactory results.
However, there are many interesting problems.
Chapter I. d-genus and the Apollonius Method In this chapter we present the classification theory by The main technique is the hyperplane section method using induction on the dimension. d-genus.
§1. Characterizations of projective spaces First of all, as the most typical example of the Apollonius method, we recall the proof of the following. (1.1) Theorem (cf. [Gor], [KobO]).
Proof.
(V, L) = (1n, 0(1)).
We use the induction on
is easy, so we consider the case of
Then
ILI.
and
V has only Cohen Macaulay
Assume that
Then
singularities.
be a
(V, L)
n = dim V, Ln = 1
polarized variety such that h0(V, L) 2 n + 1.
Let
The case n = 1
n.
Take a member
n k 2.
Ln-1D = Ln = 1.
If
D = D1 + D2
D
for
Ln-1Di
Di, then
non-zero effective Weil divisors Ln-1D 2 2
since
L
is ample.
and reduced as a Weil divisor.
for the homomorphism
By the above observation scheme
D
has a
On the other hand, D
V
such that
8: OV[-L] - OV Supp(D)
and
is irreducible
D
Therefore
natural structure as a subscheme of OD
> 0
Coker(8)
defining
D.
is irreducible and the
is reduced at its generic point.
Cohen Macaulay singularities, hence so does
V D.
has only Therefore
D
has no embedded component and is reduced everywhere, so
D
is a variety.
0V -+ OD
0.
Now we use the exact sequence Taking
®L
and then
0 --i OV[-L]
HO, we get an exact
___+
sequence
0 --+ HO (V, OV ) -* HO (V, L) - HO (D, L D ).
h0(D, LD)
z
h0(V, L) - 1 s n.
hypothesis to
Hence
Applying the induction
(D, LD), we obtain
(D, LD) - (lpn-1, 0(1))
23
§ 1: Characteristics of projective spaces
Therefore the restriction map h0(V, L) = n + 1. 0 H (V, L) - H 0(D, LD) is surjective. For any point x and
D, there is
H0(V, L).
from
Hence
T(x) # 0, and
with
T E H0(D, LD)
its global sections. such that
So
L = p*0(1). If
the restriction of
L
to
p-1(y)
Therefore
So
L.
p: V _ n y e Pn
is trivial, which is a finite
p
deg(p) = Ln = 1, hence
We also have
birational.
is spanned by
for some
dim p-1 (y) > 0
contradicts the ampleness of morphism.
OV[L]
gives a morphism
ILI
comes
T
Since D E ILI, this
x % BsILI.
BsILI = 0, or equivalently,
implies
on
p
is
is an isomorphism by virtue of
p
Zariski's Main Theorem.
The exact sequence
(1.2) Technical remark.
OV[-L] - OV If
V
OD --> 0
0 ->
is important in the above proof.
is a non-reduced scheme, it may have embedded compo-
nents and
OV[-L] -+ OV
is not always injective.
Without
the Cohen Macaulay assumption, we have the above exact sequence at the first step, but then
may have embedded
D
components, and we cannot go further.
However, working
carefully in a more general setting, we can eliminate this assumption (cf. §5).
(1.3) The theorem (1.1) yields the following interesting result. Corollary (cf. [KobO]).
complex manifold such that
K
Let
-(K + nL)
is the canonical bundle and
be a polarized
(M, L)
is ample, where
n = dim M.
Then
(M, L) =
(]pn, 0(1)). Proof.
Hi(M, tL) = 0
By Kodaira's vanishing theorem (0.4.11) we have for
i > 0
and
t 2 -n.
Therefore
X(t) =
I. A-genus and the Apollonius method
24
2i(-1)'hi(M, tL) = 0
is a polynomial in
x(t)
Moreover
vanishing theorem.
and
by the Riemann
Ln = 1
So
.
Since
x(0) = 1.
n, this implies
of degree
t
(t + n)/n!
x(t) = (t + 1) Roch theorem.
-n s t < 0
for
h0(M, L) = x(1) = n + 1
by the
Hence (1.1) applies.
(1.4) Corollary.
manifold such that
Let
be a polarized complex
(M, L)
Ln = 1
KLn-l < -n.
and
(M, L) =
Then
(1Pn, 0(1)). We have
Proof.
H'(M, -tL) = 0
by Kodaira's vanishing theorem. for
ho (M, K + tL) = 0
Hence
X(-t) = 0
t t n
x(t) = (t + 1) ... (t + n)/n! (-1)-nx(-n-1) = 1.
Then
Ln-1D
Let
since
hn(M, -tL) =
(K + tL)Ln-1 < 0.
h0(M, K + (n+l)L)
Hence
.
be a member of
D
i < n
and
This implies
KLn-1 + n + 1 < 1.
=
D = 0, so
implies
Moreover
0 < t s n.
for
t > 0
for
Since
K = -(n + 1)L
in
_
IK + (n+l)LI.
is ample, this
L
Pic(M).
Therefore
(1.3) applies. Remark.
In Chapter II, we will generalize these results
by applying Mori-Kawamata theory. (1.5) Corollary.
If a K&hler manifold
M
is a
deformation of IPn' then M = IPn. Clearly
Proof.
Hodge theory. Let
H2(M; Z) _ Z
Therefore
M
that there is a deformation family x e X
such that
x # o.
Then
LX = 0(1) h0(M, L)
M. = M
L = Lo
for any a
for some
Pic(M). (Mx) o
and
extends to a family x # o.
We have
h0(Mx, LX) = n + 1
by
h0'2(M) = 0
is projective and
be the ample generator of
L
and
Pic(M) = Z.
We may assume
parametrized by MX = Fn (Lx)
Ln = LXn = 1
for any
such that and
by the upper semicontinuity
§2: Basic notions in the Apollonius method
25
Hence (1.1) applies.
theorem (0.1.11).
The rigidity of
(1.6) Remark.
under arbitrary
IP3
deformation was proved by Nakamura [Nkm] by a similar Of course, his argument is much more complicated
method.
than ours, since he cannot assume the ampleness of
L.
§2. Basic notions in the Apollonius method (2.0) Let n
D
and let
be a polarized variety of dimension
(V, L)
be a member of
Suppose that
ILI.
irreducible and reduced as a subscheme of is called a rung of
D
case
a polarized variety of dimension (V, L)
study
is reflected in that of via
(V, L)
n - 1. (D, LD),
(D, LD)
is
The structure of
hence we can
using induction on
(D, LD)
is
In such a
V.
The pair
(V, L).
D
This is
n.
the main idea of the Apollonius method. (2.1) Let
D
be a rung of
have an exact sequence
®[tL], we obtain
is determined by
xJ .
e Z
xJ de=1
= 1, we write
since x,7(V,
X(V, tL).
x(t) E Z
L)
j a 0.
Then
X(V, tL) =
for each
t.
These integers
are important invariants of
t[J] - (t - 1)[J] = jt[J-1],
for every
we have
(V, L).
Since
xj(D. LD) = xj+l(V' L)
By the Riemann Roch Theorem we have
xn(V, L) = Ln, which is called the degree of denoted by
X(D, tLD) _
t[J] = (t+j-l)!/(t-l)! = t(t+l) ... (t+j-1)
Setting t101
Then we
Thus, the Hilbert polynomial of
x(V, tL) - X(V, (t - 1)L).
and
as above.
0 -+ OV[-L] - 0V -+ OD --' 0.
Taking cohomologies after
(D, LD)
(V, L)
d(V, L).
We set
(V, L)
g(V, L) = 1 - xn-1(V, L)
call this the sectional genus of
(V, L).
Thus
and is and
d(D, LD) =
I.0-genus and the Apollonius method
26
If
g(D, LD) = g(V, L)
and
d(V, L)
for a rung
D
of
has only Gorenstein singularities and K
V
(V, L).
is the
canonical bundle, we have the following 1)L)Ln-1.
2g(V, L) - 2 = (K + (n -
sectional genus formula:
This follows from the Riemann Roch Theorem.
When
n = 1, we
g(V, L) = h1(V, 0V).
have
(2.2) The d-genus of
is defined by the formula
(V, L)
4(V, L) = n + d(V, L) - h0(V, L). We have for a rung
d(V, L) - 4(D, LD) = dim Coker(r) t h1(V, 0V) D
(V, L), where
of
H0(V, L) -. H0(D, LD).
is the restriction map
r
In particular
and the equality holds if and only if such a case
tivity of
4(D, LD) t d(V, L) is surjective.
r
is said to be a regular rung.
D
In
The surjec-
is important from
rt: H0(V, tL) -+ HO(D, tLD)
the ring theoretic viewpoint. (2.3) Theorem (cf. [F4;(3.1)] & [F2;Prop. 2.2]).
be a polarized variety and let
(V, L) JaLl
defined by
D
be a member of
for some
6 E H0(V, aL)
Let
a > 0.
Let. 1,
k be homogeneous elements of the graded algebra G(V, L) = ED tk0H0(V, tL)
images in
G(D, LD)
is generated by
Proof.
Let
generated by Then the
8
A
8
and the
j's.
Set
Then
G(V, L)
At = A n H0(V, tL).
G(D, LD)
Hence it suffices to show
exact sequence
Suppose that
j's.
be the subalgebra of
and the
be their
nj's as an algebra.
rt(At) = HO(D, tLD), for r/j's.
nk
,
nl'
via the restriction.
is generated by the
G(D, LD) G(V, L)
and let
is generated by
Ker(rt) C At.
The
H0(V, (t - a)L) -* H0(V, tL) -* H0(D, tL)
§2: Basic notions in the Apollonius method
Ker(rt) =
implies
(2.4) A line bundle
t.
is said to be simply generated
L
In this case the rational map
H0(V, L).
is generated by
G(V, L)
if the graded algebra
just the map
Therefore we infer
(t - a)L).
by induction on
At = H0(V, tL)
27
defined by
pItLI
is
ItLI
followed by the corresponding Veronese
pILI
Hence
L
is very ample if it is ample and
simply generated.
L
is sometimes said to be normally
embedding.
generated in such a case.
Using this notion we get the following useful result. (2.5) Corollary.
surjective and if
LD
If
-
H0(V, L)
H0(D, LD)
is
is simply generated, then
is
L
simply generated.
(2.6) In the situation in (2.3), G(D, LD) phic to the quotient of
modulo
G(V, L)
is isomor-
Moreover we
6.
have the following.
Theorem (cf. [F4;(3.2)] & [F2;Prop.2.4]).
D, a, j and
be as in (2.3).
,j
G(D, LD)
are derived from
gl(')1,
'
.
V, L,
aj = deg j =
Set
Suppose that all the relations among the
deg i)j.
'(k) =
r(j's in _
,7k) = 0, where the gi's are weighted homogeneous
gr('(1'
'
polynomials in
Y11 *'*1 Yk
with
, Xk
such that
with
deg X0 = a
f1(0, Y1'
'
Yk)
and all the relations among are derived from
and
Then, there
deg Yj = aj.
exist weighted homogeneous polynomials X1,
Let
fi,
,
deg Xj = aj
gi(Y1'
6, 11
f1 = ... = fk(d' E1,
'
Yk)
, k in
fr
for
in
X0,
j > 0
for each G(V, L)
, k) = 0.
Thus, roughly speaking, any relations among the
nj's
i
I. A-genus and the Apollonius method
28
extend to relations among
and the
8
The proof of
Ed's.
this theorem uses a graded version of Nakayama's Lemma and is due to Mori, who used this method to prove the following. If
(2.7) Corollary ([Morl]).
is a weighted
(D, LD)
complete intersection in addition, then so is (2.8) Remark.
ampleness of
exact sequence
In theorems (2.3) and (2.6), the
is not necessary.
L
(V, L).
Moreover, if we have the
0 -+ OV[-aL] -+ 0V -- OD -+ 0
h0(D, 0D) = 1, we do not need to assume that
and if D
is
irreducible and reduced.
§3. Iteration of the Apollonius method (3.1) Definition.
of subvarieties of is a rung of
Vi
L
V = Vn
Vn-1 D
is called a ladder of for each
(Vi+1, Lj+l)
the restriction of each rung
V
A sequence
to
V1
j k 1, where
V
if
(V, L) L1
is
It is said to be regular if
VJ.
is regular in the sense (2.2).
Of course we have
dim Vi = j
Moreover, g(V, L) = g(VJ, L
)
and
d(V3, L) = d(V, L).
= g(Vi, L1)
is the arithmetic
i
genus of
Vi, which is why
g(V, L)
However, such a ladder does not always exist, while
genus.
g(V, L)
is always defined.
As for the d-genus, we have 2
is called the sectional
2 d(V1, L1)
d(V, L) 2 .. 2 d(VJ, Li)
and the equalities hold if and only if
the ladder is regular.
Letting
r
H0(Vi, L) -+ H0(Vi_i, L), we have
be the restriction map 4(V J, L
- 4(Vj_1, Lj_1)
)
i
= dim Coker(r), which reflects the gap in the structures of (VJ, Li)
and
(Vi _11 LJ_1)
as we saw in §2.
called 'deficiency' in classical geometry.
This was
Similar
29
§3: Iteration of the Apollonius method
phenomena can be observed for a divisor curve
d(V1, L1) k 0
Obviously
V1.
equality holds if and only if
4(V1, L1)
on the
IL1J
in this case and the
H0(V1, L1) -> H0(D, LD) = sd V1 = 11.
This is in turn equivalent to
is surjective. Thus
D E
is also a kind of deficiency.
d(V, L)
is
the sum of these deficiencies and hence is sometimes called However,
'total deficiency'.
is well-defined even
4(V, L)
if there is no ladder.
(3.2) Using the Apollonius method successively, we can generalize many results on curves to results on First of all, we have
having a ladder. Remark.
We conjecture that
(V, L)
g(V, L) a 0.
g(V, L)
for arbitrary
0
polarized variety, even if there is no ladder.
If
is
V
char(s) = 0, this is proved by using Mori-
smooth and
Kawamata theory on minimal models (see Chapter II).
However,
the problem is unsolved in general (see §19 in Chapter N), especially when
char(s) > 0.
(3.3) Proposition. ladder. 0
and
so
L
(V, L)
We have
Hence
g(V, L) = g(V1, L) = 0
When
4(V1, L) = 0, and
L1
is
By induction using (2.5), we infer that
is simply generated for each Remark.
g(V, L)
4(V, L) a 4(Vj, L) 2 4(V1, L) 1 0.
A(V, L) = 0, the ladder is regular and V1 = 1P1.
has a
is simply generated.
simply generated. Lj
if
If in addition the equality holds, then
Proof.
When
4(V, L) 2 0
j.
d = 0, we can prove the assertion
without assuming that a ladder exists (see (4.12)). (3.4) Proposition.
Let
(V, L)
be a polarized variety
I. A-genus and the Apollonius method
30
with
Suppose that
g(V, L) = 0.
Then
variety.
itself) is a normal
for any
H1(V3, -tL) = 0
j 2 2,
char(s) = 0, this follows from Mumford's
When
t > 0.
(Vi )
4(V, L) = 0.
We claim
Proof.
V
Vi (including
such that each rung
has a ladder
(V, L)
vanishing theorem (0.4.14).
To prove the claim in general,
we use the induction on
Since
Therefore
V1 = IP1.
is true for
j = 2
h1(VJ, -tL)
induction hypothesis.
We have the exact
is normal.
V i
s
H1(VJ-l, -tL)
--i
h1(V3, -(t+l)L)
On the other hand
by the
H1(Vj, -tL) = 0
by Serre's vanishing theorem and Serre duality,
t >> 0
since
we have
is a rational surface and the claim
V2
by [Ek2;Th.1.6].
Hence
j > 2.
for
g(V, L) = 0,
H1(Vi, -(t+l)L) -i H1(Vi, -tL)
sequence for
J.
Hence
h1(Vj, -tL) = 0, as desired.
From this claim we infer that
hl(V, 0)
s
h1(Vn_1, 0)
h1(V1, 0) = 0, so the ladder is regular and
4(V, L) _
4(V1, L1) = 0.
The assertion is not true without the assump-
Remark.
For example, let
tion of normality.
V
be the non-normal
variety obtained by identifying two points on
H
plane
V,
g(V, [H]) = 0
and
(V,
[H])
has a ladder,
d(V, [H]) > 0.
(3.5) The following result is useful when Theorem. ladder.
Let
(V, L)
Assume that
2) BsILI = 0
if
J
is small.
be a polarized variety having a
g = g(V, L) 2 4(V, L) = J.
1) the ladder is regular if
3) L
A hyper-
containing neither of these points is an ample
divisor on but
1P n.
Then
d = d(V, L) k 2d - 1,
d s 2d,
is simply generated,
g = d
and
Hq(V, tL) = 0
for
§4: Existence of a ladder
t, q with
any
31
0 < q < n, if
d > 2d.
Proof (cf. [F2;Th. 4.1] & [Fll;(3.6)&(3.8)]). assertion is more or less known when
The
n = 1, possibly under
For example, the assertion
different formulations.
g = d
in the case 3) is a part of Clifford's theorem (at least on See [F2;Sl] for details.
smooth curves). For
we use induction on
n k 2
Since
n.
g(Vn_1, L)
g(V, L) 2 d(V, L) 3 4(V n_1, L), the theorem applies to (V
n-11
L).
Assume that d k 2d'+l.
Therefore
So
d > 4(V n_1, L) = d'
d = d'
if
ILIV n-1
ILn-lI
g = d
by 1) and by the induction
(2.5) proves the simple generation.
G(V, L) -+ G(Vn_1, L) g hq(V, tL)
for
is surjective, we have
hq-1(Vn_l, tL) = 0
Hence
hq(V, tL) s
Since
hq(V, (t-l)L)
This is true also for
q = 1.
since
1 < q < n,
by the induction hypothesis. hq(V, LL) = 0
Serre's vanishing theorem. 0 < q < n.
BsILI _
and so
This proves 2).
d a 2d.
In case 3), clearly hypothesis.
g i d > 4'.
proving 1).
By 1) we have B5ILn-1I _ 0
by 3). This contradicts
g = 4'
Then
in case 1).
Therefore
for
£
>> 0
hq(V, tL) = 0
by for
This proves 3).
Remark.
Thus a linear basis of
generator system of
G(V, L)
H0(V, L)
in case 3).
When
becomes a d z 24 + 2,
by induction using (2.6), we can show that the relations among them are derived from those of degree two (see [F2] for details).
§4. Existence of a ladder Here we will establish sufficient conditions for the
I.0-genus and the Apollonius method
32
existence of a ladder of (V, L). Technically this is one of the most difficult part of the theory. (4.1) To begin with, let us consider the easiest case.
is smooth and
Suppose that
char(A) = 0, V
base points.
Then a general member D
D
is a rung.
of
ILI
It is connected since
by Bertini's theorem. So
ILK
L
has no is smooth is ample.
Proceeding similarly we get a ladder.
In general, the main obstruction for the existence of a BsILI, and the singularity of
ladder is the base locus When
V.
char(.) > 0, we have additional technical troubles. The result below is very useful when 4(V, L) > dim BsILJ
(4.2) Theorem.
variety
dim 0
(V, L), where
particular
d z 0
is small.
4
for any polarized
is defined to be
always, and
d = 0
implies
-1.
In
BsILI = 0.
For the proof, we use the following facts. (4.3) Theorem.
variety
Y
Let
such that
A
Bs A = 0.
the rational map defined by
member of
A.
be a linear system on a normal
A
Let
and let
W be the image of D
be a generic
Then
1) D
is irreducible if
2) D
is reduced if A
dim W k 2. is complete.
Proof. 1) is a famous result due to Zariski [Zr;p.30]. When char(.) = 0, 2) is clear by Bertini's theorem. char(.) > 0, we need to be a little bit more careful.
When See
e.g. [Fll;(2.7)] for details. (4.4) Lemma.
a variety
D1 U D2 Then
Y with
Let
D
be an effective ample divisor on
dim Y = n
and suppose that
for some proper closed subsets
dim(DI n D2) a n - 2.
Di
of
Supp(D) _ Supp(D).
§4: Existence of a ladder
33
This is clear when
Proof.
divisor is connected. When
2, since every ample
n
n > 2, take a general hyperplane
section and use induction on
See [Fl;Lemma 6.1] for
n.
details.
(4.5) Proof of (4.2), Step 1.
We will prove the
following more general assertion by induction on 4(V, A) > dim Bs A [A] = L
for any linear system
A
such that
is ample.
Here
Assume
n = 1.
v: V' - V
The normalization
n 2 2.
Hence
finite morphism.
n + Ln - (1 + dim A).
is defined to be
4(V, A)
This assertion is proved easily if
is ample and
v L
We claim Indeed, if
dim Bs A
dim Bs(v A) _
is generated by
JAI
B5IAI = A n Bs A, so
*
v A.
dim B5IAI s dimjAJ - dim A.
A
- dim A = 6 = 1, there is a member
dimJA
such that
dim BSIAI
A
and
A.
2 dim Bs A - 1
In general we use the induction on
ample.
is a
So it suffices to prove the inequality for
dim Bs A.
IAI
n = dim V.
of
Therefore since
A
is
6.
By these observations we reduce the problem to the case in which
is normal and
V
(4.6) Step 2.
homomorphism
Let
3
A
is complete (so
be the image of the natural
BsILI.
Let
n: Y -* V
V
with center
the normalization of the blowing-up of Then
be J.
is the principal ideal of an effective Cartier
a i
divisor
This ideal defines
OV(H°(V, L)) ® [-L] -> 0V.
a natural scheme structure of
*
A = ILI).
E
surjection
Y
on
such that
n(E) = BsILI
0y(H0(V, L)) 0 [-n*L] -- n*3
system M on
Y
such that
Bs M = 0
and
The
as a set.
gives a linear *
E + M = n ILI.
I. A-genus and the Apollonius method
34
p: Y -i1PN (N = dimILI)
Let
W
and let
Then
be its image.
M = p*IHI
H = 0W(1),
for
is the rational map defined by
p.n-1
and
be the morphism defined by M
We treat
ILK.
the following five cases separately. This means
(4.7) Case 1, dim BsILI = n.
4(V, L) = n + d > n
so
by the ampleness of
(4.8) Case 2, dim BsILI = n - 1
X
Let
be a general fiber of
dim W = 1.
p: Y -> W
and set
w =
Ln = LYn= L n-lE + L n-1M = L n-lE + wL n-1 X. Y Y E X is ample on n(E) and dim n(E) = n - 1, we have
L
Ln-lE > 0.
Ln-1X x 1, so
Similarly
HO(1PN00(1)) Hence
and
L.
Then
deg W.
Since
h0(V, L) = 0,
--+
h0(V, L)
HO s
The bijection
d > w.
factors through
(Y, 0(M))
H0(W, H).
Thus we have
h0(W, H) t 1 + w t d.
A(V, L) Z n, as desired. (4.9) Case 3, dim BsILI < n - 1
and
dim W = 1.
The proof is similar to that in case 2. have
Ln-lE Z 0
and
d 2 w.
Therefore
(4.10) Case 4, dim BsILI < n - 1 Let
S
be a generic member of M
corresponding member of of
V, since
V
is normal.
Hence D
D
dim BsILI So
S
ILI.
This time we
4(V, L) 2 n - 1. and
dim W > 1.
Then D = n(S) Note that
< n - 1.
D
and let
as subsets = M
(MI
1-Macaulay since
V
is normal.
So
no embedded component, and is reduced everywhere. 4(V, L) 2 4(D, ILID)
and
induction hypothesis to for
ILI.
since
is irreducible and reduced by (4.3).
is irreducible and generically reduced.
is locally
be the
BsILI = BsILID.
Moreover D
has
We have
Applying the
SLID, we get the desired inequality
35
§4: Existence of a ladder
(4.11) Case 5, dim BsILI = n - 1 Let
be a generic member of
S
corresponding member of
D = F + G Then
for
this implies
Hence
be the
D
H0(G, LG))) = 1
We have
ILIG).
by the normality of
Moreover
4(V, L) > 4(G,
F n G c BsILIG,
Since
n - 2 t dim BsILIG < 4(G,
-
and
is the fixed part of ELI.
by ( 4 . 4 ) .
dimILIG = dimILI - 1.
d(G, LG).
F
G = n*S, where
dim(Ker(H0(V, L)
Hence
M
dim W > 1.
As a Weil divisor, we have
ILI.
dim(F n G) 2 n - 2
and
V.
Ln = Ln-1F + Ln-1G > Combining these
ILIG).
4(V, L) k n, as desired.
inequalities we get
Thus we have completed the proof of (4.2). (4.12) Corollary.
very ample if Proof.
is simply generated and hence
L
d(V, L) = 0.
We use induction on
be the normalization.
n = dim V.
Let
v: Y --+ V
Then
h0(Y, L y ) a h0(V, L) and Since LY is ample, we have the
4(Y, L ) s 4(V, L) = 0. Y Any generic member equality by (4.2). irreducible and reduced by (4.3).
D
of
ILY1
Then we have
is
4(D, LD)
4(Y, L
Y ) = 0, so LD is simply generated by the induction hypothesis. Therefore LY is simply generated by (2.4). Thus
hence
) is generated by H0(Y, LY ) = H0(V, L) and Y is surjective. The injectivity G(V, L) --+ G(Y, L ) Y
G(Y, L
is clear and the assertion follows from this. Remark.
Thus
4(V, L) = 0
implies that
Actually we will classify all such pairs
(V, L)
(4.13) Next we consider the case in which finite set.
V
is normal. in §5. BsILI
By (4.2), this is indeed the case when
d = 1.
p: Y -+ W be as in (4.6), and
Let
a: Y -+ V, E, M
set
d = d(V, L), 4 = 4(V, L).
and
is a
Then we have the following.
I. A-genus and the Apollonius method
36
Proposition.
each point on when
is
has only Cohen Macaulay singularities at
V
finite and that
dim W = n, B = BsILJ
Suppose that
Suppose in addition that
B.
Then
char(s) > 0.
d k 2d - 1
has a ladder.
(V, L)
For a complete proof, see [F2;Prop.3.4] and [Fil; (3.3)].
Here we sketch the outline of it when
Let
S
be a generic member of
corresponding member of
M
Moreover
irreducible and generically reduced.
at
B, D
and let
be the
D
Then, as in (4.10), D
ILI.
reduced at any point not on
char(St) = 0.
B.
Since
D
is
is Cohen Macaulay
Therefore it is
has no embedded component.
is a rung of
D
D
reduced everywhere.
Thus
check that
satisfies the hypothesis of the
(D, LD)
is
We can
(V, L).
proposition, so we can proceed to obtain a ladder. Let the notations be as above and
(4.14) Lemma.
suppose that
point of of
ILI
point of
B
is finite, V
dim W < n, B
Then a generic member
d z 2d - 1.
and that
is a rung of
(V, L),
and
complete proof, see [Fll;(3.2)]. Y
the Chow ring of
and let D
f: V' - V Ep
is smooth at each
D
since
For a
char(s) = 0.
LE = 0
We remark that
B = n(E)
f ILI
p
be the blowing-up at a point
if and only if
for some
m > 1.
in
is a finite set.
p
be the exceptional divisor lying over p.
is singular at
part of
D
B.
Here we sketch the proof when
Let
is smooth at each
mEp
Assuming
on
Then
is in the fixed m > 1, we will
derive a contradiction. *
First of all, we show that
L'
= f L - mEp
B
is nef on
§4: Existence of a ladder
V.
Indeed, if
Z ¢ BsIL'I
is a curve on
Z
and
37
L'Z Z 0.
Therefore
E Z < 0 since [Ep]E = 0(-l). p P In either case L'Z a 0, as desired. Y
is a morphism is a part of E)n-1
L(L -
n': Y -+ V'
= Ln > 0
(0.4.7), we have
= mwEpX k mw 2 2w.
Y
Mn
=
(LY)n
=
LYMn-1
EpX a 1.
Hence
On the other hand
B.
The irreducibility of
when n > 2.
E'
with
Ep(L - Ep - E') = -Ep2 = 1,
-
LMn-1
mE*AMn
I
Combining them we get d s 2d - 1.
is smooth at each point on
D
follows from
When n = 2, we have
effective divisor
LMn-1
d =
d k 2w k 2(d - d + 1), which contradicts D
MMn-lEA
0 g 4(W, 0W(1))
n - 1 + w - h0(V, L) = w - d + d - 1.
Thus we conclude that
By
dim W = n - 1.
k
=
LMn-1
Note that
and hence
mEp
Then
= f.n'.
it
Ep = (n') Ep. on
d -
such that
*
Y, so
on
L'Z > 0.
if necessary, we may assume that there
E, where
= d - mwE*X
LZ = 0
by the normalization of the graph of the
f-1.n
rational map
Ep, then
Z c Ep = IPn-l, then
If
and
Replacing
not in
V'
E =
n'*E' = 0. w = 1.
so
dim W = n - 1
E*
p + E'
Hence
for some *
*
wEAX = EpM =
Therefore
n*(E + X) = n*X, which is irreducible, as asserted.
D = The
rest of the assertion is obvious. (4.15) Now we obtain the following. Theorem. d a 2d - 1.
Let
Suppose that
empty) and that (V, L)
V
be a polarized variety such that B = BsILI
is finite (possibly
is smooth at each point on
B.
Then
has a ladder.
Proof. (V, L)
(V, L)
(4.13) applies if
has a rung
same hypothesis as
D
dim W = n.
by (4.14) and
(V, L).
If
(D, LD)
dim W < n,
satisfies the
So we can complete the proof by
I. A-genus and the Apollonius method
38
induction on Remark.
n.
The ampleness of
fact it is enough to assume
is not necessary.
L
In
d(V, L) > 0 (see [Fll;(3.4)]).
On the other hand, the smoothness of
V
at
B
is
indispensable in general (cf. [F25]), but we have the following partial result. (4.16) Theorem.
such that that
Let
4(V, L) = 1
be a polarized variety
(V, L)
and
g(V, L) < d(V, L).
Suppose
char(.) = 0, V has only 2-Macaulay singularities (cf.
(0.2.4)) and that
point on
B.
V
Then
is locally Cohen Macaulay at each (V, L)
has a ladder. There
For the proof, consult [F25;(1.2)].
V
is
assumed to be locally Macaulay everywhere, but it is easy to generalize as above. changed.
If
n = 2, nothing need be
Indeed, if
D
n > 2, any general member
of
is a
ILI
rung by the argument in [F25].
By (0.2.6),
2-Macaulay at any point not on
B, so we get a ladder by
induction.
D
is locally
The details are left to the reader.
§5. Classification of polarized varieties of d-genus zero We will classify all the polarized varieties of d-genus zero by the Apollonius method. The result is well-known in classical geometry.
(5.1) Throughout this section
variety with
4(V, L) = 0.
ample by (4.12). Hence
g(V, L) = 0
Moreover and
L
(V, L)
is a polarized
is simply generated and very
g(V, L) k 0
Hq(V, tL) = 0
and (3.5) applies.
for any
q,
0 < q < n. Clearly
(V, L) = (1P, n 0(1))
if
d = Ln = 1.
t
with
§5: Classification of polarized varieties of A-genus zero
When Pn+l
d = 2, V
is a possibly singular hyperquadric in
L = 0V(1), since
and
39
L
is very ample.
(5.2) Now we will study the case where first we assume that
and let K
is a smooth manifold M
V
At
d s 3.
be its canonical bundle. (5.3) Claim. Proof.
h0(M, K + nL) = d - 1.
This is obvious when
general member
D
of
KD
is
[K + D]
h1(M, K + (t - 1)L) = hn-1(M, (1 - t)L) = 0 Moreover
duality and (3.5.3).
is very We have
D'
by Serre
since
IK + (t - 1)LI _ 0
(K + (n - 1)L)Ln-l = 2g - 2 < 0.
Therefore we obtain
H0(M, K + nL) = H0(D, KD + (n - 1)LD) sequence of cohomologies.
n > 1, any
L
is smooth since
ILK
Its canonical bundle
ample.
When
n = 1.
by using the exact
Thus we complete the proof by
induction.
(5.4) dim BsIK + nLI < n - 1.
D n BsIK + nLI = BsIKD + (n - 1)LDI
Since
above argument, this is proved by induction on (5.5) Claim.
unless
(F2,
(M, L)
Proof.
since
When n = 2, we have
(K + 2L)2 a 0
(K + 2L)2 = 0
This implies
by (5.4).
Moreover pg = 0
We have
since
h1(M, 0) = 0
smooth rational curve with
K2 = 9
by
C
We have
of
ILI
is
is a
K2 + c2(M)
=
by Noether's formula (cf. (0.3.9.3)), and
c2(M) = 2 + b2(M). unless
C2 > 0.
K2 s 8
Actually M
KL < 0.
a rational surface since a general member
12X(M, 0) = 12
n.
0(2)).
KL + L2 = 2g - 2 = -2.
(3.5.3).
by the
and
Since
b2 > 0, we have
b2 = 1.
(K + 2L)2 = 0
In the latter case
M = P2
by
I. A-genus and the Apollonius method
40
Moreover
the classification theory of rational surfaces. and hence
K + 2L = 0(1)
L = 0(2).
(5.6) Claim. (K + nL)2Ln-2 = 0
Let
be a general member of
S
(K + 3L)2L = (KS + 2LS)2(S) = 0
sequence
ELI. Then, by (5.5),
unless
(S, LS)
We will rule out this exceptional case.
(F2, 0(2)).
Set
n a 3.
By induction we reduce the problem to the case
Proof. n = 3.
if
H = K + U.
Then
HS = 0S(1).
By the exact
h1(M, -tL - 2H) s h1(M, -(t + 1)L - 2H)
that since s
for any
h1(S, sH) = 0
(M, L) # (F2, 0(2))
(5.7) BsIK + nLI = 0.
rational mapping f
Ln-2A1A2 > 0
Hence
and (5.6).
Moreover
A
1
This yields
Pic(M).
IK + nLI
by (5.4).
is ample.
n A2 = 0
and
is a curve.
Al, A2 of If
of the
W
A
1
IK + nLI.
n A2 # 0, then
This contradicts (5.5) BsJK + nLI _ 0.
dim W = 1.
(5.8) W = IP1
fiber X
L
since
>> 0
Hence
Moreover, the image
defined by
dim(A1 n A2) t n - 2
i
in the sequel.
Take two general members
Proof.
for
Similarly we obtain
h0(M, 2H - L) > 0. Hence L = 2H in LS2 = 4 = L3 = 8H3, which is absurd. We assume
t,
h1(M, -2H) t
h1(M, -LL - 2H) = h2(M, K + 2H + £L) = 0
h0(M, L - 2H) k h0(S, 0S) > 0.
of
and
(X, LX) - (]Pn-1, 0(1))
for every
f.
Set
w = deg W.
0 s 4(W, 0W(1)) = 1 + w - (d - 1), hence
w i d - 2.
Proof.
Then
for any
Therefore
s.
by Serre duality and the vanishing theorem.
Then
we infer
0M[-2H] -* OM[L - 2H] --' OS -> 0
0
We have
W c IPd-2
by (5.3).
§5: Classification of polarized varieties of 0-genus zero
41
On the other hand wLn-1X = (K + nL)Ln-1 = d - 2. Ln-1X = 1
and
Ln-1X = 1
implies
(X, LX)
1P1, while
since
(1Pn-1, 0(1))
ti
W
This implies
4(W, 0(1)) = 0.
Therefore
L
is
very ample.
is a locally free sheaf of rank n
(5.9) 8 = f*(0M[L])
on W by (5.8) and (0.1.11). 8, namely
scroll of
(M, L)
Moreover ti
is the
(M, L)
(F(8), H(8))
H(8)
where
is
the tautological line bundle. F1
It is well-known that every vector bundle on direct sum of line bundles (cf. (Grol]). sum factor that
Al
LZ = A1.
of
For each direct
S. there is a section
Hence
is a
of
Z
f
such
deg Al > 0.
Summing up, we now obtain the following. Let
(5.10) Theorem.
with n = dim M a 2
and
be a polarized manifold
(M, L)
4(M, L) = 0.
Then
is
(M, L)
isomorphic to
1) (1Pn, 0(1)), 2) a hyperquadric in
pn+1
with
L = 0(1),
F1, which is a direct
3) the scroll of a vector bundle on
sum of line bundles of positive degrees, or 4) (1?2, 0(2)) (this is often called the Veronese surface).
When n = d = 2, (M, L)
Remark.
be of type 3) as well as type 2).
can be considered to
Otherwise the four
classes above are disjoint.
Every scroll of an ample vector bundle actually of 4-genus zero. Roch Theorem.
only if
8
We have
8
on
F1
is
This follows from the Riemann-
d a n
and the equality holds if and
is the direct sum of
0(1)'s.
(5.11) We now study the case in which
V
is singular
I.0-genus and the Apollonius method
42
4(V, L) = 0.
and
by
We embed
V
We introduce the following notation.
ILI.
x, y
(5.12) Given two points denote the line passing them. X * Y
P = FN (N = n + d - 1)
in
X, Y
A point on it is called a
X c P, let
For any
be a linear subspace of
T
P
Ridge(X n T) _ 0
R # 0, X
such that
if
dim T =
R = 0).
See e.g. [Fl;54].
is said to be a generalized cone over
(5.14) Lemma.
X
Let
be a subvariety of
be a point of X not in
Ridge(X).
Then
The equality holds only if X
deg X - 1.
R = Ridge(X)
X = (X n T) * R.
and
The proof is elementary and easy.
x
by
This is a
N - 1 - dim R and T n R = ¢ (hence T = P
If
x e X,
X.
(5.13) Proposition.
Then
P, let
of
such that
x * X = X}.
Ridge(X) = 1.x e X
linear (possibly empty) subset.
and let
x * y
x * y
We define x * x = x and X * ¢ = X
y e Y and x # y. convention.
vertex of
P, let
For subsets
be the union of all lines
We set
on
X n T.
and let
P
deg(x * X) s
is smooth at
x.
This is well-known and easy to prove. Let
(5.15) Theorem.
for
d(V, L) = 0 unless
over a submanifold
x
Moreover h0(V, L).
of
V.
M with
Then
Then V
P
with
Sing(V) = Ridge(V)
is a generalized cone
4(M, LM) = 0.
x % Ridge(V)
dim W = n + 1
deg W < deg V - 1 So
be a subvariety of
Moreover
Suppose that
Proof.
point
L = 0V(1).
is linear.
V
V
for some singular
for W = x * V.
by (5.14) and
h0(W, °W(1))
d(W, 0(1)) < d(V, L) = 0, which is absurd.
§6: Polarized varieties of 0-genus one: First step Sing(V) = Ridge(V).
Thus we get
Next take
as in (5.13) and set
T
M
4(M, LM) = 0, for
Then
M = V n T.
is obtained by taking hyperplane
sections successively.
Ridge(M) = 0
So
implies that
M
Now the assertion is clear.
is smooth.
(5.16) Remark. (V, L)
43
(M, LM)
is determined uniquely by Theorems (5.15) and (5.10)
modulo isomorphism.
together give a complete classification of polarized varieties of d-genus zero and describe precisely their structures. They are known as 'varieties of minimal degrees' in classical projective geometry (cf. [GH], [EH]).
The
results (5.10) and (5.15) have been known from ancient times, though classical case. old ones.
is assumed to be very ample in the
L
Our argument is a little different from the
I hope it is more easily understood by those with
modern mathematical background - thanks to the power of the sheaf cohomology theory. (5.17) Corollary.
V
is normal and has only rational
Cohen Macaulay singularities if Remark. general.
The singularity of
L
V
is not Gorenstein in
Ridge(V) # 0, then
Indeed, if
generated by
4(V, L) = 0.
Pic(V)
and hence the canonical sheaf
invertible unless
is
wV
is not
d = 2.
§6. Polarized varieties of 4-genus one: First step Throughout this section let variety with
4(V, L) = 1
(6.1.1) B = BsILI
(6.1.2) if
V
and
(V, L)
n = dim V.
be a polarized Then we have:
is at most a finite set by (4.2).
is smooth at
B, then
(V, L)
has a
I. A-genus and the Apollonius method
44
ladder by (4.15).
(6.1.3) Suppose that B
point of Then
and that
(V, L)
is locally Macaulay at each
V
is locally 2-Macaulay everywhere.
V
has a ladder if
Ln = 1.
This is proved by the argument in (1.1).
(6.1.4) Let the singularity of
V be as in (6.1.3).
has a ladder if
Then, by (4.16), (V, L)
and
char(St) = 0
g(V, L) < Ln (6.2) In the rest of this section we assume that has a ladder.
(V, L)
This is not always true, but we only
have partial results without this assumption. g(V, L) = g 2 1.
assume
We also
This is in fact true if
only 2-Macaulay singularities.
V
has
The proof is based on (3.4)
and can be found in [F11;(5.2)]. By these assumptions we can apply (3.5).
In particular
we have: 1) (V, L)
has a regular ladder.
2) BsILj _ 0
if
3) g(V, L) = 1
d = Ln x 2.
and
L
(6.3) Definition.
is simply generated if
A polarized variety
d k 3.
(X, L)
is
called a Del Pezzo variety if the following conditions are satisfied: (a) 4(X, L) = 1.
(b) g(X, L) = 1. (c) X
has only Gorenstein singularities and the canonical
sheaf
wx
is
0x[(1 - n)L], where
(d) Hq(X, tL) = 0
for any
t, q
n = dim X.
with
0 < q < n.
In classical geometry, 'Del Pezzo surface' means an
I.0-genus and the Apollonius method
45
algebraic surface whose anti-canonical bundle is ample (or very ample, in ancient times).
Our Del Pezzo variety is a
natural higher dimensional polarized version of this notion. (6.4) We have the following relationships between the conditions (a) " (d) in (6.3).
1) (b) follows from (c).
2) When X
is smooth and
char(.) = 0, (d) follows
from (c) by Kodaira's vanishing theorem (0.4.11). 3) (a) follows from (C) and (d). Hence
(X, L)
may be called a Del Pezzo manifold if
is smooth, char(St) = 0
KX = (1 - n)L.
and
To prove 3), we use the method in (1.3). X(X, tL)
for
is a polynomial in
dtn/n!, where
term
0 > t > 1 - n
of degree
t
Moreover
d = Ln.
and
n
X(t) aer
with the main
X(0) = 1, X(t) = 0
X(1 - n) = (-1)n.
Therefore
X(t) = (t + 1)...(t + n - 2)(dt2 + (n - 1)dt + (n - 1)n)/n! and
h0(X, L) = X(1) = n + d - 1.
4) (c) implies (d) if
L
This implies (a).
is very ample.
The proof is technically complicated and can be found in [Fll;(5.12)].
5) (a) and (c) follow from (b) and (d). For a proof, see [F11;(5.9)].
6) (c) and (d) follow from (a) and (b) if (X, L)
d z 3
and
has a ladder.
Indeed, we get (d) by (3.5.3) and then use
5) above.
7) (c) and (d) follow from (a) and (b) if x locally 2-Macaulay and if
X
(X, L)
has a ladder.
is
I. A-genus and the Apollonius method
46
The proof is complicated, but is found in [Fll;(5.11)].
A polarized variety
(6.5) Corollary.
4(X, L) = g(X, L) = 1
with
(X, L)
is a Del Pezzo variety if one of the
following conditions is satisfied. 1) L
is very ample.
2) X
is smooth.
3) char(.) = 0
(X, L)
is locally Cohen Macaulay.
In case 1) we have
Proof.
In case 2),
X
and
(X, L)
d z 3, so (6.4.6) applies.
has a ladder by (4.15).
has a ladder by (6.1.3) or (6.1.4).
In case 3),
So (6.4.7)
applies in these cases.
(6.6) Here we provide a 'cheaper' proof of (6.5.2) in the case where induction on
A
ILI
C'
By (6.2) any general member
n = 2.
is smooth if
Indeed, if
This is true even if
d > 1.
is another general member, then
a finite set by (4.2).
Since
C
C'
by Bertini's theorem.
C n C'
is
Hence
C
Thus
C
is a smooth elliptic curve
d.
We claim while the map
bijective.
Indeed, otherwise, b1(X) 2 2
H1(X, 0) = 0.
h: H1(C; Z) - H1(X; Z)
Lefschetz theorem.
Since
Therefore
is surjective by the
H1(C; Z) = Z ® Z, h
Alb(C) = Alb(X).
fiber of the Albanese map
a: X --* Alb(X).
1P1
for any general
F
must be
F
be any
Then
F n C
Let
a simple point by the above reasoning and thus
F ti
d = 1.
B = C n C' = BsILI, so it is smooth everywhere
is smooth at
So
C
= d = 1, this is a
simple point and the intersection is transverse.
for any
We use
C.
n.
Suppose first of
is the complex number field
is
FL = FC = 1.
with F n BSILI = 0.
More-
§6: Polarized varieties of A-genus one: First step
47
over every fiber is irreducible and reduced since ample.
From this we infer that
(X, L)
is the scroll of the vector bundle However, since
Alb(X) = C.
C1(8) = L2
and
our assumption and we conclude
(K + L)L = 2g - 2 = 0
,
H0(X, K + L)
K + L = 0
Pic(X).
in
and
on
h0(X, L)
This contradicts
H1(X, 0) = 0.
and by Serre duality we infer
H1(X, K)
d = a*OX[L]
is ample, we have
8
is
is a 1P1-bundle, and
a
4(X, L) = 2 (cf. (At)).
By the exactness of
L
H0(C, K(C)) -+ IK + LI
Since
# 0.
is ample, this implies
L
Thus we prove (6.3.c), and by
Kodaira's vanishing theorem we get (6.3.d). n > 2.
Now we consider the case member of
D
be a general
By (6.2.2) and by (4.14), we infer that
ILI.
We can easily check that
is smooth.
Let
(D, LD)
D
satisfies the
conditions (a) and (b) in (6.3), so it is a Del Pezzo manifold by the induction hypothesis. the sequence --+
H0(X, K + (n - 1)L) --* H0(D, K(D) + (n - 2)L)
H1(X, K + (n - 2)L)
we get
IK + (n - 1)LI n = 2.
case
By the exactness of
and by Kodaira's vanishing theorem, # 0.
This implies (6.3.c) as in the
We obtain (6.3.d) by the vanishing theorem.
(6.7) Let
(V, L)
be as in (6.2) and we will study
dividing the cases according to the value of
(V, L)
Suppose first
d = 3.
is very ample by (6.2.3).
Then Hence
h0(V, L) = n + 2 V c Pn+l
and
(6.8) Suppose that (6.2.3). one.
So
d = 4.
A one-dimensional rung (Vi, L)
Again V1
L
and
L
is a hypercubic.
Any polarized manifold (M, L) n+1. 4(M, L) = 1 is a hypercubic in Corollary.
Ln.
with
d = 3
is very ample by
is of arithmetic genus
is a complete intersection of type (2, 2)
I. A-genus and the Apollonius method
48 in
By induction using (2.6), we infer that
1P3.
(V, L)
is a complete intersection of two hyperquadrics in Any polarized manifold
Corollary.
in
with
(M, L)
d = 4
is a complete intersection of type
4(M, L) = 1
and
Fn+2
(2, 2)
Ipn+l. is very ample also in the case
(6.9) L
Moreover
d > 4.
is a Del Pezzo variety by (6.4.6).
(V, L)
is never a complete intersection.
But it
The classification of
such varieties will be treated in §8 and §9. (6.10) Now we study the case
p: V -,Fn with
by (6.2.2), we have a morphism Clearly
Since
d = 2.
p 0(1) = L.
is a finite morphism of degree two.
p
there is no upper bound for
g(V, L)
BsILI = 0
Note that
in this case.
(6.11) We study the structure of such a double covering
f: X -+ Y
OY-algebra and
a sheaf of
f
Suppose that
Note that
char(.) # 2.
assuming
d = f*OX
is
X = Jfte-c(d).
Then
is flat.
d
is locally free of
rank two.
So we have a trace homomorphism
t: d -> 0Y.
Then
is twice the identity, where
is the natural
t.t
homomorphism d = 0y ® A cation of
OY
A = Ker(t).
gives a homomorphism
algebra structure of this homomorphism = 0Y[-F], P
char(&) # 2, this implies
0Y-modules where
as
4
Since
d.
f.
The
B
of
12FI.
F
A
such that
and
A
This is the
X is determined by the triple (Y, F, B)
up to an isomorphism over
will be denoted by
No2 , 0Y.
For the line bundle
defines a member
branch locus of
A:
The multipli-
is determined by the sheaf
4
0.
c
Y.
RB,F(Y).
Hence such a double covering Since
B E 12FI, F
is often
49
§6: Polarized varieties of A-genus one: First step
determined by
is in fact flat if
f
Remark.
In such a case we write
B.
RB(Y)
also.
is smooth and if
Y
X
has only Cohen Macaulay singularities (cf. (0.1.12)). (6.12) We describe the structure of precisely by using local coordinates. open set C
of
Y
of
U
such that
An
OY-base
gives a fiber coordinate of the total space of
'NU
the line bundle
On the other hand
FU.
.NU c dU
So we have a natural map
C E H0(f-1(U), OX). over
Take a small enough
is trivial.
.NU
more
RB F(Y)
and hence
f-1(U)
- FU
This is an embedding onto the image, which is
Y.
defined by
C2 = p, where
is the function
q,
on U.
j4(CeC)
The above local structures are patched globally as Let
follows.
on
C2 = fAlt CY
defining ('2
Let
as above.
U2
E H0(U2, OY)
the divisors
2
be a fiber coordinate of
B
Then
FU . 2
(f2p)
U2 n Uµ, where Let
F.
T2 = f2'U 2P µ
be a covering by small open sets
Y = U2U2
be defined by
is a 1-cocycle T E H0(Y, 2F)
and let
be the corresponding function such that U2 n Uµ
on
Then X
X2 = (C22 = P2)
in
is obtained by patching FU . 2
From these observations we infer:
1) The relative canonical bundle 2) If
K(X/Y)
is
f F.
is smooth, X has only hypersurface singularities
Y
and they lie exactly over the singular points of (6.13) Corollary.
with
d(V, L) = 1
ladder,
Then 2g + 2
V
(V, L)
d = 2.
be a polarized variety
Suppose that
is locally Cohen Macaulay and
V = RB(IPn)
and
and
Let
L
for some hypersurface is the pull-back of
B.
B
0(1).
has a
(V, L)
char(A) # 2.
of degree (V, L)
is a
I. A-genus and the Apollonius method
50
weighted hypersurface of degree projective space
1P(2, 1,
V
1).
,
over the singular locus of
2g + 2
is singular exactly
B.
(6.14) Now we consider the case
D1, , Dn
members
of
a finite set by (4.2).
We get a ladder by setting n: V' -, V
n L - E = H, where
that P.
E
Since
1P n-1 and
isomorphism and E on
V'.
is the line bundle
is ample on
X
cally reduced.
point
x
of
ladder of
p.
V'
X
If
such
0(1)
on
is an
Thus, for every
x
1
X = p- (X).
for
i
EX
>> 0.
Since
is ample on
X.
is irreducible and generi-
is everywhere reduced for any general
since it is the one-dimensional rung of a Clearly
(V, L).
Remark.
flat and
P
X
be
1P n-1
[L - E]E = 0(l), PE: E --i P
Moreover
dim X = 1.
E
and
P
p: V'
LX - EX = [H]X = 0, this implies that Hence
B, let
In L - El
is a simple point for
iL - E
Recall that
n Dn-J.
Then D a
is a section of
P, E n X = EX
B.
and let D be the proper
B
H
is
is smooth at
V
be the blowing-up at
Hence we have a morphism
_ 0.
(1 D'.
on
DJ
So
n Dn
is a simple point
VJ = D1 n
the exceptional divisor over transform of
B = D1 n
Ln = 1, B
Since
Take general
d = 1.
Then
ILI.
and the intersection is transverse.
Let
in the weighted
V
g(X) = g(V, L).
is locally Cohen Macaulay, then
is locally Macaulay for every fiber
X.
f
is
So
X
has no embedded component and hence is reduced everywhere. (6.15) Now we assume that Macaulay.
V
is locally Cohen
In subsections before (6.18), we also assume
g(V, L) = 1. Set
9 = OV,[2n*L]
and
Y = p*9.
Then
h0(Xx, 9 x)
2
§6: Polarized varieties of 0-genus one: First step
for every fiber
and X
Xx
over
x E P
51
deg 0x = 2LX = 2
since
is of arithmetic genus one. Moreover
by global sections.
Hence, by (0.1.11),
surjective.
morphism
f: V'
T, we have a
f H _ 9 (cf. (0.1.4)).
such that
---' W
is
p 3" -+ 0
is the scroll of
(W, H
So, if
is locally free
3'
of rank two and the natural homomorphism
is spanned
9x
fx
This is a finite double covering since the restiction
is so over every
Xx -> Wx = F1
The natural homomorphism
on
x
P.
p*OV,[2L - E] - p*OE[2L - E]
is a zero map since
E n X
for every fiber
Therefore, using the exact sequence
X.
is the base point of 1[2L - E]XI
0 -+ OV, [2L - 2E] --p OV, [2L - E) -+ OE[2L - E] -> 0,
we get
P*OV,[2L - E] = p*OV,[2L - 2E] ti p*OV, 0 [2H] ti OP[2H ].
Next we use 0 -+ OV , [ 2L - E] -+ 0 - OE -* 0.
We have
R1p*OV,[2L - E] = 0
for every
fiber so
0 - OP[2H ] -+ 3 -+ Op -
Hence
Xx.
S = 0P(2) a Op
since
(6.16) Now we assume (6.11).
Then
We have
S c B
V'
*
and is
IHC - 2p HEI
p: W -+ P.
a section of
char(.) # 2
RB,F(W)
V'
S
so that we can apply B E
for some
since f: x Xx -. W x
is smooth along
B = S + B1
is exact,
0
Note that
H1(P, 0(2)) = 0.
is the unique member of
S = f(E)
B
HI(Xx, 2L - E) = 0
since
since
V'
12FI, F e Pic(W).
is smooth along
for some other component
B1
with
generated by
H.
[B1] = bH
and p H .
H
+
S n B1 = 0.
Then
c = 0
since
general
[B] = (H. - 2HE) + 3HC
x.
So
since
is
[B1]S = 0.
is ramified at four points for
Moreover b = 3
fx
B1.
Pic(W)
since
Hence
E.
is singular exactly over the singular locus of We may set
E n Xx
is ramified at
and
F = 2HC -
H.
I. A-genus and the Apollonius method
52
Thus the structure of
Conversely, for any member
explicitly.
® 0)
W = 1P(2H
(V, L)
is described very
(V, L)
B1
RS+B1,2HCH (W)
S.
(6.17) Remark.
In the above case (6.16),
weighted hypersurface of degree six in
..., 1).
using (2.7).
e(V) + n - 1 = e(V') = 2e(W) - e(S) - e(B1).
is a divisor in
W, so the total Chern class is computed
as in the case of hypersurfaces in obtain
is a
is smooth, the topological Euler number can be
V
computed by B1
(V, L)
1P(3, 2, 1,
n
This can be proved by induction on If
on
as above, we obtain a polarized variety
by blowing down the divisor on
lying over
13Hj
of
when n = 3.
e(V) = -38
1PN.
In particular, we
This is used in
[F5-1] to
prove the following fact:
M be a smooth threefold such that
Let
ample and the cohomology ring H(M; Z)
H (1P3; Z) = Z(h]/h4.
is isomorphic to
Then M = F3.
This result is generalized in the case (cf. [F6],
[L.
is
-K(M)
n = dim M g 6
W]).
(6.18) Now we study the case
g = g(V, L) s 2.
Unfortunately the general case is difficult and we do not have a complete answer, even if sketch our result when every fiber
Xx
p
V
is smooth.
Here we
is hyperelliptic, which means,
is a double covering of
F1.
See [F5-11I]
for details.
(6.19) Let
restriction 8 = p*w
wx
w to
be the canonical sheaf of
Xx
V'.
is the canonical sheaf of
is locally free of rank
g
and
p 6 -, w
Then the
X
x
is
§6: Polarized varieties of A-genus one: First step surjective.
We have a morphism
f H(s) = w.
Its restriction
canonical map of
fx: Xx -*IPg-1
that
is hyperelliptic.
p
is a 1P1-bundle over P = 1Pn-l
is a finite double covering.
f: V' --i W
OP[eH ]
is the
This is a double covering onto a
Xx.
Hence the image W = f(V') a section of
such that
f: V1 --,IP(6)
Veronese curve by the assumption that
and
53
W = F(Y)
0, where
and
is
and we have an exact sequence
p: W --> P
S --> Op -
S = f(E)
is a vector bundle such
Y
is the integer with
e
0 -+
[S] S = 0(-e).
Thus we are in a similar situation to the case where g = 1, but this time there are several possibilities since f
is not always ramified over (6.20) Now we assume B e
for some
S.
Then
char(A) # 2.
V' = RB F(W)
We have three possibilities:
12FI.
'
a) S c B.
b)SnB=0. c) S ¢ B and S n B# 0. (6.21) Case a).
B = S + B1
for some member
V
S n B1 = 0. of
B1.
Similarly as in (6.16), e = 2 of
B1
such that
1(2g + 1)H,l
is singular exactly over the singular locus
(V, L) is a weighted hypersurface of degree
in IP(2g+1, 2,
e = 1
and
*
effective divisor addition to
4g + 2
1, , 1).
(6.22) Case b).
E
such that
E, we can blow down
points by (0.1.6).
double covering
f
z ¢ B
*
E*
f S = E +
for some
*
E n E
_ 0, E
*
In
= S.
*
and
E
S
to smooth
By these blow-downs we get a finite V1'
- W'
phically onto a hypersurface such that
and
where
z
ti
Bb
IPC.
B
is mapped isomor-
of degree
2g + 2
is the image point of
on S.
Fn So
I. A-genus and the Apollonius method
54
V
is the blowing-up of
z
and the exceptional divisor is identified with
have
L = [H.]V - E .
at a point
RB,(ff'n)
Thus
zl
lying over We
E .
is a birational
(V, L)
modification of a polarized variety of the type in (6.13).
if
n a g + 2, there is a line
in
£
that the intersection multiplicity of point on
is even.
i n Bb
where the smoothness of
zl a Z1.
the proper transform the ampleness of
that
Bb
at each
LZi = (H. - E )Zi t 0 V
on
such that HCZ = 1.
of
Z1.
for
This contradicts
The arguments in [F5-E;§18] apply at
is smooth at any point on
B
1(2g + 2)Ha + 2bH,l
H(kH ® 0)
bundle
Z1, Z2
B n S.
n = 2, P = Ek = IP(kH$ ® 0), S e
particular B E
and
such
z
L.
(6.23) Case c).
first, since
£
passing
Now, by the evenness,
Then
Zi
Fn
Indeed,
is assumed, but the proof is
B
has two components
We may assume
g + 1.
This is Lemma (17.5) in [F5-III],
valid without this assumption. (fb)-1(j)
n
(V, L) exists only if
Such a pair
on
where
Ek
Ha
JHa + aH,I
and
is the tautological line
b, a
and
In
are integers such
-e = k + 2a, gk + (g - 1)a + b = 1.
Moreover
a 2 0.
However, unlike the case in [F5-III;(l8.5)], we do not have
2b 2 -k
since some big multiple of the unique member may possibly be contained in
JHa - kH,)
In this exceptional case C n S = 0
and
B
B.
is singular along
B
So
is reduced (or equivalently if
k > 0.
V
is normal),
we can rule out the above possibility and we obtain b = 1, a = 0
C.
a = 0, but we cannot go further in general.
Such an example does really exist even if If
of
C
and
e = 0
as in [F5-IQ;§18].
k = Of
Therefore
§7: Results of Lefschetz type X F ,
P = 1P
P -+1P1 a
V
Thus
is
55
B E 1(2g + 2)Ha + 2H
j
and
is the fiber of
S
over a branch point of the double cover is rational since any general fiber of
B -*1P1.
V' - 1P
1P
(6.24) Problem.
Study the case in which
A = d = 1
is not a hyperelliptic fibration.
and
p
§7.
Results of Lefschetz type We review several useful results which relate the
structure of
V
and
when
D
is a rung of
D
To
(V, L).
begin with, recall the following classical theorems. (7.1) Theorem.
Let
A
be an effective ample divisor
on a compact complex manifold M with A -- M be the inclusion. 1) wi(t): :ti (A) -i ai(M) a n-1(t)
is bijective for
is bijective for
i < n - 1
is bijective for
i < n - 1,
is injective and its cokernel is torsion free.
4) Hp'q(t): Hq(M, QM) - Hq(A, QA) if
A
is bijective for
is smooth or if p = 0.
tive for p + q = n - 1 A
and
is surjective.
3) H1(t): H1(M; Z) -* H1(A; Z)
< n - 1
i < n - 1
is surjective.
Hn-1(t)
Hn-1(t)
t:
Then
2) Hi(t): Hi(A; Z) -i Hi(M; Z) and
Let
dim M = n.
HP q(,)
p + q
is injec-
under the same condition.
is assumed to be smooth in the following assertions.
5) Pic(t): Pic(M) --' Pic(A)
is bijective if
n > 3.
This
is injective and its cokernel is torsion free if n = 3.
6) Pic(t): Pic0(M)
Pic0(A)
is an isomorphism if
n > 2
and is injective if n = 2. 7) Alb(t): Alb(A) -+ Alb(M) is an isomorphism if n > 2.
It
I.0-genus and the Apollonius method
56
is surjective and its general fiber is connected if
n = 2.
Outline of proof. Using Morse theory [Mil] we prove 1). 2) and 3) follow from 1).
A
Hodge theory yields 4) if
is
When p = 0, we use Kodaira's vanishing theorem.
smooth.
5) and 6) follow from 3) and 4). Remark.
7) follows from 2) and 4).
There are several generalizations of the above
results (see [Gro2], [Gro3], [Ha3], [Badl]).
(7.2) The structure of a variety is often described by a morphism defined on it having a special property.
Such a
morphism can be defined by the pull-back of a very ample linear system on its image.
relation between
V
Thus, in order to study the
D E ILI, we need various extension
and
theorems for morphisms, linear systems and line bundles.
For this, several vanishing theorems are very important for the technical background. found in [Sol].
Such a systematic approach is
We review here some slightly improved
versions of results in [Sol]. (7.3) Lemma (cf. [Sol;Lemma IB] & [F4;(2.1)]). and let
be an ample effective divisor on a scheme S a coherent sheaf on where
3"(t)
such that any
t > 0.
denotes
S
such that F ® 0S[tA].
Let
be
be an integer
i 1 0
is surjective for
Hi+1(S, 3) = 0.
For the proof, use the exact sequence Hi(A
T
A
Ker(Y - FA) = 3'(-l),
H1(S, F(t)) - H1(A, 3"(t)A) Then
Let
Y(t)A) -* Hi+l(S, 3(t - 1))
H1(S, 3'(t)) -*
Hi+l(S, S(t))
and
Serre's vanishing theorem as in (3.5.3). (7.4) Lemma (cf. [F4;(2.2)]).
ample divisor on a locally
Let
A
be an effective
k-Macaulay variety
V
and let
§7: Results of Lefschetz type
6
57
be a locally free sheaf on i < k
such that Then
and
Let
V.
be an integer
i
H1(A, S(t)A) = 0
for any
t g 0.
H1(V, 6) = 0.
H1(V, d(-i)) = 0
Proof.
for
by duality and by
>> 0
i
Using the exact sequence
Serre's vanishing theorem.
we prove
H1(V, 8(-t-1)) --b H1(V, 6(-t)) --, H1(A, 6(-t)A),
the assertion by similar arguments.
A
Let
(7.5) Lemma.
be an ample effective divisor on
a locally
2-Macaulay variety
bundle on
V
H1(A, (F - tA]A) = 0
such that
H0(V, F) -> HO(A, FA)
Then
t 2 1.
that the conclusion is equivalent to
Proof.
ILAI.
ILIA =
t 2 1.
Remark.
FA = 0
Suppose that
h0(V, F)
and similarly
if
Note
Then
Pic(V) -+
is injective.
Pic(A)
Then
by (7.4).
be as in (7.5) and suppose
for any
HI(A, [-tA]A) = 0
that
V, A
Let
for any
is surjective.
H1(V, F - A) = 0
This is clear since
(7.6) Corollary.
F be a line
and let
V
2
h0(A, FA) = 1
I-FI
by (7.5).
F E Pic(V).
Hence
# 0,
IFI
F = 0.
So
# 0.
for some
The assumption of this corollary is satisfied and if
char(.) = 0, dim V > 2
A
is normal.
However,
unlike the case in (7.1.5), the cokernel is not always torsion free, even if (7.7) Lemma.
dim V > 3.
Let
and
S
A
be as in (7.3) and let
be a linear system of Cartier divisors on
S
such that
dim pA(A) < dim A
A n BsA = 0.
Suppose that
morphism
pA
defined by
AA.
morphism
PA
defined by
A
Then
BsA = 0
for the and the
is an extension of
PA
such
A
I. A-genus and the Apollonius method
58 that
PA(V) =
PA
(A).
This is proved by a similar argument to that in (5.7). See [F4;(2.7)] for details. Let
(7.8) Corollary.
A
be an ample effective divisor
on a locally 2-Macaulay variety Let
F be a line bundle on
V
V
with
n = dim V a 3. BsIFAI = 0,
such that
Fn-lA = 0
and
H1(A, [F - tAJA) = 0
BsIFI = 0
and
PIFI(V) = PIFI(A).
for any
Then
t > 0.
For the proof, use (7.5) and (7.7). Let
(7.9) Corollary.
assume in addition that
A
A
and
V
be as in (7.8) and
is smooth and
char(A) = 0.
Let
f: A --' W be a surjective morphism onto a projective scheme
W with
dim W g n - 3.
FA = f H for some
Suppose that
H on
and some ample line bundle
F E Pic(V)
W.
Then f
V - W.
extends to a morphism Outline of proof.
By an argument using Kodaira's
vanishing theorem and Leray spectral sequence, we infer that H1(A, F - tA) = 0
for any
t > 0, so (7.8) applies.
See
[F4;(2.9)] for details. (7.10) Corollary.
and suppose that morphism
V
Proof.
V
Let
is smooth.
Then
f
extends to a
W.
We may assume
For a very ample line bundle such that
and W be as in (7.9)
A, V, f
dim W > 0 H
on
W,
and hence
we have
n > 3.
F E Pic(V)
FA = f H by (7.1.5), so (7.9) applies.
(7.11) Remark.
We can sometimes extend a morphism by a
similar method using vector bundles instead of line bundles,
since a vector bundle generated by its global sections gives
§7: Results of Lefschetz type
59
a morphism to Grassmann varieties.
See [F3,7] & [Mukl,2].
(7.12) Now we give another application of (7.5). Lemma (cf. [F4;(3.5)]). A E JaLl
variety and let is locally
projective space
,
in
dr)
V
is a weighted complete d
,
r
in the weighted
)
and that
, wn+r)
]P(a, w1,
H1(A, tLA) = 0
Proof. so
F(wl,
Suppose that
a > 0.
n = dim V 1 3.
is a weighted complete intersection of type
(V, L)
(dl,
(dl,
be a polarized
(V, L)
for some
2-Macaulay, (A, LA)
intersection of type
Then
Let
H0(V, tL) -f H0(A, tLA)
wn+r).
,
for any integer
by [Morl],
t
is surjective by (7.5).
Hence
(2.6) applies.
(7.13) Corollary. Let
A be an ample effective divisor
on a locally 2-Macaulay variety (.n-1,
that
0(1))
n = 2
some integer =
When
and
F e Pic(V).
F = [A].
Suppose that
Proof.
A n-1{A)
for some
Then
a.
Hence
a nFn.
n = 2
and
Then
H1(V, 0V) = 0.
a = Fn = 1
F = [A], V V
Hence
This implies
n > 2, or
Suppose that
Set
[A]A = 0(a)
by (7.6), so
[A] = aF
larities since it is smooth along so
(A, FA)
(V, F) = (Ipn, 0(1)).
n > 2.
Moreover
Noether's criterion.
such that
V
for
an-1
=
and (7.12) applies.
is a rational surface by has only isolated singuA.
Hence
V
is normal,
4(V, F) = 4(A, FA) = 0.
(V, F) = (]P2, 0(1)).
(7.14) Now we consider the following problem: Let
A be an ample effective divisor on a locally
2-Macaulay variety vector bundle
8
V
and suppose that
on a smooth variety
an exact sequence 0 -> q
9'
A = ]P(8) W.
for some
Does there exist
---> 8 -* 0 of vector bundles
I. A-genus and the Apollonius method
60
on
such that (V, A) = (IP(Y), F(6)) ?
W
(7.14.1) Remark.
under one of the following hypotheses:
Yes
is
extends to a morphism
is locally Cohen Macaulay, the answer
V
and if
f: V -i W
fA: A -+ W
If
a) [A]A
Ax
for any fiber
= 0(1)
of
fA.
x
b) r = rank(s) 2 3
and
Indeed, for any
x E W, Ax = fAl(x)
divisor on the locally Macaulay variety (V
X,
[Ax]) = (Fr00(1))
(8 ® 1V
is an ample Vx = f-1(x).
Hence
by (7.13).
Then
3 E Pic(W).
for some
is surjective.
Pic(V) --+ Pic(Ax)
So
(fA)*0A[A] =
Y = q 0 f*0V[A]
has
the desired property.
(7.14.2) We have the affirmitive answer if smooth, char(t) = 0 Indeed, fA tion map
and if
extends to
Pic(V) -* Pic(A)
V
is
r = rank(s) k 3.
f
by (7.10) and the restricHence
is surjective by (7.1.5).
(7.14.1.b) applies.
(7.14.3) When
r = 2, the problem is more subtle, but
we have several partial results [Bad2-5], [FaSS), [Fas2l, [SatS], [Satl].
(7.15) Now we consider the extension of blow-down. Lemma (cf. [F4;(5.5)]).
Let
A
be an ample effective
divisor on a compact complex manifold
Suppose that
M.
is the blowing-up of a projective manifold C
which is a submanifold of B
M
center
and
bundle over
r k 3.
as a divisor
is isomorphic to the blowing-up of N with
such that
Proof.
with center
of codimension
Then there is a manifold N containing B
C
B
A
A
is the strict transform of
The exceptional divisor C.
E
on
B
A
on is a
N. IPr-1
Using the Leray spectral sequence we obtain
§7: Results of Lefschetz type
H(E, [E - tA]E) = 0 H1(A, [-tA])
-
61
for any
H1(A, [E - tA]) - H1(E, [E - tA]E)
We have
t > 0.
Moreover
(7.1.5).
E = DA
E -+ C
very ample line bundle H L E Pic(M)
over we get
on
LE
by (7.1.5).
a
x
HA = LA
for some
for any
Hence
t > 0
E --> C
Ex
and
C
on
by using
extends to
is a hyperplane in
Since
C.
More-
HC.
Applying (7.14.1.b), we infer that
Fr- bundle over
over any point
D, we take a
is the pull-back of
H1(E, LE - tA) = 0
by (7.8).
Then
by
by (7.5).
IFI
extends to
B.
the Leray spectral sequence.
D -* C
D E
for some
Now,, in order to show that
F E Pic(M)
for some
[E] A = FA
and by
H1(A, [E - tA]) = 0
Kodaira's vanishing theorem, we get for
By the exact sequence
t k 0.
D
is
Dx = Fr
= 0(-1), we have
[E]E
x
= 0(-l).
[D]D
Hence, by (0.1.6),
D
can be blown down
x
smoothly along the direction manifold
N
and we obtain a
as desired.
Remark. r = 2.
D --> C
There are several partial results in the case
See [Fa],
[So5], [FaSl].
(7.16) The divisor
on N
B
is not always ample in
case (7.15), but we have the following criterion. Lemma.
Let
M
along a submanifold HC
such that
M
for some
over
C.
be the blowing-up of a manifold N Let
C.
Suppose that
is ample.
y > 0, where
Then H
H be a line bundle on N
E
is ample on
B
is ample on
is the exceptional divisor N.
For a proof, see [F4;(5.7)].
situation (7.15), then
ftHM - E
If
C
is a point in the
is ample by this criterion.
(7.17) We have seen that various conditions are imposed
I.0-genus and the Apollonius method
62
on the structure of such that variety.
(V, L)
if
ILI
contains a member
D
is isomorphic to a given polarized
(D, LD)
Sometimes these conditions turn out to be so
restrictive that no such pair
(V, L)
exists where
V
is
smooth, or even no normal pair at all, except the cone over (D, LD).
See
[Sol], [Bad2], [F9, 14], [Lviti3].
It often happens that, because of such restrictions,
the induced polarization for a given variety
LD
must be of very special type
D, provided
V
In his
is smooth.
series of papers [Badl "S. 4], Badescu observed such phenomena
for a number of varieties Fn-1,
1Ps x ]pt, IP1-bundles
D, for example, in the cases over curves, etc.
D =
Here we just
present the following sample result. (7.18) Theorem.
fold M D
Let
D = 1Pn-1
as an ample divisor.
be contained in a maniM .
If n > 2, then
n
and
is a hyperplane on it. Indeed, if
A = C, Pic(M) --* Pic(D) = Z
by (7.1.5), so (7.13) applies.
is surjective
By Grothendieck's Lefschetz
theorem (see e.g. [Gro3], [Ha3]), the same argument works unless
n = 3 = char(.).
This final exceptional case was
proved by Tango [Ta] using a direct approach, and later by Badescu [Bad2] using techniques of lifting to char(.) = 0. See (8.12) ti (8.15) for the details of this method.
§8. Classification of Del Pezzo manifolds Throughout this section let (M, L) be a smooth Del Pezzo variety as in (6.3) with n = dim M a 2 and d = Ln. (8.1) First we review the classical result concerning the case
n = 2.
if and only if
A polarized surface
K = -L
(S, L)
is Del Pezzo
for the canonical bundle
K.
63
§8: Classification of Del Pezzo manifolds If S'
S
by contracting
L' = L + E
E
to a point.
comes from
Pic(S')
for the canonical bundle ample by (7.16).
we have If
E, we get another surface
contains a (-1)-curve
Thus
K'
and
of
LE = -KE = 1,
Since
K' = -L'
Moreover
S'.
(L') 2 = L2 + 1.
S
is relatively minimal, then
or a Hirzebruch surface
Z'1
S = L'0.
or
S = IP2
S
S = Z0 z IP1
by the ampleness of
From these observations we infer that IP2
is
L'
is a Del Pezzo surface and
(S', L')
IP1-bundle over a curve. In the latter case
from
Pic(S')
in
is a X
F1
-K.
is obtained
S
by blowing up several points successively, unless Note that these points cannot be infinitely near
since any Del Pezzo surface contains no (-2)-curve. F2 and no six of Similarly no three points are colinear on them lie on a conic.
Note that every (-1)-curve
E
namely a smooth rational curve with
on
S
is a 'line',
LE = 1.
Conversely,
any such 'line' is a (-1)-curve.
(8.2) In the rest of this section, we study the case n k 3.
We assume
d x 5
in §6.
Therefore
L
lifting. [Fnlti3],
was studied
d s 4
is simply generated and very ample.
We first consider the case (8.12) to the case
since the case
char(St) = 0, and proceed in
char(3t) > 0
by using the technique of
The results are known to Fano and Iskovskih (cf.
[Isl, 2]), at least when n = 3
and
char(.) = 0.
Their method is based on the observation that a Del Pezzo 3-fold contains many lines and is covered by them. approach here is slightly different from theirs. (8.3) Claim.
d ;g
8.
Our
I. A-genus and the Apollonius method
64
Assuming
d k 9, we will derive a contradiction.
Taking general members of problem to the case
Then a general member
n = 3.
is a Del Pezzo surface.
ILI
d a 9.
since
Then any general member
n = 3.
S
The restriction mapping
is injective by the Lefschetz Theorem
r: Pic(M) ---, Pic(S)
In order to study
(7.1.5).
of
By (8.1), (S, L) = (IP233H)
is a Del Pezzo surface.
ILI
S
This contradicts (7.18).
(8.4) Suppose that of
if necessary, we reduce the
ILI
Im(r), we follow the argument
in [F5].
Recall that P
M c IPd+1
since
L
is very ample.
be the space parametrizing hyperplanes in
Let and set
IPd+l
P*
((x, h) E M X
F
corresponding to
x e Hh), where
I
h e P*.
Then
F
Hh
is the hyperplane
is a IPd-bundle over
M
P*
f: F --
and the fibers of
be the open subset of
B e Pic(S)
If
such that
is
be the image
comes from
and hence is
Pic(F)
G
S = f- 1 (0).
U
Let
be the monodromy
Im(ju).
Pic(M), then
comes via
B
On the other hand, if
G-invariant.
G-invariant, then
Let
ILI.
parametrizing smooth members of
n1(U, o) - Aut(H2(S; Z)) = Aut(Pic(S))
representation and let
B
are members of
o E U
and take a point
ILI /.c:
P
*
C1(B)@ E H2(S; Q)
comes from
H2 (F; IQ)
by Deligne's invariant cycle theorem [D;(4.1.1)].
H2(F; Q)
is generated by
0 *(l)
and
H2(M; Q)
since
F
P
is a
IPd-bundle over
H2(M; Q)
M.
since the restriction of
comes from
c1(B)Q
Therefore 0
(1)
to
S
is
P
trivial.
B E Im(r)
Hence
B
is mapped to a torsion in
Coker(r), so
by (7.1.5).
Thus, B E Im(r)
if and only if
B
is
G-invariant.
§8: Classification of Del Pezzo manifolds
65
In particular the canonical bundle of Moreover
G.
is stabilized by
S
preserves the intersection products of
G
Pic(S).
(8.5) Suppose now that case 1
A general member
n = 3.
We first consider the
d = 8.
P1
S
of
E
be the (-1)-curve on
is
ILI
1P1
x
by (8.1).
1
Assume that
Let
S = E1.
is the unique element in
[E]
Therefore
[E]
in (8.4).
Hence
Pic(S)
is invariant under the monodromy action as for some
= FS
[E]
F E Pic(M).
It is
H1(S, E - tLS) = 0
for any
E = DS
D E
Therefore we have
for some
(D, LD) = (1P200(1))
JFI
by (7.5).
and
[D] D = 0(-l)
(L + D)3 = (L + D)2L = (L + D)L2 = 9. where
[L + D] S = 3HS
the birational map Pic(M), so
L + D = 3H
HS = Ha + Hp
(M, H)
ti
of
S = F1 x F1.
and K = -2L = -4H
ruled out, and
Therefore
in
by (7.1.5).
More-
Therefore
Pic(M).
by (1.3). n = 4.
This contradicts (7.18).
n > 4
A general member Hence
(T, LT)
This case is thus
cannot occur either.
(8.6) Suppose that n = 3.
via
LS = 2Ha + 2H
is a Del Pezzo 3-fold as above.
(F3, 0(2)).
0(1)
comes from
HS
Then
H E Pic(M)
for some
(1P3, 0(1))
ELI
On the other hand
H E Pic(M).
Secondly we consider the case T
Hence
which is absurd.
Thus we conclude
L = 2H
by (7.13).
By (7.1.5)
for some
Hence
t > 0.
is the pull-back of
HS
S --'1P2.
9 = (L + D)3 =
over
S.
E2 = -1 = KE.
with
easy to see that
so
or
A general member
d = 7. S
of
First we consider the case ILI
is a Del Pezzo surface
of degree seven, and hence is the blowing-up of
F2
at two
I. A-genus and the Apollonius method
66
points. The proper transform
E
these points is a (-1)-curve on E
contracting
]P2
We get
X IP1
S.
IP1
S, namely
are exactly three (-1)-curves on
E
and those Since the
which are blown up.
IP2
by
It is easy to see that there
to a point.
over the two points on
passing
of the line on
monodromy action in (8.4) preserves the configuration of (-1)-curves, we infer that Hence, as in (8.5),
on M so
such that
M
E = DS
a Pic(S)
is
L + D
G-invariant.
for some effective divisor
(D, LD) = (1P200(1))
and
is the blowing-up of another manifold
comes from
by (7.16).
a Pic(M1)
Lt
We have
at a point
Mb
Moreover
(IP3, 0(2))
Kb
by (8.5).
of
Thus
and
Therefore
Mb.
M
is ample on
Lb
= d + 1 = 8
(L1')3
the canonical bundle
and
D
[D]D = 0(-1),
is the exceptional divisor (cf. (0.1.6)).
D
and
[E]
Kb
=
Mb
-2L6
for
(Mb, Lb) =
is the blowing-up of
IP3
a point.
Secondly we consider the case n = 4. member
T
(7.15), M
of
is a Del Pezzo 3-fold as above.
ILI
is the blowing-up of another manifold
point, and N contains
as a divisor.
IP3
Such a pair
ample by (7.16). (7.18).
Thus the case
n 2 4
(8.7) Suppose that
(N, IP3)
d = 6.
Again
n = 3
member
S
of
at three non-colinear points
p1' p2' p3.
lying over
at first.
of
is the blowing-up
ELI
Let pi
Then a general
Ei
be the (-1)-curve
and let
Zi
be the
proper transform of the line on
IP2
N
By at a
Moreover it is
does not exist by
cannot occur.
assume
IP2
Any general
at
§8: Classification of Del Pezzo manifolds
passing the two points other than on
67
Then the (-1)-curves
pi.
are exactly these six curves, which form a cycle.
S
Aut(Pic(S))
analyzing the structure of
By
and using (8.4), we
infer that there are the following three possibilities (see [F5;S5] for details):
1) The pull-back H
of
2) H - E.
comes from
3) Pic(M)
is generated by
for some
Pic(M)
Take
Case (8.7.1).
comes from
0 2(1)
Pic(M).
i.
L.
F E Pic(M)
FS = H.
such that
LS - FS = 2H - E1 - E2 - E3 = f 0(1), where
Then
F2
birational map onto
for any
H1(S, FS - tLS) = 0 Therefore
(7.5).
finite, so
which contracts the t > 0, so
S n BsIF1 _ 0
and
BsIFI
IL - F1
yield a morphism
p Ha = F
p H
and
L = p (Ha + H
is ample.
of
on
JaHa + bHPI
Hence
p: M -->F2 x ]P2
= L - F.
p
(L - F)3 =
tions of the mappings
such that
and
is a member
W = p(M)
for some
M ---, Fa
a, b.
The restric-
M - F2 to
b > 0. We have
and
= 3(a + b)deg(p), so
IP2,
some
W
r.
W
and
is defined by the equation
Here
H3{W}
a = b = deg p = 1.
012
and
r > 0
Hence
6 = L3{M} =
homogeneous coordinates (a:a:a)
since
W
are
S
both birational morphisms, so they are surjective. a > 0
and
IFI
is a finite morphism since
Therefore
Fa X F2
by
is at most
BsIF1 = 0 = BsIL - Fl, so
This implies
Hence
Zi's.
0 s (L - F)3, but
(L - F)3 + F3 = L3 - 3LF{S} + 3F2{S} = 0. F3 = 0.
is the
(FIS = IFS1
Similarly we have
F3 2 0.
f
For suitable (#0: 1:p2) :Er=0ajij
is irreducible.
= 0
of
Pa 2
for
Therefore
is normal and hence M . W by Zariski's Main Theorem.
Hence
r = 2
since
W
is smooth, so
(M, L)
is the scroll
I.0-genus and the Apollonius method
68
of the tangent bundle
and
F2.
Then
F E Pic(M).
is the contraction of
As in the case (8.7.1) we have
Z1.
Moreover, by (7.8),
by (7.5).
LS - FS = 2H - E2 - E3 =
f: S -- P1 X P
f (H77 + He), where E1
of
By symmetry we may assume that H - E1
Case (8.7.2).
extends to some
0
IFIS = IFSI
extends to
pIH-E I: S *--b F 1
a morphism
have and
IL - FIS = ILS - FSI
Therefore
Then
IL - FI.
S n BsIL - FI = 0
Therefore
(L - F)3 = 0
dim W t 2
F1 X P
77
and
p
such that
77
yield a morphism
p
L = 0(H + H77 + He).
L
0: M --+ P
We have
is birational.
0
Therefore
is ample.
P1 X Ft
X
6 = L3 =
Hence
H77 + HC)3 =
It is finite since
p
(L - F)3 = 0.
since
On the other hand W D p(S) = P1 X Fl, so W Now,
We also
Let W be the image of the map
BsIL - FI = 0.
defined by
by (7.5), so
is at most finite.
BsIL - FI
implies
F = p H,.
such that
(L - F)3 = L3 - 3LF{S) + 3F2(S) = 0.
and
F3 = 0
: M -i P
p
0
is an
isomorphism by Zariski's Main Theorem. Case (8.7.3). We will show that this case cannot occur. M c P7
Recall that
and
For any general hyperplane Su = M n Hu
infer that
open condition and
S
is a line in
E1
Hu
in
P
containing
E1.
lying on M lines.
Then
Each
is smooth.
and meeting dim A 2 2.
a > 0.
E1, we
Su
is a Del Pezzo E1, two of
Hence there are infinitely many lines in E1.
Since
Let
A
Pic(M)
every non-zero effective divisor on M for some
P7 d=
is smooth, since smoothness is an
surface, and contains five (-1)-curves besides which meet
P
So
A
is ample and
P
be the union of such is generated by is a member of A n Z1 0 0.
L,
IaLl
Hence
§8: Classification of Del Pezzo manifolds
there is a line
Therefore
on M which meets both
L
and
E1
Z1.
for the hyperplane H with M n H = S.
£ c H
Clearly
69
£ c S
and
is a (-1)-curve on
l
is no such curve meeting
and
E1
But there
S.
Thus we get a
Z1.
contradiction, as desired.
Next we consider the case
(8.8)
n = 4, any general member
on
t > 0.
T
Hence
So
If
T
and
f
is a member of
LT = [Ha + Hf]T.
extends to M
L.
Moreover
Similarly
f: M 2
P2 a
x lP2
(Ha + H4(Pa X P2) = 6,
Since
f
comes
for any
by (7.8).
is finite since
L
is
is an isomorphism by Zariski's Main Theorem.
is of the type (8.7.2), then
LT = H
[Ha]T
H1(T, Ha - tLT) = 0
Thus we get a morphism
f*[Ha +
is birational.
ample.
and
T -->lP2
extends.
F2
such that
f
Pa X F2
by (7.1.5) and
Pic(M)
from
is a Del Pezzo 3-fold
ILI
is of the type (8.7.1), T
T
IHa + HMI
of
When
It is of the type (8.7.1) or (8.7.2).
as in (8.7). If
T
d = 6, n s 4.
+ H77 + HC.
Each map
by (7.9).
Thus we get a morphism
such that
L = f (H
+ H77 + He).
T = P
T - P1
X P17
x lP
extends to M X Fl X F1
f: M --+ P
f has positive
But
dimensional fiber, contradicting the ampleness of
L.
Thus
this case is ruled out.
When n = 5, any general member D isomorphic to a morphism
Pa X F2
of
is
ILI
by the above observation.
f: M -+ Pa X F2
such that
(7.9), which yields a contradiction.
ruled out, and the case n > 5
L = f*(Ha + H
by
Thus this case is
cannot occur either.
(8.9) It now remains to consider the case case is studied fully in [F5-II].
We obtain
d = 5.
This
But the proof is terribly
I. A-genus and the Apollonius method
70
Here we just sketch the results.
complicated. Let
linear
parametrizing
be the Grassmann variety Gr(5,2)
C
C5 (or equivalently lines in
C2's in
embedded in
by the PlUcker embedding, and
1P9
is a Del Pezzo 6-fold of degree any Del Pezzo manifold with this variety
5.
This is
F4).
(G, 0(1))
Our main result is that is a linear section of
d = 5
In particular n g 6.
G = Gr(5,2).
The proof in [F5-II] is based on a certain birational
correspondence between M
and
described as follows. F2 C P5 Let C = F1 X F5
D =
along
JPa
theoretically in
Let
be the blowing
P#
be the exceptional divisor over
be the proper transform of
D#
and let
E
C,
When n = 6, this is
be the Segre embedding and let
be a linear embedding.
C 1Pa
up of
Fn.
D
on
C
Scheme-
P#.
is the intersection of three hyperquadrics
C
The linear system generated by these hyperquadrics
D.
yields a morphism D# --+F2.
Actually this is a
1P3-bundle
can be blown down smoothly along this direction.
and
D
Let
P# -> M6
be the blow-down.
Then
2Ha - E = L
for
P
some
L E Pic(M6), and it turns out that
Pezzo 6-fold of degree
(M6, L)
is a Del
5 (cf. [F5-II;(7.12)]).
A similar construction can be carried out in the cases n = 4 C
and
n = 4, we start from a Veronese curve
When
5.
of degree three in
rational scroll
C
case we blow up
Fn
transform of
D
D = 1P3.
When n = 5, we start from a
of degree three in along
suitably.
D = ]P4.
In either
C, and then blow down the proper See [F5;(7.8) & (7.10)] for
details.
The situation is a little different when
n = 3.
There
71
§8: Classification of Del Pezzo manifolds
is a certain birational transformation between M
and
Q3
but another transformation from a smooth hyperquadric
of
By blowing up
Q3.
be a smooth hyperplane section
D
Let
is more important.
]P3,
Q3 along a certain curve
D
in
C
D, we get a
and then blowing down the proper transform of
birational transformation onto a Del Pezzo 3-fold of degree See [F5;(7.4)].
five.
In the proof of the main result, we reconstruct birational transformations as above from the side of
This is done inductively on n
M.
and requires a lot of
computation.
From this we infer that every Del Pezzo n-fold
of degree
is isomorphic to the others for each fixed
5
C c D c Fn (or
Indeed, the triples
as above are all
Q3)
Therefore, from this
projective equivalent to each other.
very observation, it follows that
n.
M
is a linear section of
Gr(5,2). below.
See [F7] for technical details.
We may assume ILK
on
is ruled out by the argument outlined
n > 6
The case
must be
Then any general member
n = 7.
on
8
t > 0
and
(7.5).
We have
M.
for any
t k 0.
H0(M, 6) = H0(G, 6G)
is surjective for any
H0(M, 8[tL]) 0 H0(M, L)
for any as in
H0(G, 6[tL]G) 0 H0(G, LG) -+
We also see that
H0(G, 8[(t + 1)L]G)
to a vector
G
Hi(G, 8[-tL]G) = 0
Hence
i = 0, 1.
of
By using various vanishing theorems
Gr(5,2).
G, we extend the universal bundle on
bundle
G
--'
HO(M, 8[(t + 1)L])
This implies that
6
global sections, so we get a morphism extending the identity of
t a 0.
G.
Hence
is surjective
is generated by p: M --> Gr(5,2) = G
As in (8.8), this yields a
I. A-genus and the Apollonius method
72
contradiction by an argument in [Sol;Lemma I-A].
Likewise the Grassmann variety
Remark.
Gr(n, r)
cannot be an ample divisor on another manifold unless r = 1, n - r = 1
(n, r) = (4, 2).
or
(8.10) Remark.
See [F7;(5.2)].
d = 5, there is a completely
In case
different approach due to Mukai.
His method uses the
restriction of the universal bundle on
M y Gr(5,2)
embedding
Gr(5,2).
The
can be recovered by reconstructing
This approach works for many other
this vector bundle.
types of Fano manifolds (cf. [Mukl,2]). (8.11) Summing up, we now establish the following. Theorem.
Let
(M, L)
n = dim M k 3
and
d = Ln.
be a Del Pezzo manifold with When
char(A) = 0,
(M, L)
is
of one of the following types. 1) d = 1 6
and
(M, L)
is a weighted hypersurface of degree
in the weighted projective space
IP(3, 2, 1,
,
It
1).
has a structure as in (6.14) - (6.16). 2) d = 2
and
(M, L)
is a double covering of
along a smooth hypersurface of degree
pull-back of
branched
is the
L
4.
IPn
0(1).
3) d = 3
and
M
is a hypercubic in
4) d = 4
and
M
is a complete intersection of two hyper-
quadrics in 5) d = 5
Ipn+2.
and
M
is a linear section of
6) d = 6 and M is either for the tangent bundle
6
]pl x IP1 of
and
M
8) d = 8
and
(M, L) = (]P3, 0(2)).
L
Gr(5,2) c 1P9.
x Ipl,
1P2
x
F2
or
IP(0)
F2
7) d = 7
Remark.
L = 0M(1).
IPn+l.
is the blowing-up of
is determined by
M
IF3
since
at a point.
K = (1 - n)L.
73
§8: Classification of Del Pezzo manifolds
char(A) = p > 0.
Now we study the case where
(8.12)
The conclusion will be the same as in (8.11) if
d
3.
Following Badescu's idea [Bad2], we use the technique of lifting to characteristic zero.
Here we review the argument
in [F11].
(8.13) Key lemma.
d = L3 a 5
with
Then the pair
Let
D
and let
be a Del Pezzo 3-fold
(M, L)
be a general member of
is liftable.
(M, D)
This means that there is a smooth morphism over the ring
Spec(W(.))
W(it)
9
A( -p S
a =
of Witt vectors together
on
A(
such that the fiber of
over the closed point of
S
is isomorphic to
with an effective divisor (A(, 9)
JLJ.
Note that
(M, D).
is a discrete valuation ring of
W(R)
characteristic zero with residue field
A.
The proof consists of several steps. 1) By (8.1), D at
points with
r
F1 X P1
is
0(D)
2) A general member We obtain
Hence
r = 9 - d .g 4.
for the tangent bundle
curve.
or the blowing-up of
C
of of
H1(D, 0(D)) = 0
D.
is a smooth elliptic
ILDI
H1(C, 0(D)[tL]) = 0
for any
using the exact sequence 0 -* OC -- 0(D)C Hence
h1(D, 0(D)[tL])
this implies
t
3) We have an exact sequence -s 0
of vector bundles on
bundle of step 2).
So
M.
Hence
4) Let
3
for any
0(M)
H1(D, O(M)[tL]D) = 0 H2(M, 0(M)) = 0
be the
by
By step 1),
t a 0.
O(M)D -+ LD
0 --> 0(D)
D, where
t > 0
0C[L] ---> 0.
h1(D, 0(D)[(t - 1)L]).
H1(D, 0(D)[tL]) = 0
1P2
is the tangent
for any
t a 0
by
by (7.3).
OM-ideal defining
D.
0(M)
can be
I. A-genus and the Apollonius method
74
identified with the sheaf of derivations of a natural homomorphism 0(M, D)
0(M) -- Yt*mM(3/32, 0M/3).
Let
M
This is locally free on
be its kernel.
and we
0 - 0(M, D) - 0(M) -+ OD[L] --> 0.
have an exact sequence Hence
OM, so we have
H2(M, 0(M, D)) = 0
by step 3).
5) By the obstruction theory for the liftability due to Grothendieck [Gro2;expose IQ],
formal completion
is liftable over the at the closed
S = Spec(W(.))
of
S^''
(M, D)
point.
This formal lift is algebraizable since
ample.
Thus the proof of the Key lemma is complete. Let
(8.14)
The generic fiber field
of
A'
over
Let
*
(M , D
3-fold with
L
polynomial implies H1(M, L) = 0
)
[D*], (ML*) is
(M', L')
a Del Pezzo
implies
S.
The invariance of the Hilbert g(M', L') = 1.
and
Moreover
h0(M, L) = rank f*Y = h0(M', L'),
= g(M', L') = 1
(8.15) Lemma.
is a smooth family of
(4, Y)
(L')3 = d
d(M', L') = d(M, L) = 1. g(M , L
=
q = [9],
polarized manifolds over
*
be the algebraic closure of
Si
d = (L*) 3.
Indeed, for
*
as above.
be the scalar extension of
)
Then, for
Si*.
S
is
is defined over the quotient
(M', D')
W(St). *
and let
St'
be the family over
(Al, 9)
D
Hence and
(L*)3
=
*
*
4(M , L
so
(L')3 = d, )
= d(M', L') = 1.
There is a natural commutative diagram
Pic(M)
--> Pic(D)
1*
1*
Pic(M ) -' Pic(D ), where the vertical maps are bijective. The proof consists of several steps. 1) Let
m
be the maximal ideal of
W(A).
We have an
75
§8: Classification of Del Pezzo manifolds
exact sequence
0 --> OM --> 0 j
x
--> O j_ i- 0,
where
Ojx
is the sheaf of multiplicative groups consisting of invertible sections of
for any
j.
Similarly 2) 2
So H1jx
0jM = 0A/mj0A.
Hence
Pic(AZ)
Pic(O)
Pic(D)
for
ti Pic(M) for
AZ = A x S
rb = A xS S.
So, for any
is ample on
F e Pic(AZ), there
is an exact sequence A --* B --> F -f 0 on and 8
B
Replacing 2
bundles
d
extends to
and
0
by
on
y: 4 - R.
A
St, we extend
A.
3' = Coker(y)
Pic(A) -4 Pic(AZ)
is obvious.
Similarly
3) The map is smooth.
to vector
B
is an extension of So
M.
Pic(g) -+ Pic(g)
It is injective since
F
3 e Pic(A).
is bijective.
is surjective since
A - M' = M
divisor defined by the principal ideal
4) We have
206
is surjective, while the injectivity
Pic(A() -+ Pic(M')
Pic(g) -* Pic(D')
and
A
Then the homomorphism A - B
and is invertible in a neighborhood of
So
such that
AZ
are direct sums of line bundles of the form
e Z.
Thus
H1(Oj_i
m0 A.
A(
is a Cartier Similarly
is bijective.
Pic(D') = Pic(D)
by the preceding steps.
rank(Pic(D')) = 10 - d = rank(Pic(D )).
Moreover, the
above bijection preserves the intersection pairings by its definition 1) '. 3).
Hence the determinant of the matrix
formed by the intersection numbers with respect to a Z-base of
Pic(D')
is
Therefore
(-1)9-d.
Pic(D') -> Pic(D
)
is
bijective.
*
5) Pic(M ) --i Pic(D * step 4), G = Gal($ /St')
acts trivially on
* )
is injective by (7.1.5). *
By
acts trivially on Pic(D ). So * * Hence Pic(M') _- Pic(M ). Pic(M ).
G
I.0-genus and the Apollonius method
76
Combining these observations we prove (8.15). Let
(8.16) Lemma.
with n > 3
and let
be a Del Pezzo n-fold
(M, L)
D be a smooth member of
Then
ILI.
Pic(M) = Pic(D).
for any
Hi(D, tLD) = 0
Pic(D), where Moreover
11
t E Z, i = 1, 2.
theorem (cf. [Gro3]).
Suppose
Note also
at three points.
IP2
H - Ei comes from Pic(M)
H, Ei E Pic(D)
this we infer that
D.
Then, by
or F(e).
Fl X F1 X IP1
IPl X IP1 X IP1,
by (8.15), where
along
d = 6.
be as in (8.14).
is the blowing-up of
When M*
M
This completes the proof.
Let M
n = 3.
is either
D
that
Pic(111) =
by Grothendieck's Lefschetz
(8.17) Let us now consider the case
(8.7), M
So
is the formal completion of
Pic(M) = Pic(1)
first that
Hence
is also a Del Pezzo manifold.
(D, LD)
Proof.
are as in (8.7).
M = F1 X F1 X F1
From
by the same argument
as in (8.7.2).
When M M * P(©)
*
ti F(6), H
by (8.15), so
Pic(M)
by the same reasoning as in (8.7.1).
Next suppose Then
ILK.
comes from
n = 4.
(D, LD)
Let
D
be a general member of
is a Del Pezzo 3-fold as above.
(8.16) in place of (7.1.5), we obtain M *
F2
X
F2
Using by the
argument in (8.8). The case
true when
n > 4
is easily ruled out.
Thus (8.11) is
d = 6.
(8.18) Similarly we establish (8.11) also in the case d k 7.
The proof is simpler than in the case
details, see [Fli].
d = 6.
For
§9: Polarized varieties of A-genus one: Remaining cases
d = 5, the problem is more subtle.
(8.19) When
It is easy to show that
Suppose that
n = 3.
generated by
L
char(.) = 0,
the normal bundle
is
0 ® 0
77
Pic(M)
and that there are many lines on
is
If
M.
M
of a general line on
.N
by the generic smoothness theorem.
This is a key
observation in the classical theory of Fano-Iskovskih.
By
blowing-up along such a line we recover a birational correspondence with a smooth hyperquadric as in (8.9).
If 3
I= 0(-I) ® 0(l), we still get a birational map onto
,
but its structure is more complicated than in the case X
0 ® 0. When
char(A) > 0, however, it may be possible that
N ti 0(-l) ® 0(1)
for every line on
M.
As a matter of
fact, we can rule out this possibility by analyzing the
structure of lines on M
very precisely, but the proof is
very lengthy and is omitted here.
We encounter new problems in higher dimensions too. However, with some effort, we can adapt the argument in [F5-II] and get the same result as in the case
I hope that
char(.) = 0.
Mukai's method (8.10) works also in the
positive characteristic case and gives a simpler proof. Thus, (8.11) is true even if few cases in which
char(.) > 0, except for a
char(.) = 2 a d.
§9. Polarized varieties of A-genus one: remaining cases Throughout this section let (V, L) be a polarized variety with n = dim V a 2 and A(V, L) = 1. Set d = Ln and g = g(V, L). The results are summarized in (9.18). (9.1) We assume that
indeed true either if
V
(V, L)
has a ladder.
This is
is smooth at each point of
BsILI
I.0-genus and the Apollonius method
78
g < d, char(s) = 0
(cf. (4.15)), or if
Cohen Macaulay at each point of
is locally
V
and
and 2-Macaulay
BsILI
everywhere (cf. (4.16)).
There are examples which have no ladder though
A = 1.
But such polarized varieties are not treated in this book. (9.2) In §6 we assumed that
g s 1.
This is true if
is locally 2-Macaulay (cf. [Fll;(5.2)] and (3.4)). consider the case
assumption Let
g s 0, so
j > b.
that
and
(5.17).
n Dn-j
and the
Vb, so
L)
Dn-b a
for each Vb
v-1(Vb) = Vb.
is locally Macaulay by
Therefore
V'
Vb.
By (0.2.8), V
Now, perturbing
is not normal if
for the normalization
such
In particular
j 2 b.
little if necessary, we get a ladder of
Thus, V
ILI
Moreover,
dim(Sing(V) n Vb) s dim Sing Vb
since
Di's are ample.
each rung is normal.
Then
is surjective for
is locally Macaulay along
V
dim Sing(V) < n - 1 < b - 1
s 4(V, L) = 1.
D1,
On the other hand
V = Vn D by the
A(Vb, L) = 0.
H0(V
--+
Therefore we have
Hence
V1 = 1P1
0 = 4(V1, L1) s
H0(V, L)
Vi = D1 n
BsILJ c Vb.
along
Then
(V, L).
be the maximal number with
b
d(Vj, L) = 1 any
be the normalization and let
be a ladder of
D V1
Here we
g s 0.
v: V' -+ V
Let
of
(9.3) We studied the case
(V', L
4(V', L) = 0
g s 0, and
is normal
v-1(Di)
a
) such that V by (3.4).
4(V', L) = 0
V.
g k 1
and
d s 4
in §6.
In the remainder of this section we study the case where is very ample.
V
By (6.2.3), L
is very ample if
L
and
g k 1
d 2 3 (provided a ladder exists, as always). (9.4) We will review the theory in [F17].
Since
L
is
§9: Polarized varieties of A-genus one: Remaining cases
very ample, we have V
that
V c
Fn+d-2
L = 0V(1).
We assume
is singular, since the smooth case was studied in
We further assume that
§8.
and
79
is neither a hypercubic nor
V
a complete intersection of two hyperquadrics.
Using (5.13), we reduce the problem to the case where Ridge(V) = 0.
This is equivalent to saying that
is not
V
This condition is assumed
a cone over any other variety. in the rest of this section.
(9.5) Take a singular point
on
v
which is the union of all the lines passing point on
V (cf. (5.12)).
Ridge(V) _ 0.
Hence
for
section M Moreover
of
Ha
let
R = Ridge(W)
ifi
by (5.14).
By (5.15), we have
and r
v e R
and
=
det
dim R = 0.
4(M, 0(1)) = 0.
be the blowing-up of
be the pull-back of
scroll of
since
and for some smooth linear
Note that
W.
and another
dim W = n + 1
and w = d - 2.
dim M = n - r
(9.6) Let
v
Moreover w = deg(W) s d - 2
A(W, 0(1)) = 0
W = R * M
Then
W = v * V,
Set
V.
Then
°W(1).
6 = 0M(1) ® 0M (r+l) on
W
M.
along
R
is the
(S, Ha)
The unique member
*
of
f
where a
is the exceptional divisor of
IHa - f 0M(1)1
and
is the map
f
a:
T - W
give an isomorphism D= R x M.
aDI
R
LJ -L M al
W
f*BI
B E Pic(M).
For a proof, see [F17;(8)].
Fn+d-2
C
(9.7) The proper transform for some
D
Note that the restrictions of
TiJ --b M.
D
IHa +
and
a
?
of
V
Moreover
is a member of BsIBI = 0.
I.0-genus and the Apollonius method
80
(9.8) Key Lemma.
contains the fiber over
V
If
contains no fiber of
11
n : D -* R. D
x E R, we can show that
x e Ridge(V), contradicting the assumption in (9.4).
See
[F17;(9)] for details of the proof of the Key Lemma.
(9.9) The type of the polarized manifold is classified in (5.10).
There are three cases:
is a Veronese curve of degree
(c) M
(M, 0M(1))
d - 2.
(v) (M, 0M(1)) = (IP2, 2Hso d = 6
r = n - 2.
and
is the scroll of a vector bundle
(s) (M, OM(1))
on
6
IP77.
We will proceed further by case-by-case arguments.
M
type of
depends on the choice of
v E Sing(V).
(9.10) In the case (9.9.c) where we have
(d - 2)H P, IPn-l-bundle
over
infer that
V E
M
IHa + 2f H
The
T, and
on
OM(1) _
and
M = IP1
is a
T/
Using (9.8), we
and hence smooth.
is the normalization of
Moreover
V.
L) = 0.
The divisor
nD
defined by an equation where
(#O:8l)
on
is
D = R x M = pa-1 x IP1
£0(a)P02 + £1(a)fl0fl1 + £2(a),612 = 0,
is the homogeneous coordinate of
M
and the
Gi's are linear forms in the homogeneous coordinate (a0: :ar)
of
implies
Then
R.
This
n - 1 = r t 2.
The singular locus of
singularities along
R.
V
is
R
and
V
turns out to be normal and
Pezzo variety.
V
has non-normal
See [Fl7;p.153] for details.
(9.11) In the case (9.9.v), we see Moreover
by (9.8).
(GO = tl = 12 = 0) _ 0
V E
IHa + f HPI.
(V, L)
See [F17;p.154] for details.
is a Del
§9: Polarized varieties of A-genus one: Remaining cases
The divisor
n D
D = 1Pa x F2
on
is defined by Li's are linear
10(a)fl0 + L1(a)fll + t2(a)P2 = 0, where the
forms in
n - 2 = r t 2
Hence
ar.
a0,
81
by (9.8).
For
further details see [F17;p.155].
(9.12) In the case (9.9.s), set is generated by
Pic(M)
of the map
M --1P77.
HC
HC = 0M(1).
and the class
H77
Then
of a fiber
It turns out that there are the
following three cases:
(so) B = 2Hwhere B
is the line bundle as in (9.7).
(si) B = HC - (d - 4)H27.
(su) B = 2H
- (2d - 6)X17.
For the proof, see [Fl7;p.156]
(9.13) In the case (9.12.so), V
Thus the situation is that of (9.2).
g = 0.
unlike the case (9.9.c),
1
Analyzing the divisor r t 2
is not normal and
in this case.
However,
is not the normalization of [/ n D
on
D
V.
as before, we get
See [F17;pp. 157-160] for further
details.
(9.14) In the case (9.12.si), (V, L) Pezzo variety. follows.
The vector bundle
In each case
r
8
is a normal Del
is classified as
is bounded by (9.8) as before.
1) 8 = 0(4, 2) (def [4H,] ® [2H72]), n - 2 = r = 0 and d = 8. 2) 6 = 0(3, 2), n - 2 = r = 0 and d = 7. 3) 6 = 0(3, 1), n - 2 = r s 1 and d = 6. 4) S 0(2, 2), n - 2 = r s 1 and d 6. 5) 8 0(2, 1, 1), n - 3 = r = 0 and d = 6. 6) 8 = 0(2, 1), n - 2 = r s 2 and d = 5. 7) 8 = 0(1, 1, 1), n - 3 = r g 2 and d = 5.
I.0-genus and the Apollonius method
82
The type of singularity of precisely.
V
is described very
See [F17;pp. 161-170] for details.
(9.15) In the case (9.12.su), it turns out that 8
0(1, 1)
and M = 1P1 x 1P1
with
HC = H77 + H.
changing the role of the two rulings of
d = 4, Hence,
M, we have
(9.12.so).
(9.16) Summing up, we obtain the following. Theorem.
d(V, L) = 1
V
Let
for
be a subvariety of
L = 0V(1).
1PN
Suppose that
V
such that is singular,
is not a cone over another variety, and is neither a hypercubic nor a complete intersection of two hyperquadrics. Then
(V, L)
zation
V'
is a normal Del Pezzo variety, or the normaliof
V
is a smooth variety with
Indeed, (V, L)
4(V', LV,) = 0.
is Del Pezzo in the cases (v) and (si),
while the latter is true in the cases (c), (so), (su). the case (c), (V, L)
In
is a non-normal Del Pezzo variety.
(9.17) Theorem (cf. [F17;(2.9)]).
normal Del Pezzo variety as in (9.16).
Let Then
(V, L)
be a
(n,d) = (2,8),
(2,7), (2,6), (2,5), (3,6), (3,5), (4,6), (4,5) or (5,5).
This follows from the results in (9.11) and (9.14). Now we summarize the results in the chart of the next page.
§ 10: Polarized manifolds of A-genus two (9.18) Chart.
83
Classification of singular polarized d-genus one.
varieties of
(V, L)
YES
has a ladder ?
I
1N0
g(V, L) > 0 ?
???
g s 1
g = 0
I
(6.14) ti
d-4
(9.2)
k5
(6.10) (6.7)
ti
(6.23)
d-3
d- 2
d- 1
L
(6.8)
is very ample
(6.13) (9.9)
(c)
(9.11) g=l
non-normal Del Pezzo
normal Del Pezzo
non-normal
is not very ample
(s)
(v)
(9.10) g=1
(so) (9.13) g=0
L
generalized cone
1
(9.12)
(su)
(si) (9.14) g=1
(9.15) d=4 same as (so)
normal Del Pezzo
§10. Polarized manifolds of 4-genus two Here we study the case d = 2. Since the technical difficulties increase as d becomes larger, we consider only smooth polarized varieties for the sake of simplicity. Moreover we assume char(s) = 0. Our method does not work The results are summarized fully without these assumptions. in the chart (10.12). (10.1) Key Lemma.
with
Let
(M, L)
dim M = n 1 3, d = Ln 2 2
and
be a polarized manifold d(M, L) = 2.
Then any
I.0-genus and the Apollonius method
84
general member
D
of
regular rung, namely
4(D, LD) = 2 D
The smoothness of
Moreover D
is smooth.
ILI
and
ILDI = ILID
can be proved by the method in
§4, but the argument is far more complicated. [F12;(4.1)] for details.
4(D, L) s d(M, L)
See
As for the last assertion, we have
in general.
d(D, L) S 1, then
If
Hence
H1(D, OD) = 0
by the preceding results.
h1(M, -L) = 0
by Kodaira's vanishing theorem.
ILID =
ILDI
is a
hI(M, OM This implies
4(D, L) = 4(M, L).
and
By virtue of this lemma we reduce many problems to two-dimensional cases by the Apollonius method. (10.2) Theorem.
Let
be a polarized manifold
(M, L)
with n = dim m 2 2, d(M, L) = 2
and
g(M, L) s 1.
is the scroll of a vector bundle
(M, L)
elliptic curve
Then
over an
6
C.
A proof can be found in [F16;(0.2)].
Another proof can
be obtained by using the theorems (12.1) and (12.3) in the next chapter. Remark.
We obtain
h0(M, L)
by using Atiyah's theory [At].
=
h0(C,
48)
Therefore
= c1(6) = Ln
2 = 4(M, L) = n
in this case.
(10.3) In the rest of this section let polarized manifold with g = g(M, L) k 2. case
d = 1
n = dim M k 2, d(M, L) = 2
We assume further
be a and
d = Ln > 1, since the
is difficult to study.
(10.4) When results.
(M, L)
dim BsILI
> 0, we have the following
See [F16;(1.14) & (1.17)].
1) Y = BsILI
is a smooth rational curve with
LY = 1.
§ 10: Polarized manifolds of 0-genus two
n: M' --p M be the blowing-up along
2) Let
be the exceptional divisor over 3) Let
85
M' -
W be the image of the morphism *
defined by
in
Then
L - El.
E
Bsin*L - El _ 0.
Then
Y.
and let
Y
Fn+d-3
dim W = n - I, deg W = d - 1
4(W, 0(1)) = 0.
and
4) E
is a section of
5) p
is flat and every fiber of
6) If
s 1.
y
is a polarized manifold of the above
type for any general member
D
of
ILI.
7) There exists a morphism
0: M --> Y = ]P1
y E Y, the fiber
over
any
(My, Ly)
y
L
Y 8) d a n.
such that, for is a polarized
d(My, Ly) = 4(My, L ) = 1 y is the restriction of L.
and
Moreover, if
is a trivial
variety with where
is determined
y
g = (d - 1)y. (D, LD)
n a 3,
is an irreducible
p
reduced curve of arithmetic genus by the formula
E = W.
and
p: M' --> W
fibration and
d = n, then
is the Segre product of
(M, L)
and a fixed polarized manifold
FXN and
0
g(My, Ly) = y,
(N, A).
(1P27 ,
This means
H7)
M
L = [HM + AM.
(10.5) By (10.4.7), we can proceed further as in the d = A = 1 (cf. §6).
case
Theorem (cf. [F16;(2.4)].
and suppose in addition that covering
f: M' -> P
B
of
l3H.I
f
on
IHc - 2HI, where
member of
is
with
y = 1.
of the scroll
8 = 0E ® [-2EJE
bundle
Let things be as in (10.4)
S + B1, where
E = W.
Then there is a double
(P, H) of the vector S = f(E)
H, = [-E]p. B1
is the unique
The branch locus
is a smooth member of
S n B1 = 0.
(see the diagram in the next page)
1. 0-genus and the Apollonius method
86
Y-E
S
n
n
n n
M
f--
x
W
f
V
M' -* P - W
This corresponds to the results in (6.15) and (6.16). If
y a 2
and if every fiber of
is hyperelliptic, we
p
have similar structure theorem to those in (6.21), (6.22) See [F16] for details.
and (6.23).
There are also cases in which general fibers of
are
p
These cases seem more difficult to study
non-hyperelliptic.
and are left to the reader.
(10.6) In the rest of this section we assume that
We will study the cases
dim BsILI 9 0.
d > 4, d = 4, 3, 2
separately.
(10.7) Suppose (4.15) and L
L
d > 4.
Then
has a ladder by
(M, L)
is simply generated by (3.5.3). In particular Moreover
is very ample.
g(M, L) = 2
and
H1(M, tL) = 0
for any t E Z, 0 < i < n. Next we claim
bundle K Indeed, if
of
D
ample.
We have
This is proved by induction on
M.
of
ELI
such that
p E D, since
KD
n.
of
L
is very
h1(M, K + (n - 2)L) = hn-1(M, (2 - n)L) = 0
IK + (n - 1)LID = IKD + (n - 2)LDI
bundle
for the canonical
p E BsIK + (n - 1)LI, we can find a smooth
member
and
BsIK + (n - 1)LI = 0
D.
by the choice of
p
for the canonical
is a base point of this linear system
D, which contradicts the induction hypo-
thesis.
By induction we have also we have a morphism Let
X
f: M - 1P1
be any smooth fiber of
h0(M, K + (n - 1)L) = 2. with f.
So
K + (n - 1)L = f H Then we have
Ln-1X
.
=
§ 10: Polarized manifolds of A-genus two
87
(K + (n - 1)L)Ln-1 = 2g(M, L) - 2 = 2.
is very
The latter does not occur since the
]Pn-1's.
X
canonical bundle of h0(X, LX) = n + 1
of rank
L
is a hyperquadric or a disjoint union of two
ample, (X, Lx) linear
Since
Kx = (1 - n)LX.
is
f*0M[L] _ 8
and
Therefore
is a locally free sheaf
n + 1.
There is no multiple fiber F = 2Y, [Y]Y
F
of
Indeed, if
f.
would be numerically trivial and we would have
1 = x(OX) = x(OF) = x(OY) + x(OY[-Y]) = 2x(OY), which is absurd.
Thus, every fiber
hyperquadric since
is an irreducible reduced
Mx
Ln-1Mx = 2
is very ample.
L
and
From this we infer that the natural homomorphism *
f 8 - OM[L]
is surjective, so we have a mapping
with pH = L
(P, H
onto the scroll
For every point x on IP1, the
map
the embedding of the hyperquadric of
12HC + bH,J
we have
on
for some
P
d = HnM = 2Hn.+1 + bHn_H
hand, we have
p
of
x : Mx
'
-- Px
Thus
Mx.
b e Z.
M
b + e = 3, which implies
ti ]Pn
= 2e + b.
e = d - 3
We can further classify
and
IP
is
is a member e = cl(S)
On the other since the
-(n + 1)HC + (e -
is
P
over
Setting
K = [(1 - n)HC + (b + e - 2)H ]M
canonical bundle of
p: M --i P
Hence
b = 6 - d.
as follows.
(M, L)
(10.7.2) When n = 2, one of the following conditions is satisfied.
0) M
JP
x ]P
- E, where
E
is the sum of the (-1)-curves
L = 2H27 + 3H
over these points. 1) M
(12 - d) points and
is a blowing-up of
is
at
d g 12.
I1 (def IP(H
® 0)), or is a blowing-up of
a point lying on the (-1)-curve on
1'1.
I1
at
L = 2Ha + 2H - E,
I. A-genus and the Apollonius method
88
where
Ha
E
0 (if
d
is
is the line bundle
on
0(1)
M = 1) or the last (-1)-curve.
ti 0(4, 3, 2) (deP [4H ] ® [3H ] ® [2H J)
and
® 0)
1P(H
Note that
or
0(4, 3, 1)
in this case.
and
® 0)
2) M ti 12 = 1P(2H
(10.7.3) When
L = 2Ha + H.
8 = 0(5, 3, 1).
n = 3, there are five cases.
1) d = 5, 6 = 0(1, 1, 0, 0). 2) d = 6, d ti 0(1, 1, 1, 0). of
1Pa
x 1P1
M
is a finite double covering
and its branch locus
is a smooth member of
B
12Ha +
3) d = 7, 6 = 0(1, 1, 1, 1).
along a smooth curve
C
M
is the blowing-up of
1P3
which is a complete intersection of
two hyperquadrics.
4) d = 8, d = 0(2, 1, 1, 1).
smooth hyperquadric
Q3
in
5) d = 9, 8 = 0(2, 2, 1, 1).
M IF4
is the blowing-up of a along a smooth conic curve.
M = 1P1 x 11.
(10.7.4) When n > 3, (M, L) of
(IP1, 0(1))
and
(Q3, 0(1)).
must be the Segre product Thus
n = 4, d = 8
and
6 = 0(1, 1, 1, 1, 1).
For a proof of these facts, see [Iol].
We can obtain
another proof by using the facts in (15.20) in Chapter II,
which were proved in [F21].
Thus, we have a complete classification when (10.8) Suppose and (3.5.2).
Let W = p(M) w = 4
d = 4.
So we have and set and
We have
BsILI _ 0
p: M -+IFa+l Then
w = deg W.
deg p = 1, or
such that
d > 4.
by (4.15) p H6 = L.
4 =
w = deg p = 2.
(10.8.1) In the former case, p
is birational and W
89
§10: Polarized manifolds of A-genus two
is a hypersurface of degree
g(M, L)
So
4.
W
The equality holds if and only if
g(W, Ha) = 3.
is normal, and
in this case by Zariski's Main Theorem.
If
g(M, L) = 2,
has a similar structure to those in (10.7).
(M, L)
M = W
In
fact, using-(10.2) further, we obtain the following. Theorem.
Let
be a polarized manifold with
(M, L)
n = dim M z 2, d(M, L) = 4 that
and
d(M, L) = 2
defines a birational morphism
ELI
and suppose
p: M -.,n+1
Then one of the following conditions is satisfied. a) M = p(M)
is a smooth hyperquartic.
b) g(M, L) = 2, n = 2, M 8
points and
is a blowing-up of
L = 2X77 + 3H
c) g(M, L) = 2, n = 3
and
to those in (10.7.3) with d) g(M, L) = 1, n = 2 bundle
and
X
(M, L)
has a similar structure
8 = 0(1, 0, 0, 0). (M, L)
is a scroll of a vector There is an exact
C.
sequence (#): 0 -- T1 -+ 8 -4 T2 -> 0 for some with
deg(.T
di) Fl dii) Fl
at
IP
1P17
- E (cf. (10.7.2.0)).
on an elliptic curve
8
g(M, L) = 3.
T
e Pic(C)
= 2, and we have either 3"2
and the sequence
(#)
splits, or
and the sequence
(#)
does not split.
g2
The proof uses the results in X15 and will be given in (15.22).
When n = 2, the corresponding result can be
deduced also from the classification theory of quartic surfaces (cf. [Umz], [Ur]). (10.8.2) When
w = deg p = 2, p
of a possibly singular hyperquadric
is a double covering W.
The structure of
such double coverings are studied in [F13].
If
n s 3, W
turns out to be smooth and the observations in (6.11) and
90
I. A-genus and the Apollonius method
In particular the branch locus is a smooth
(6.12) apply.
hypersurface section and is connected.
n = 2, there are
If
several possibilities, as follows. If
Let
v
W
is singular, W
is a cone over a conic curve.
Then
be its vertex.
is an isolated fixed
p-1(v)
point of the sheet changing involution of branch locus consists of surface section of
W
locus of
p
over
The
W.
and a smooth connected hyper-
v
(M, L)
In particular
(*11) in [F13].
If
W.
M
is said to be of the type h1(M, 0M) = 0.
is smooth, we have
W = 1P1 X 1P1.
The branch
is either a smooth ample divisor (a:
type
(1(1,1)+) in [F13]), or the sum of several fibers of one ruling (def type (1'(1,1)0)
In the latter case
for some hyperelliptic curve
M = 1P1 x C
C.
deg p = 1, there is no upper bound for
Unlike the case g(M, L)
in [F13]).
in the above cases where
(10.9) Suppose
d = 3.
deg p = 2.
In this case
BsILI
may and
may not be empty. (10.9.1) When M ->IPn.
If
BsILI _ 0,
ILI
gives a triple covering
n 3 4, such a triple covering is of triple
section type by [Lazi] (or [F20]).
This means that
embedded in the total space of a line bundle over triple section.
Fn
as a
which are not of triple section type.
Their structures have yet to be studied.
See also [Mirl],
for triple coverings.
(10.9.2) When
dim BsILI = 0, we have the following
results (cf. [F23]). 1) BsILI
IPn
is
However, when n s 3, there are many smooth
triple covers of
[F20], [To1-3]
M
consists of one simple point
p.
§ 10: Polarized manifolds of 0-genus two 2) Let
x: M' -+ M
of
curve such that 4) S = p(E)
p
with
Then
p.
yields a morphism
Every fiber X
p
be the blowing-up at
the exceptional divisor over 3) jx*L - El
91
and let
Bsln L - El _ 0.
p: M' - Fn
dim X > 0
E be
of degree two.
is an irreducible
EX = 1.
is a hyperplane in
Pn
Furthermore, it turns out that there are the following three possibilities: *
a) p S = 2E + D
for some effective divisor D
with
dimp(D) < n - 1. p*S
E*
+ D, dimp(D) < n - 1, and * * divisor such that p(E ) = S, E n E _ 0. * * * b)
= E +
E*
is a prime
C) P S = E + E + D, dimp(D) < n - 1, p(E ) = S and E n E * is a hyperplane in
E 2:
Fn-1.
The above cases are studied thoroughly in [F23] and it is found that the situation is similar to that of the corresponding cases in (6.20). In particular we have g(M, L) + 3
in case b) and
n
3
2n s
in case c), while case
a) can occur in any dimension. (10.10) When
d = 2, we have only partial results.
consists of two points, which may be infinitely near.
BsILI
When
g(M, L) = 2, we can say a little more (see [Lan)), and
there are several interesting examples.
However the general
case is still difficult to study.
(10.11) Thus we have classified polarized manifolds (M, L)
with
4(M, L) = 2 (assuming char(.) = 0) except the
following cases, which are yet to be studied.
1) d = L n = 1.
92
I.0-genus and the Apollonius method
2) d = 2
and
is a finite set.
BsILI
Fn
3) Triple coverings of 4) dim BsILI = 1
not of triple section type.
and a general fiber of the mapping
p
in
(10.4) is non-hyperelliptic.
(10.12) Chart. Classification of polarized manifolds of d-genus two.
A (M, L) = 2
g
dim M = 2
&
2
(10.2)
elliptic scroll
d 1 2
dim BsILI=1
dim BsILI;90
d=2
d=3
d=4
(10.7) [i01]
triple cover Fn of
(10.4)
da5
Is
p
1
type (10.9.2) [F23]
(10.5) [F16]
(10.8.1)
birational to a hyperquartic
(10.8.2)
double cover of a hyperquadric [F13]
hyperelliptic ?
Chapter II. Sectional Genus and Adjoint Bundles Throughout this chapter we assume that K is the complex number field C, but most of the results are true if char(A) = 0. How they can be generalized in positive characteristic cases is an interesting problem. The contents of this chapter are relatively independent of those in Chapter I. It is not necessary to have read Chapter I, but it could be helpful. §11. Semipositivity of adjoint bundles (11.1) Throughout this section let dim M = n s 2.
ized manifold with bundle of
nef for
Let
K
be the canonical
We will consider whether or not
M.
be a polar-
(M, L)
K + tL
is
t > 0.
The theory of adjoint bundles was first developed by BsILI = 0 (cf. §18 in Chapter B) using
Sommese in the case
However, our approach is technically
Apollonius method.
independent of his, since we do not assume
BsILI = 0.
Our main tool is Mori-Kawamata theory, and the result Thus, if
(0.4.16) is especially important.
nef, there is an extremal curve the contraction morphism (11.2) Theorem. (IPn, 0(1)).
of
0: M --* W
K + nL
and
(cf. (0.4.16)).
R
is nef unless
In particular K + tL
is not
(K + tL)R < 0
with
R
K + tL
(M, L) =
is always nef if
t > n.
The proof will be given in (11.6). Remark.
The nefness of
from Mori's theory [Mor3].
extremal curve [Mor3;(1.4)].
Z
with
K + tL
follows
Indeed, if not, there is an
(K + tL)Z < 0
This is impossible since
(11.3) Lemma.
t > n
for
Suppose that
K + tL
and
-KZ g n + 1
by
LZ > 0.
is not nef and let
II. Sectional Genus and Adjoint Bundles
94
0, W, R
be as in (0.4.16).
such that any
X be a subscheme of M with
Let
birational.
Suppose in addition that
x = O(X)
Then
is a point.
B E Pic(M)
and any
q s dimO-1(x)
Hq(X, B
with
q = dim 0-1 (x) = dim X.
ample line bundle H
on W
subsystem of
X
Dl, , Dn_q
general members
V0 = M.
is a subscheme of HP(VJ, B + sH) = 0
Vn_q
Then if
for
(B - K)R 2 0.
It is enough
Take a very
Take
x.
Vi = GD1 n
and set
codim VJ = J.
Moreover X
is large enough.
t
for any
0 A
of
= 0
be the linear
A
We claim
p > 0, j 2 0, s >> 0.
We prove this claim by induction on
is nef and big on M
B - K + sH
)
consisting of members passing
JHI
n iDa, and
and let
is
dim X > 0
Proof (cf. [F18;(2.3)], [Mor3;(3.25.1)]). to consider the case
0
for
j.
By (0.4.16.3),
s >> 0.
So the
assertion follows from the vanishing theorem (0.4.12) when j = 0.
0 --' OV
When
i-1
j > 0, we have an exact sequence
[-iH] -b OV -p OV j-1 j
-i 0
since the defining
Di's form a regular sequence at each point.
equations of
Hence the assertion follows from the induction hypothesis. Having proved the claim, we next use the exact sequence -' OX -p 0,
0 --> 3 - OV
where
i
is the ideal defining
n-q
X
in
Vn_q.
Since
dim(Supp(i)) s q, we have
hq(X, B + sH) g hq(Vn_q, B + sH) = 0 above claim. point.
Thus
On the other hand HX = 0 hq(X, B
(11.4) Lemma.
X
)
with
s >> 0
since
by the O(X)
is a
= 0, as desired.
Assume the hypotheses of (11.3) and
suppose in addition that 0-1(x)
for
X
is an irreducible component of
dim X = dim 0-1(x) = q > 0.
be a desingularization of
X.
Then
Let
a: Y - X
Hq(Y, BY) = 0
for any
§ 11: Semipositivity of adjoint bundles
line bundle
on M
B
such that
Proof (cf. [F18;(2.4)]).
95
(B - K)R z 0.
We have an exact sequence
0 --i OX -+ 9*0Y - W -f 0 of sheaves on X where supported on
dim(Supp(T)) < q.
Hence
Sing(X).
hq(g*0y[B]) s hq(X, B) = 0
by (11.3).
{x e X
I
Next consider the
H1+j(Y, By) (cf. (0.1.8)).
dim g
(x) > 0)
Therefore
E2'J = H1(X, Rig*OY[B])
Leray spectral sequence with converging to
Let
S =
Then
E = g 1(S).
and set
dim(Supp(RJ9*0Y)) g dim E - j
< q - j
by (0.1.9).
j = q - i > 0.
We have
E2'0 = 0
E2 'J
for
= 0
preceding observation, so i + j
Let
(11.5) Lemma.
by the
with
i, j
0 be as in (11.3) (in particular
birational) and suppose that x
for any
= 0
Ex1,'3
Hence
which yields the desired assertion.
= q,
point
is
1W
on
Then
W.
for some
q = dim 0-1 (x) > 0
(K + qB)R z 0
B E Pic(M)
for any
with BR > 0. Proof (cf. [F18;(2.5)]).
Assuming
(K + qB)R < 0, we
will derive a contradiction. Take a component X as in (11.4) and let by (0.4.16.3), so Similarly
hand
-[K + qB]y
is nef and big. for
i < q for
hq(Y, K + tB) = 0
x(Y, K + tB) = 0
for any
and
t 2 0
Hence
for at most
x = 0
is nef and big.
Therefore t s q.
On the other Hence
By the Riemann-Roch of degree
t
q
0-ample
is
by (11.4).
0 a t s q.
Theorem, this is a polynomial in BY > 0.
BY
is ample and
BX
h1(Y, [K + tB]Y) = 0
be its smooth model. B
Y
0-1(x)
of
values of
q
since t,
contradicting the above observation. (11.6) Proof of Theorem (11.2). is not nef.
We take
R
and
0
with
Suppose that K + nL (K + nL)R < 0
as
II. Sectional Genus and Adjoint Bundles
96
is birational, we have
0
If
above.
n > dim 0-1(x) > 0
x e W, but this contradicts (11.5).
for some Thus
F
Let
dim W < n.
be a general fiber of
Its canonical bundle is the restriction of -(K + nL)
F
is ample on
Moreover
K.
By (1.3), this
by (0.4.16.3).
(F, LF) . (,n, O(1)), F = M
implies
0.
W
and
is a point.
Thus we have completed the proof.
This result is still valid if
Remark.
M
is allowed
to have certain mild singularities, e.g. rational Gorenstein singularities (cf. [F18]), or even arbitrary log-terminal See [KMM] (or [Rd4], [Mor4])
singularities (cf. [F24;§3]).
for the notion of such singularities.
Similarly the
subsequent results in this section can be generalized to the singular case, but we usually need stronger assumptions on K + tL
singularities for
(11.7) Theorem.
K + (n - 1)L
with smaller
Suppose that
K + nL
Then
is nef.
is nef except the following cases: Fn+l
is a hyperquadric in
a) M
t.
L = 0(1).
and
b) (M, L) = (F2. 0(2)).
is a scroll over a smooth curve.
c) (M, L)
Suppose that
Proof. R
and
0
(11.6), 0
(K + (n - 1)L)R < 0
with
general fiber Hence
for every fiber reduced.
is not nef.
Take
As in
as before.
cannot be birational by (11.5). Then
dim W > 0.
Suppose that
(11.6).
K + (n - 1)L
F
0, so
of
W X
We have
0.
h0(X, LX)
opr
(F, LF) =
is a curve. of
dim F g n -
Moreover
Therefore a
X
,
0(1))
Ln-1X
=
as in
Ln-1F = 1
is irreducible and
h0(F, LF) = n
semicontinuity theorem (0.1.11), so
for any
1
by the upper
d(X, LX) 4 0
and
§ 11: Semipositivity of adjoint bundles
Hence
(X, LX) = (1Pn-1, 0(1)).
free sheaf on W
and
97
is the scroll of
(M, L)
M
dim W = 0, every curve in
proportional to
If
R.
L ti aA
Pic(M), then K + (n - 1)aA
(M, A) = (F200(1)).
is numerically
for some
and A E
a 2 2
is not nef and
This is possible only when
by (11.2).
Thus
S.
c) is satisfied.
the condition When
is a locally
= 0*0M(L]
48
(n - 1)a s n
n = a = 2
and
b) is satisfied.
Thus
If neither of the above cases occur, every line bundle on
M
is numerically an integral multiple of In fact
infer K ti -nL.
K = -nL
Hence we
L.
by (0.4.16.3), so
(M, L)
a) by a result in [Kob01, the proof of which is
is of type
similar to that in (1.3). (11.8) Theorem. that
Suppose that
Then K + (n - 2)L
n > 2.
K + (n - 1)L
is nef and
is nef except in the
following cases.
a) There exists an effective divisor (Fn-1,
(E, LE)
and
0(l))
E on M such that
[E] E = 0(-l).
bO) (M, L)
is a Del Pezzo manifold with b2(M) = 1, or
(F3, 0(j))
with
in
F4
with
j = 2 or 3, (1P4, 0(2)), or a hyperquadric
L = 0(2).
bl) There is a fibration
0: M --* W over a smooth curve
w
with one of the following properties: for every fiber
bl-V) (F, LF) = (1P2, 0(2))
bl-Q) Every fiber in
Pn
F
of
0
member of
is locally free, M 12H(8) + BPI
restriction of
H(8).
of
0.
is an irreducible hyperquadric
having only isolated singularities.
0*0M[L]
F
is embedded in
for some
Moreover, P = F(8)
B E Pic(W) and
L
8 =
as a is the
II. Sectional Genus and Adjoint Bundles
98
b2) (M, L)
is the scroll over a smooth surface.
For the proof, we use the following fact. (11.8.0) Lemma.
Let
onto a projective variety
X
algebraic variety
f: X --* Y be a morphism from an
be a Cartier divisor on X there is a curve f(z)
in
Z
such that
codim f(E) > 1.
EZ < 0
such that
Supp(E)
E Then
and
is a point.
We use induction on n = dim X
tive.
X
We may assume that
Outline of proof.
is projec-
m = dim f(E).
and
m > 0, we take a general hyperplane section
If
Let
Y.
X
and replace
and
Y
f-1(A)
by
and
A.
n > 2, we take a general hyperplane section replace
X
by H and
Y
and
problem to the case where
f(H).
If
A
on
m = 0
on X
H
Y and and
Thus we reduce the
and m = 0, and use the
n = 2
classical index theorem.
This lemma is true even if
Remark.
X
is a non-KAhler Moishezon variety.
lemma only in the case where
f(E)
char(.) > 0, or if
Here we use this
is a point, but we give
a general version for the convenience of later use. (11.8.1) Beginning of the proof of (11.8). (n - 2)L 0: M --' W
and suppose that K + A
for some
a prime divisor such that is a curve
C
by (0.4.16.2). O(Z)
in
is birational.
dim W = n, 0
= n - 1
45-
E
Hence
is a point, so
x E W by (11.5).
such that
Z c E.
R
and
EC < 0.
0
Let
Therefore
for any curve Thus
Moreover E
be
By (11.8.0), there
E c '-1(x).
EZ < 0
Take
is not nef.
A =
(K + A)R < 0.
as in (0.4.16), so
(11.8.2) When
Set
Z
contracts
ER < 0
such that E
to a
§ 11: Semipositivity of adjoint bundles
99
point, but nothing else.
By (11.5) we have
(K + (n - 1)L)R k 0.
the equality must hold. B e Pic(M)
some
since
Indeed, otherwise, 0 < BR < LR (K + (n - 2)L)R < 0.
for
LR = aHR
So
This contradicts (11.5) since
a 2 2, H E Pic(M).
for some
We claim that
0 > (K + (n - 2)L)R = (K + a(n - 2)H)R. K + (n - 1)L
By this claim [K + (n - 1)L] E = 0.
hi (M, K + tL + £H)
have
(11.3), where on
For any
W.
Hence
hi(E, tLE) = 0 X(E, tLE) = 0
H
=
comes from
i > 0,
t k 0
hl(E, K + tL) = 0 for for
for
i > 0,
t = 1 - n,
as in
since
(1Pn-1,
(E, LE) =
Thus we have
0(1))
and by (1.1).
[E]E = 0(-l)
by the
a) of the theorem.
(11.8.3) When w = dim W < n, let Then
x(t) _
is a
x(t)
Ln-lE = 1
Therefore
n - 1.
so
t z 0,
x(0) = 1. Hence
and
-1
[K + (n - 1)L] E = 0, we get
adjunction formula.
(K + (n - w + 1)L)x
X
be a general is nef by (11.2).
n - w + 1 >n-2 and ws 2.
Hence
(11.8.4) When w = 0,
H
>> 0, we
This implies
i > 0, t z 1 - n.
h0(E, LE) = x(l) = n, so
0.
£
is the pull-back of an ample line bundle
polynomial of degree
fiber of
and
Hence
hi (M, K + tL + £H - E) = 0
x(t) = (t + 1)...(t + n - 1)/(n - 1)!
Since
Pic(W).
Pic(M) = Z
K = kH, L = £H.
be the ample generator and set
k + (n - 1)i a 0, k + (n - 2)1 < 0 k k -(n + 1) unless
by (11.2).
Therefore
by assumption. L = 1
and
Then
Moreover
k = 1 - n
(n, -k, £) = (3, 4, 2), (3, 4, 3), (3, 3, 2)
(4, 5, 2).
Thus we have
Let
by (0.4.16.3).
or
b0).
(11.8.5) When w = 1, take
A E Pic(M)
attains the smallest positive value.
Then
such that
LR = LAR
AR and
100
II. Sectional Genus and Adjoint Bundies
KR = kAR
for some
k + (n - 2)t < 0 Hence
k,
by applying (11.2) to
k = 1 - n
(X, AX) = (F200(1))
A
0, since
of
and
L
x
As in the case
= 0(2).
for every fiber
F
Thus we have
O-ample by (0.4.16.3).
is
by (0.4.16.3),
[K + 3A]x = 0
(F, AF) = (F2, 0(1))
(11.7;c), we have
X.
unless (n, -k, t) = (3, 3, 2).
(5-i) In the latter case so
k + (n - 1)t k 0,
We have
E Z.
k k -n
and
and
t = 1
t
bl-V).
in
Fn
comes from
as in (11.7;a).
L = 0X(1)
and
F
of
most two prime components. then
F1C > 0
for some curve
F = 2Y, then
YR = 0
4(F, LF) t d(X, LX) = 0, so Y = OM(L)
Thus
*
homomorphism
n + 1
on
p: M -+ P 6.
such that
is irreducible
p
B E Pic(W).
the tangent space of
M
p
in
y
PX = Fn.
on
P.
is
I2H + BPI
Therefore
PX
x
Hence
I = Sing(FX),
and the tangent space of
coincide in the tangent space of
(P, H)
over each point
is a member of
At every point
M
The natural
H = L, where
FX = 0-1(x)
itself is an embedding and
is a
6
By (0.1.4), we have
The restriction of
is the embedding of
for some
is
comes from
F
W.
is surjective.
0 6 -, Y
the scroll of
p
C
By (0.1.11),
*
on W
F1 # F2,
is a hyperquadric.
(F, LF)
6 = 0*1.
and
locally free sheaf of rank
a morphism
has at
By the upper semicontinuity theorem we have
and reduced.
Set
But
F2.
E Pic(M)
[Y]
This cannot occur either.
Pic(W).
for
for
and hence this cannot occur.
R
and
Ln-1X = 2, F
=
in
C
K + (n - 1)L
Since
F = F1 + F2
If
numerically proportional to If
Ln-1F
Since
0.
is a hyperquadric
[K + (n - 1)L] F = 0
by (0.4.16.3),
Pic(W)
every fiber
k = 1 - n, X
and
t = 1
(5-li) When
§ 11: Semipositivity of adjoint bundles
[e(P)/e(M)]y = [e(P)/e(PX)]Y
101
for any curve
This contradicts
(2H + B)Y = [F]Y = 0.
Y
X, hence
in
HY > 0.
Thus
I
cannot contain a curve, namely every fiber has only isolated Now we have
singularities.
(11.8.6) When
w = 2, we have
X
for any smooth fiber F
for every fiber
bl-Q).
of
of
0.
D
in
C
We claim
dim F = n - 2
Indeed, otherwise, k(D)
0.
point for some prime divisor is a curve
(X, LX) = (Fn-2, 0(1))
D
such that
on
is a
By (11.8.0), there
M.
Hence
DC < 0.
DR < 0
by
(0.4.16.2). However, there are curves in other fibers which do not meet Thus of
F.
D, so this case cannot occur. dim Fi = n - 2
for any irreducible component
Take a large integer
ample and take general members
Then N = D1 n
n Dn-2
t
such that
scheme of dimension zero consisting of di = Ln-2Fi.
of
JGLl.
Moreover, by applying
is smooth.
Fi, we infer that
Bertini's theorem on
is very
£L
D1, -1 Dn-2
Fi
is a smooth
N n F.
to-2di points, where
Now, let us consider the restriction map
0N: N - W, which is proper and finite over
x = O(F).
If
we take a small enough neighborhood (with respect to the metric topology) U of
x, then each connected component
contains only one point of
ON-1 (U)
The restriction Set
of
(]Pn-2, 0(1))
Then
So
F
is proper finite for every A. deg(ON) _ 2Z mA 3 2. in-2di since
.fin-2di.
On the other hand (X, LX)
for any nearby fiber
Combining these we obtain A.
= N n F.
0Z: UZ --* U
mz = deg(OA).
##((,1)) _ #(N n F)
ON-1 (x)
UZ
X, so
2i di = 1, and
is irreducible, Ln-2F = 1
and
deg(ON) _ to-2
ml = 1 0l
for every
is an isomor-
phism by (an analytic version of) Zariski's Main Theorem.
II. Sectional Genus and Adjoint Bundles
102
Hence
is smooth.
U
Therefore
d(X, LX) = 0
,J (F, LF)
is locally free on
0*OM[L]
scroll of
S.
Hence
by (0.1.11).
for every fiber
(Fn-2, 0(1))
is flat by (0.1.12), and
0
F.
Thus we have
is smooth and
W
Clearly
W.
(F, LF) = 6 _
is a
(M, L)
b2).
Thus we have completed the proof of (11.8).
1) The result (11.8) was obtained by
(11.9) Remark.
Ionescu [Io3] and by the author [F18] independently by different methods.
2) The argument in (11.8.6) works to prove that (M, L)
is a scroll over
W, if
equidimensional, W
is normal and if
(M, L)
smooth and
over an open dense subset of
0
W
is
is
is a scroll
See [F18;(2.12)].
W.
(11.10) Theorem (11.8) can be viewed as a polarized version of the following classical fact (cf. (0.5.5)). Theorem.
The canonical bundle
algebraic surface a) S
S
K
of a smooth
is nef except the following cases:
contains a (-1)-curve.
bO) S = F2 bl) S
IP1-bundle over a smooth curve.
is a
(11.11) As a polarized version of the theory of rela-
tively minimal models of surfaces, we have the theory of minimal reduction as follows.
This notion was introduced by
Sommese in a slightly different context.
A polarized manifold
(M, L)
E
minimal if there is no divisor a divisor
manifold
E M'
exists, then at a point
M
will be said to be as in (11.8;a).
If such
is the blowing-up of another
x, and
E
is the exceptional
§ 11: Semipositivity of adjoint bundles
L'
E Pic(M'), and
L + E = L'M
Moreover
divisor (cf. (0.1.6)).
is ample on
L'
103
for some
M' (cf. (7.16)).
Note
and K + (n - 1)L = [K' + (n - 1)L'] M for the canonical bundle K' of M'. In such a case we say that
(L' )n = Ln + 1
that
(M, L)
is the simple blow-up of
such that
(E', L') = (Fn-1, 0(1))
for the proper transform
(L' - E)Y f 0
So
x ¢ E'
and
E'
lifts isomorphically on
for some
since
positive
LY =
E Y > 0 i
and
i
since
Ei Y < 0
[Ei]E
Ei
since
of the
Indeed, if
= 0(S)
Y C E..
L.
M.
# Ej, there is a curve
Clearly
n a 3.
6, so
= 0(-l)
[Ei]E
E.
We
of a line in
Y
type (11.8;a) are disjoint from each other.
E. n E.
E'
[E']E, = 0(-l).
If in addition n 3 3, exceptional divisors
E. n E. # 0
x.
x, contradicting the ampleness of
passing
E' = 1P n-1
and
x E E', we would have
Indeed, if
x % E'.
claim
at
is not minimal, there is a divisor
(M', L')
If
(M', L')
Y
in
for some
On the other hand
Y C Ei.
Thus we get a
contradiction.
When n = 2, it is possible that contract
Ei
to a point, the other
Ei n Ej # 0.
Ej
If we
does not remain a
(-1)-curve.
When
n a 3, let
n: M - M0
contraction of all the divisors Then M0.
L + 2i Ei = n LO
Ei
be the birational of the type (11.8;a).
for some ample line bundle
By the above observation we infer that
minimal.
This process
reduction of
(M, L).
M -> M0
L0
(M0, L0)
on is
will be called the minimal
Of course the map
n
is the blowing-
up at several points.
When
n = 2, we get similarly a map
n: M - M0
such
H. Sectional Genus and Adjoint Bundles
104
that
is a sequence of simple blowing-ups and
a
In fact
is minimal.
is the blowing-up at several points
n
(not infinitely near) on
M0
However, unlike the
again.
n k 3, such a process
case
(M0, L0)
is not unique in general.
a
So we should say 'a minimal reduction of
(M, L)' in this
Unfortunately, this theory is not powerful enough to
case.
An improved version will be
study polarized surfaces. discussed in §14.
(11.12) As a polarized version of Mori's theory on
threefolds [Mor3], we have a result describing the case in which
K + (n - 3)L
is not nef and
In general, if have
R
moreover
and
K + tL
is not nef for some
as in (11.1) with
0
dim W = n
n > 3 (see [F18;Th.4]). t > 0, we
(K + tL)R < 0, and
dim W g n - t (see [F18;(3.3)]).
or
(11.13) Survey. There is a beautiful application of the theory of adjoint bundles to the structure of projective manifolds containing many rational curves (cf. [Lane])' Let
(M, L)
of dimension
n.
be a polarized manifold defined over For a positive integer
Hilbert scheme of smooth rational curves that 6
LZ = 6.
X E M, let
For
for a general point curves in
X6
on
pa a 0
I
surjective at a general point on SZcF
IV
M
such
is covered by such
ZCM
> 0.
6
x e Z).
is surjective, the differential map
of normal bundles
M
p6 a 0.
for some
((x, Z) E M X Xa
in
pa = dim C6(x)
Set
x.
Thus,
M.
if and only if
Suppose that flag scheme
x
Z
be the
X a
be the subscheme of
C6(x)
consisting of curves passing
6, let
C
F.
Let
Since
F
be the
n: F
M
0(F) - a 0(M)
is
Hence the homomorphism
is surjective at any
105
§ 11: Semipositivity of adjoint bundles
general point on a general fiber
Z
of F ---b QCs, where
is identified with the corresponding curve Clearly
A
is trivial, so
NZcF
a=
NZCM
global sections at any general point on
a(Z)
in
Z
M.
is generated by Hence it
Z = F1.
is a direct sum of line bundles of non-negative degrees.
So
(n - 1) + c1(,N) = h0(.N) = dim Xa = dim F - 1 = n + pa - 1 and hence of
pa = cl(.N) = -KZ - 2
K
for the canonical bundle
M.
From this, by (11.2), we obtain (M, L)
ti
(IPn, 0(1)).
the property
Thus
pa s na - 2
is characterized by
(Fn, 0(1))
pa = (n + 1)6 - 2.
unless
Using (11.7) and (11.8),
we get similar characterizations of other varieties too. See [LanP] for details.
[Sat2] studies similar
problems in positive characteristic cases.
(11.14) We have a similar situation in the study of Severs varieties (cf. [Zk2], [LazV]). 1N Let M be a submanifold in with that
where
M
dim M = n
is not contained in any hyperplane and
union of all secant and tangent lines of
This last
M.
M
condition is equivalent to saying that
is mapped isomor-
phically onto its image under the projection from a general point on such cases.
M
IPN.
Sec(M) # IPN
M, namely, the
is the secant variety of
Sec(M)
such
Zak proved
-
IPN
IPN-1
3n s 2N - 4
in
is called a Severs variety if the equality
holds, and Zak classified all such varieties.
It turns out
that there are only four such varieties up to a projective equivalence, and their dimensions are is a Veronese surface
2, 4,
8
and
16.
in
fact
(M, 0(1))
when
n = 2, which is a classical result due to Severs.
(IPa, 2Ha)
F5
In
II. Sectional Genus and Adjoint Bundles
106
Zak's ingenious proof is based on various geometric Among them, the following fact is of funda-
observations.
mental importance.
For a general point {x E M
I
The line passing
tangent line of
Mj.
and
x
Then
Sec(M), let
on
u
Qu
QU
be the set
is a secant line or a
u
(n/2)-dimensional
is an
smooth hyperquadric.
A proof of this fact can be found in [FUR) too. Actually, Zak proved the above assertion for all points on
Sec(M) - M, and this stronger version plays a crucial
role in his theory.
However, the weaker version given above
is enough to study the case
n s 4.
Let us consider the case are many quadrics on
M
notation in (11.13).
Hence
(F200(2)) If
u
for example.
in this case, so KQ = -3.
is a scroll, Q
(K + 2L)Q = 1.
In view of
is the scroll of h0(M, L) = 5
and
Thus we conclude
is a section of
over
So
under the
C = 1P1.
is
(K + L)Q < 0. since
M --' C
Q2 = 1, we infer that
0(2) ® 0(1)
There
By (11.7), (M, L)
(M, L)
But then
M c 1P4, contradicting the assumption. (M, L)
(1Pa, 2Ha).
n = 4, M contains many
(1P1
but more delicate arguments, we infer case.
p2 = 1
or a scroll over a curve since
(M, L)
When
n = 2
(M, L)
(1P2
a
x
1P2, Ha + H,)
X 1P1)'s. K + 3L = 0
By similar in this
by the classifica-
tion theory of Del Pezzo manifolds (cf. (8.11)).
See [FUR]
for details. Remark.
The four Severi varieties are closely related
to the four standard
I-algebras: A, S[i]/(i2 + 1), the
quarternion algebra and the Cayley algebra. This was
§ 12: Polarized manifolds of sectional genus S 1
107
discovered by Roberts [Rob;pp.13-18]. §12. Polarized manifolds of sectional genus (12.1) Theorem. g(M, L) z 0
for any polarized manifold
The equality holds if and only if
(M, L).
s 1
A(M, L) = 0.
Suppose that
g(M, L) s 0.
Then
(K + (n - 1)L)Ln-1, hence
K + (n - 1)L
is not nef.
Proof.
So
In the cases a) and b), clearly we have
(11.7) applies.
A(M, L) = g(M, L) = 0.
In the case c),
scroll of a vector bundle
of rank
d
is the
(M, L)
n
over a curve
and K = -nL + a*(det 6 + KC), where
Ln = c1(6)
Hence
0 > 2g - 2 =
is the canonical bundle of
C
and
n
is the map
C.
KC
M -> C.
2g(M, L) - 2 = (- L + n*(det 8 + KC))Ln-1 = deg KC.
Hence
g(M, L) = g(C)
Thus
and
C = F1,
so
Thus we have completed the proof.
d(M, L) = 0.
(12.2) Remark.
is a scroll over a curve
(M, L)
If
g(M, L) = g(C).
C, we have always
(12.3) Theorem.
with
implies
g g 0
g(M, L) = 1.
Let
Then
(M, L) (M, L)
be a polarized manifold is either a Del Pezzo
variety or a scroll over an elliptic curve. Proof.
If
K + (n - 1)L
is not nef, (11.7) applies.
g(M, L) = 1, we must have
Since (M, L)
(11.7;c).
By (12.2),
is a scroll over an elliptic curve.
If K + (n - 1)L is nef, we have K + (n - 1)L = 0 *H for some
0: M -> W
(0.4.15).
assumption.
W
and an ample line bundle
On the other hand, Since
is a point.
L
Hence
H
HMLn-1 = 2g - 2 = 0
on W by by
is ample, this is possible only when (M, L)
is a Del Pezzo manifold.
(12.4) Thus we have satisfactory results in the case
II. Sectional Genus and Adjoint Bundles
108
where
I suspect that they are valid under
g(M, L) s 1.
much weaker assumptions.
See §19 for details.
§13. Classification of polarized manifolds of a fixed sectional genus: higher dimensional cases Using the theory of adjoint bundles, we will establish an algorithm for classifying polarized manifolds (M, L) with n = dim M and g = g(M, L) fixed. By this method we can prove the following result for example. (13.1) Theorem.
n
For any fixed
and
only finitely many deformation types of
g, there are except
(M, L)
scrolls over curves.
Here, 'deformation type' is defined as follows.
By a
deformation family of polarized manifolds we mean a proper surjective smooth morphism
f: .M -* T
possibly non-compact complex manifold f-ample line bundle the fiber Mt.
Then
f-1(t)
Y
on
and let
(Mt, Lt)
onto a connected but T
together with an
X.
For each
Lt
be the restriction of
t e T, set
Mt
be
L
to
is a polarized manifold for each
t,
which is said to be a member of this family. Polarized manifolds
(M, L)
and
are said to
(M', L')
be deformation equivalent to each other if there is a chain (M0, L0) _ (M, L), (M1, L1),
polarized manifolds such that
of
(Mk, Lk) _ (M', L') (Mj_l, Lj_1)
and
L.)
(M j,
belong to the same deformation family for each
j.
A defor-
mation equivalence class will be called a deformation type. (13.2) The Hilbert polynomial deformation invariant of
hand, for any fixed deformation types of
(M, L)
x(t) = x(M, tL) by (0.1.11).
is a
On the other
x(t), there are only finitely many (M, L)
with
X(M, tL) = x(t).
§13: Classification of polarized manifolds of a fixed sectional genus:
109
higher dimensional cases
This is a consequence of Matsusaka's Big Theorem [Mat]. Indeed, there is an integer such that = X(t).
mL
m > 0
depending only on
is very ample for any
Such pairs
(M, L)
with
x(t)
X(M, tL)
are parametrized by a certain
(M, L)
subset of the corresponding Hilbert scheme, which has only finitely many connected components.
Thus, (13.1) is equivalent to the following assertion.
For any fixed n polynomials
g, there exist finitely many
and
xl(t),
,
such that the Hilbert poly-
xk(t)
nomial of any polarized manifold g = g(M, L)
and
is one of the
(M, L)
xj's
with n = dim M
unless
is a
(M, L)
scroll over a curve.
We actually need to exclude scrolls
(13.3) Remark.
over curves.
n
rank of
Indeed, for any ample vector bundle
over a curve satisfies
6
C
n = dim M
and
there is no upper bound for is ample for any
A E Pic(C)
are infinitely many possible
We may assume
n 3 2
and
Then, by (11.7), K + (n - 1)L scroll over a curve. assume that
S
However,
Hence there
deg A > 0.
with
6 ® A
xj's.
(M, L)
g k 2
with fixed
n
and
by the results in §12.
is nef unless
(M, L)
is a
In the rest of this section we will
K + (n - 1)L
(13.5) Suppose that Ln
g(M, L) = g.
(M, L)
d = Ln = c1(.), since
(13.4) Now we will classify g.
g, the scroll
of genus
of
'
is nef.
K + (n - 2)L
is nef.
Then
0 < d
(K + (n - 1)L)Ln-1 = 2g - 2, so there are only
finitely many possible values for the following general result.
d.
Hence we can apply
II. Sectional Genus and Adjoint Bundles
110
(13.6) Theorem (cf. [Ko1M]). For any fixed
n, d and g,
there are only finitely many deformation types of
(M, L).
Proof (a slightly modified version of that in [KolM]).
We may assume that A = K + (n + 1)L ak = AkLn-k
We claim that
, n
if
n, d
and
is ample by (11.2).
is bounded for each
This is clear for
are fixed.
g
k = 0,
2 for each k by the ak+lak-l S ak Index Theorem (0.4.6). Hence the claim is proved by
k = 0, 1.
Moreover
induction on
k.
This claim implies that so
(K + mL)n
KkLn-k is bounded for each
is bounded for each
Therefore
m > n.
h0(K + mL) = n + (K + mL)n - A(M, K + mL)
k,
is bounded too.
By Serre duality and Kodaira's vanishing theorem we X(M, -mL) = (-l)nh0(M, K + mL)
infer m > n.
Since
X(M, tL)
is bounded for each
is a polynomial of degree
coefficients are determined by its values at -(2n + 1).
polynomials for
n, its
t = -(n + 1),
Hence there are only finitely many possible X(M, tL).
This is enough by the
observation in (13.2). (13.7) Remark.
(13.6) is enough to prove (13.1) in
the case (13.5), and assures us, at least philosophically, of the existence of certain classification theories of polarized manifolds of this type.
Unfortunately, however,
this method is not powerful enough from the practical viewpoint. It yields a list of possible Hilbert polynomials, but this list is usually too big and far from being best possible.
Moreover it gives very little information on the
deformation type itself. q = h1(M, 0M)
For example, the irregularity
should be bounded since it is a topological
§ 13: Classification of polarized manifolds of a fixed sectional genus: higher dimensional cases
111
invariant, but we need another method to accomplish this (as a matter of fact, we conjecture that
arbitrary polarized manifold
for
q a g(M, L)
(M, L)).
Thus, we should not be satisfied, but should not be too disappointed.
No method has almighty power, and the case in
which K + (n - 2)L
is nef is a kind of 'general type'.
The above method at least provides us with a starting point for a classification theory (cf. [BeBS]). (13.8) In the rest of this section, we consider the
case in which K + (n - 2)L
So (11.8) applies when n > 2.
is nef.
have the classical result in (11.10). case (11.8;a), we do not have in the case (11.10;a).
K + (n - 1)L
is not nef while
LE = 1
When n = 2, we However, unlike the for the (-1)-curve
E
Therefore the theory of minimal
reduction as in (11.11) is not enough to study polarized We study the case
surfaces.
Here we assume
n = 2
n > 2.
(13.9) In the case (11.8;a), let minimal reduction as in (11.11).
K0 + (n - 1)L0
and
K + (n - 1)L.
Moreover
number fixed
(M, L) v.
(M0, L0)
Clearly
g(M0, L0) = g
L0n M0
= d + v, where
The deformation
is blown up.
is determined by that of
(M0, L0)
g, and then decide how many points on (M0, L0).
Clearly
problem is reduced to the case where
is
is the
v
Hence we should first classify
blown up for each
be the
is nef since its pull-back on M
number of points at which type of
in the next section.
and the
(M0, L0)
M0
for
can be
v < L0n, so the (M, L)
is minimal.
By (13.5) and (13.6), it suffices to consider the cases b0), bl) and b2) in (11.8).
In the case b0) we use the
II. Sectional Genus and Adjoint Bundles
112
classification theory of Del Pezzo manifolds in §6 and §8. The other cases will be studied separately.
(13.10) The case bl) is further divided into the two cases
and
bl-V)
In case
f: M -+ W
bl-V), we have for every fiber
(IP2, 0(2))
locally free sheaf of rank of
8
and
*
L = 2H + f B
=
0 g H3 = e
H
since
6g - 6 = d + 4e.
b = (g - 1)/2 - e
on
Of course
B E Pic(W).
Let
q
be
e = c1(6), b = deg B.
Moreover
is nef.
2g - 2 = HL2
and
d
So
e
are bounded.
Hence
Moreover H = K + 2L =
by the canonical bundle formula for
q
Thus
is also bounded.
W.
Hence
In fact we
4q - 4 = 2e - 2(g - 1) < g - 1.
have
Enumerating all the possible triples obtain a classification of
(M, L)
(q, e, b), we
of the type
deformation type is clearly determined by (13.11) In case on a scroll where of
is a
is the scroll
(M, H)
is the canonical bundle of
0 = 2q - 2 + e + 2b.
n = 3.
8 = f*(0M[H])
W.
is bounded, too.
KW
(F, LF)
d = L3 = 8H3 + 12H2B = 8e + 12b
H + f (KW + det 8 + 2B) scrolls, where
3
such that
f.
and
and set
4H3 + 4H2B = 4e + 4b,
and
of
for some
W
the genus of the curve Then
F
Then HF = 0(1)
H = K + 2L.
Set
bl-Q).
W
(P, H)
B E Pic(W).
and set
of a vector bundle Moreover
L = HM.
6
Let
e = c1(6), b = deg B. Since
the canonical bundle
KP
of
P
is
The
(q, e, b).
is a member of
bl-Q), M
bl-V).
12H + BPI
over a curve q
W,
be the genus rank 8 = n + 1,
[KW + D] P - (n + 1)H
for D = det 8, so K = AM - (n - 1 )L for A = KW + D + B. Hence 2g - 2 = Ln-lA = 2 deg A = 4q - 4 + 2e + 2b, so
e + b = g + 1 - 2q.
113
§ 13: Classification of polarized manifolds of a fixed sectional genus: higher dimensional cases
We claim
s
Indeed, taking
2e + (n + 1)b s 0.
def
determinants of the symmetric matrices giving the quadric Mx
form defining
Px = Fn
in
over
x e W, we obtain a
2D + (n + 1)B
holomorphic section of
which vanishes
exactly at the points over which Mx
Hence
is singular.
s a 0. We have
d = Ln{M} = Hn(2H + B){P} = 2e + b.
Hence
(n - 1)d + s = 2n(e + b), so (n - 1)d + s + 4nq = 2n(g + 1). Therefore
d, s
and
are bounded, so
q
e = ((n + 1)d - s)/2n the possible values of
are also bounded. q, e
and
Enumerating all
b, we obtain a classifi-
and
The deformation type of
cation.
b = (s - d)/n
(M, L)
is determined by
(q, e, b). Eventually we can further determine the vector
Remark.
bundle
This is indeed the case if
8.
(13.12) In case bundle
8
of rank
b2), (M, L)
n - 1
is the scroll of a vector
on a smooth surface
ample by definition, and hence
A
det 6
def
(KS + A)A{S}
S.
for the canonical bundle
is
8
By
is ample.
2g - 2 = (KS +
the canonical bundle formula we have =
g = 2, see §15.
KS
of
S.
A)MLn-1 Hence
g(S, A) = g. For any rational curve
Z
on
decomposable on the normalization of rank(8) = n - 1 2 2.
In particular
is ample and
8
S,
So
Z.
(S, A)
AZ = c1(8Z) Z cannot be a
scroll over a curve.
We should first classify the polarized surface This is done using the method in the next section.
after we classify ample vector bundles for each pair
(S, A).
We have
8
with
(S, A).
There-
det 8 = A
d = Ln = s2(8) = A2 - c2(8)
U. Sectional Genus and Adjoint Bundles
114
by (0.3.7), so and
d
since
A2 1 2
by [B1G]. Moreover
c2(.) > 0
are bounded for any fixed
c2(d)
proof of (13.1) is reduced to the case
Thus the
(S, A).
In order to
n = 2.
get a good classification, however, we must work harder ad hoc
using
techniques.
§14. Classification of polarized surfaces of a fixed sectional genus (14.1) In this section we will classify polarized surfaces
M
E
L-weight of to a point.
bundle
a (L
of weight
)
be the contraction of
LZ k m
Note that
m.
for any (-1)-curve
is said to be a
(M, L)
for the canonical bundle
+ mK ) If
L
is called
for some ample line
In such a case
(M ,
M, m = LE
on
L + mE = n L
Then
blowing-up of
E
a: M -+ M
Let
E.
on M .
L
is
P2 or a 1P1-bundle over a curve.
(14.2) For a (-1)-curve the
K
in this case, unless
E
contains a (-1)-curve
is either
M
is nef, we can
K
If
Hence we consider the case where
argue as in (13.5). not nef.
g(M, L) = g.
with
(M, L)
K
of
M
on
M,
n
L + mK =
Z
is said
to be admissible. (14.3) If
n
is admissible, L + mK
otherwise, (L + mK)Z < 0 theory [Mor3].
Moreover
0: M -+ W
A = L + MK
is not nef.
L + mK
L + (m - 1)K
L + (m - 1)(L + mK).
*
Indeed,
by Mori's
Z
This contradicts the admissibility.
maximal number such that
fibration
for some (-1)-curve
L + (m + 1)K
other hand,
is nef.
m
Thus
On the
is the
is nef.
m(L + (m - 1)K)
is ample since
Hence, by (0.4.15), there is a
onto a normal variety W
for an ample line bundle
A
on
such that W.
Note that
115
§ 14: Classification of polarized surfaces of a fixed sectional genus 0
is determined uniquely by
cases: 0) dim W = 0. In case
1) dim W = 1.
and M
L = -mK
0),
F2
r
at
unless
M = F2, we obtain a
taking
(r - 1)
In case so
KF < 0
and
M
Thus
in the classical sense.
or a blowing-up of
There are now three
(M, L).
2) dim W = 2.
is a Del Pezzo surface is either
points (0 s r g 8).
F1
X
Hence,
F1-bundle over a curve by
times admissible contractions of weight
F
for every fiber
(L + mK)F = 0
1),
and any general fiber is
0 < LZ = -mKZ
We have
1P1.
for any component
of
m. 0,
Z2 < 0
of a singular
Z
Hence any singular fiber is of the form
fiber.
F1
Z'0 =
where each component is a (-1)-curve of
Z + Z
L-weight
m.
Contracting one of them for every singular fiber, we get a 1P1-bundle over the curve
This process is a sequence of
W.
admissible contractions of weight
is birational and
2), 0
In case
any (-1)-curve
with
E
point for some curve Hence
= -mKZ.
m.
Conversely, if
LE = m.
M, we have
on
Z
Hence
Ek
El,
W
L + mK Lb + mKb
of 0
L-weight
on
Lb
0
contractions of weight
on
m
Z Z
i j
> 0
M, which are
W.
L + 2 mEj =0L *[L
Note that
for some
+ mK] =
of
W
and hence
is a sequence of
k
admissible
for the canonical bundle is ample.
Moreover,
is the contraction of all of
is smooth and
ample line bundle
0 < LZ
Thus, there are finitely many
mutually disjoint, and them.
and
is a
(Zi + Zj)2 k 0, contradicting
Zi, Z,, then
the index theorem.
O(Z)
m.
Indeed, if
such curves are mutually disjoint.
(-1)-curves
Z2 < 0
is a (-1)-curve of weight
Z
for some curves
is a point for
O(E)
m.
Kb
II. Sectional Genus and Adjoint Bundles
116
0) - 2), the map
In any of the above cases
factors
0
M -> M .
through any admissible contraction
(14.4) Taking admissible contractions successively we such that
(M', L')
get a polarized surface
]P1-bundle over a curve, or its canonical bundle unless
is nef,
K'
Such a sequence of admissible contractions
M = IP2.
will be called an admissible
Mr = M'
M = M0 -+ Ml -
minimalization of
(M, L).
M' = El, although
We stop when
there is still a (-1)-curve in this case.
m = (mr,
be the
mj
is called the weight sequence of
ml)
,
Let
aj:(Mj_1, Lj_1) -> (M., L.).
weight of the contraction Then
is a
M'
this minimalization.
mr a . a ml.
Note that
canonical bundle of curve of
J J
J
M.
since its
J
+ mJKJ]MJ-1 = Lj_l + mjKj_1
[L
be the
K.
be the exceptional
E.
is nef on
L. + m.K.
J
pull-back
and let
M.
Then
a..
To see this, let
is nef on
Mj_1.
0 f (L. + mjKj)Ej+l = mj+1 - mj.
Hence
(14.5) Definition. The weight sequence of an admissible minimalization of spectrum of
(M, L)
will be called the adjoint
(M, L).
This is because it does not depend on the choice of the minimalization.
Indeed, in the case (14.3;0) and 1), the
weight sequence
m
is
m)
(m,
for every minimaliza-
In the case (14.3;2), every minimalization factors
tion.
through spectrum
W, so mb
m = (mb, m, of
(W, Lt').
proved by induction on Remark.
,
m)
for the adjoint
Thus the independence of
m
b2(M).
This well-definedness of
m
reason why we stop when we reach the case
is the main M'
= 1 1
in
is
§ 14: Classification of polarized surfaces of a fixed sectional genus
117
(14.4).
The adjoint spectrum is a deformation
(14.6) Theorem.
invariant of
(M, L).
We use induction on
Proof.
It is enough to
b2(M).
prove the invariance under small deformations.
f: A - T
Let
with an
be a deformation family of surfaces
f-ample line bundle
for some member over
such that (M, L) = (Mo' L o)
Z
o e T.
Let
admissible contraction of a (-1)-curve (14.2).
p
0
on M
E
as in
By the stability of (-1)-curves (cf. [Kod2]), we is a member of a family
M
may assume that and
be an
po: M --> M
For each
tion of a
(-1)-curve
:
A: A -+ A , by shrinking
is a member of
necessary.
f
pt: Mt -> Mt
t E T,
Et
Mt, and
on
A - T T
if
is the contrac-
Et = 8M
for some t
effective divisor in (14.3), L
0
on
8
+ (m - 1)K0
This implies that
is not a scroll. any is
t e T.
pt
Hence the adjoint spectrum
(m (t), m),
(Mt, Lt).
Ht = Lt + (m - 1)Kt
Lt + mKt is nef since
Then
t E T.
where
Therefore
m (t) m(t)
induction hypothesis on (14.7) Lemma.
Let
m(t)
m = (mr,
,
(M, L)
Then
1) K2 = K'2 - r, + mi,
of
(Mt, Lt)
t
by the
The proof is now complete.
(MC, LC) -i ... --> (Mr, Lr) = (M', L')
2) KL = K'L' + mr +
(Mt, Ht)
is admissible for
is independent of
b2(M).
(M, L).
is
is the adjoint spectrum of
spectrum of a polarized surface
minimalization of
As we saw
is ample by the admissibility of
po, so we may further assume that ample for any
LtEt = m.
Clearly
A.
and
m1)
be the adjoint
and let
(M, L) _
be an admissible
II. Sectional Genus and Adjoint Bundles
118
mi2,
3) L2 = L'2 - mr2 where
is the canonocal bundle of
K'
M'.
4) g(M', L') - g(M, L) = 2 mj(mj - 1)/2 k 0, and the ml =
equality holds if and only if
= mr = 1.
The proof is easy.
(14.8) The deformation type of
by the deformation type of
and the adjoint
(M', L')
We will classify the possible types of
spectrum.
for any fixed
M
is not ruled.
is nef for an admissible minimalization as in
K'
(14.7), so 2g - 2 = L2 + KL 2 d + mr + and
d
(M', L').
of
+ m1
by
Hence there are only finitely many possibilities
(14.7.2). for
(M, L)
g = g(M, L).
First we consider the case in which Then
is determined
(M, L)
m.
After enumerating them, we study the type By (14.7), d'
are determined by
d, g
and
=
L'2, K'L'
and
m = (mr,
g(M', L')
m1).
By (13.6),
this is already enough to prove (13.1) in this case. We have
(K')2.d'
s
by the index theorem.
(K'L')2
Since
K'2 Z 0, we can enumerate all the possible values of
K'2.
M
x(M) = 0
is of general type if and only if if and only if
K'L' > 0, M
K'2 = K'L' = 0.
is an elliptic surface with
K'2 > 0.
When K'2 = 0 and x(M) = 1.
In
order to proceed further we need various ad hoc techniques. In lucky cases we can determine the possible deformation types, but usually we get just a list of possible values of numerical invariants like q = h1(M, 0M), Pg = h2(M, 0M) etc. (14.9) Now we consider the case in which We assume C
of genus
M # F2.
Thus, M'
q, where
is a
(M, L) --->
M
is ruled.
1P1-bundle over a curve
(M', L')
is an admissible
119
§ 14: Classification of polarized surfaces of a fixed sectional genus
minimalization.
for some vector bundle
M = M' = )P(d)
that
where
general fiber of M -i C.
H2
since we assume that
(M, L)
F
is a
K2 = 8 - 8q,
and
and K ti -2H + (E + 2q - 2)F.
E
L ti xH + yF,
z = Ex + 2y, we
Then, setting
KL = 2x(q - 1) - z
L2 = xz, =
and
denotes the numerical equivalence and
ti
get
of rank two such
d
H = H(d)
cl(S) = 0 or 1. Set
e d=
Hence
r = 0.
Now, until (14.12), we study the case
Note that
since
x > 1
Therefore
is not a scroll.
2g - 2 = (x - 1)z + 2x(q - 1) > 2x(q - 1) Z 4q - 4, since z > 0
by
Moreover both
Thus
d = xz.
2q < g + 1
is bounded.
(x - 1)(z + 2q - 2) = 2(g - q) > 0
x
and
and hence
z = ex + 2y, we can
Using
are bounded.
z
enumerate all the possible values of fixed
q
and
E, x, y
q,
for any
The deformation type is determined by these
g.
numerical invariants, and thus we get a classification. is nef if and only if
2H - EF
(14.10) Remark.
semi-stable (cf. [Miy2;§3]).
ample in this case since
Hence
xH + yF
there are many semi-stable vector bundles on values
(q,
If
take
8 = 0 ® 0.
have
y > 0
q =
solutions Set
imply
E
But if
(E, x, y)
b = -KL.
and all the
= 0, the same is true since we can q = 0
and
= 1, we also need to
E
since there is a section
(14.11) Remark.
C
q > 0,
enumerated by the above method do
E, x, y)
really occur.
is
is actually
When
z = ex + 2y > 0.
6
d
with
HA g 0.
Let us consider how many possible g, q
we have for a fixed Then
z = 2x(q - 1) + b
2(q - 1)x2 + bx - d = 0.
one positive solution for
x, so
If
x
and
and
d.
d = xz
q > 0, there is at most and
z
are determined
U. Sectional Genus and Adjoint Bundles
120
uniquely.
z = ex + 2y,
Since
unless both
x
and
z
0,
provided
1
one positive solution positive solution
s
z > x.
However, sometimes we have two
x.
and
a
are even since
In such a case both
P.
a + P = b/2
and
addition, occasionally we have
Hence we
a§ = d/2.
(1, a,
In
- a/2) and/or
The existence of the two solutions
(1, P, a - 9/2).
and
and
b
(e, x, y) = (0, a, fi) or (0, §, a).
have two solutions
two
are also unique
q = 0, the above is true when there is only
Even if
(0, a, §)
y
In the latter case
are even.
can take the both values
d
and
a
reflects the fact that
(0, 5, a)
has
BO
1P1-rulings.
(14.12) Now we study the case where
(13.9), we reduce the problem to the case
L + 2K
is nef and
r > 0.
Then
m = ml > 1.
0 s (L + 2K)L = 4g - 4 - d.
this is already enough to prove
As in
By (13.6),
(13.1).
In order to obtain a more precise classification, we proceed as follows.
0 s (L + 2K)2 =
First we have
4K2 + 8g - 8 - 3d = 8g - 32q - 4r - 3d + 24. q, r
and
are bounded.
d
(14.13) We may fix claim
KL < 0
when
Indeed, if t >> 0
K
x(M, L + tK) > 0
KL > 0, then
# 0 If
for any
since
We
We have
for any
h2(L + tK)
((1 - t)K - L)L < 0.
t >> 0, which is absurd since
KL = 0, then
K2 < 0
is not numerically trivial.
either.
K2 = 8 - 8q - r.
since
K2 i 0.
h0((1 - t)K - L) = 0
ruled.
K2
by the Riemann-Roch Theorem.
IL + tKI
Therefore
_
Therefore
M
by the index theorem,
is
for
Hence this case cannot occur
Thus we have proved the claim.
§ 14: Classification of polarized surfaces of a fixed sectional genus
(14.14) m
is bounded for any fixed
Indeed, L + tK (14.3).
Hence
is nef for any
0 S (L + mK)2
L2, KL
0 s t t m
121
and
K2.
as in
and the assertion is clear
when K2 < 0. When K2 a 0, we have
KL < 0
(L + tK)2 = 0
for some positive real number
determined by
L2, KL
wise
and
K2.
would be ample.
in.
For any fixed
(14.15) Claim.
t, which is
Then m t t, since other-
m(L + tK) = (m - t)L + t(L + mK)
Thus we get a bound for
Therefore
by (14.13).
L2, KL, K2
and
there exist at most finitely many adjoint spectra
r
This is proved by induction on (14.16) For any fixed values of
(14.7).
and
Then we compute
Setting
and
e, x
m.
using (14.14).
g, we enumerate all the possible
q, r, d, KL, K2
above method.
r,
m = (mr,
m1)
d' = L'2 and for
y
by the
b = K'L'
(M'. L')
by
as in (14.9),
we enumerate all their possible values by the method of (14.9).
These invariants
determine the type of possible types of (11.11).
reduction
General pair
(14.17) Remark.
tion invariants, but
e, x, y, r
(M, L).
(M, L)
(M0, L0)
q,
and
m
together
Thus we can classify the
which is minimal in the sense
(M, L)
is obtained from a minimal
by simple blow-ups. q, r, d, KL, K2 e, x
and
y
and
m
are deforma-
depend eventually on the
choice of admissible minimalization. If
(Lr + mrKr)2 > 0, then the minimalization is a
composition of processes of the type of (14.3.2) only. Hence (Mr, Lr) = (M', L'), and
e, x
and
y
are determined
U. Sectional Genus and Adjoint Bundles
122
uniquely.
(Lr + mrKr)2 = 0, the final process of
But if
the minimalization is of the type (14.3.0) or (14.3.1), and e
depends on the choice of (-1)-curves.
may assume
e
In this case we
for the purpose of classification.
= 0
(14.18) Based on the above observations, we have
produced a computer program to enumerate all the possible deformation types of ruled polarized surfaces for any fixed sectional genus
g (see §21).
§15. Polarized manifolds of sectional genus two
We apply the methods of §13 and §14 to classify polarized manifolds of sectional genus two. Details and proofs can be found in [F21] and [F22]. (15.1) Throughout this section let ized manifold with
K + (n - 1)L
be a polar-
g(M, L) = 2, n = dim M a 2.
As in (12.3), (M, L) two if
(M, L)
is a scroll over a curve of genus
is not nef.
Hence we consider the case where K + (n - 1)L
is nef.
(15.2) When n = 2, we have the following result. Theorem (cf. [BeLP], [F21]). (S, L)
with
g(S, L) = 2
Any polarized surface
satisfies one of the following
conditions.
0) There is another polarized surface (S, L)
is a simple blowing-up of
1) The canonical bundle K to
L
1') S
2) K
and
of
S
(S', L')
(S', L')
such that
at a point.
is numerically equivalent
L2 = 1.
is a minimal elliptic surface and is numerically trivial and
3) There is an ample vector bundle
KL = L2 = 1.
L2 = 2.
T
on an elliptic curve
123
§ 15: Polarized manifolds of sectional genus two C
c1(S) = 1, S = F(3'), L = 3H(Y) - AS
such that
A E Pic(C)
with
deg A = 1.
So
4) There is a vector bundle such that Pic(C)
and
S = 1P(Y)
with
Y
on an elliptic curve
Y
is the blowing-up of
P
is the (-1)-curve over
Ep
bundle on
C
60 ) S = 1P1
X 1P1
deg A = 1.
with
E
is a line
.
= 12.
L2
at a point.
P2
is the pull-back of
L2 = 12
and
and
0 2(1)
is the exceptional curve.
62) S = T2 = F(3) 1P1
A
and
L2 = 1.
61) S = E1, the blowing-up of
H
S
L = 5H(f)S - 2AS - 2Ep, p
and L = 2Ha + 3H
L = 4H - 2E, where
C
c1(3) = 1,
such that
and
p
at
L2 = 4.
on an elliptic curve
P = F(f)
on
p
C
B E
for some
(c1(S), deg B) _ (0, 1) or (1, 0).
together a point
where
L2 = 3.
L = 2H(T) + BS
5) There is a vector bundle
for some
L = 2H(S) + H .
and
7) -K
for the vector bundle
is ample, K2 = 1
up of
S"
9) (S, L)
pl, p2
on
L2 = 12.
and
L = -2K.
8) There are a Del Pezzo surface and two points
3" _ [2H ] ® 0
on
S"
at these points and
L2 = 4.
(S", L")
such that
with
(L")2 = 1
is the blowing
S
L = 3L" - 2E1 - 2E2. L2 = 1.
is a scroll over a curve of genus two.
The proof is outlined in (15.3) ti (15.6) below.
(15.3) Suppose that Hence
K
is nef.
Then
0 £ KL = 2 - L2.
d = L2 = 1 or 2. If
implies
d = 2, then K ti 0 (".
index theorem, so If
KL = 0.
K
is nef, this
denotes numerical equivalence) by the 2) is satisfied.
d = 1, we have
1) is valid, or
Since
(K - L)L = 0.
(K - L)2 < 0
Hence
K " L
by the index theorem.
and Since
II. Sectional Genus and Adjoint Bundles
124
(K - L) 2 = K2 - 1, the latter implies
K2 = 0, so we have
1').
K
(15.4) Now we suppose that Clearly
is nef.
b) there is a (-1)-curve
E
on
B E Pic(C).
Then, as in (14.9), K2 = 8 - 8q
Moreover
for
e
= c1(Y) = 0 or 1.
KL = 2x(q - 1) - z
When
= 1, y = -1
d = 4,
q = 0, F
and
z = ex + 2y.
Moreover is a fiber
F
so we have 4).
has a direct sum factor of degree
In either case
In
In the second case we have
is decomposable.
4 = (x - 1)(z - 2)
q t 1.
d = 3.
L + F, where
3H '
Hence we have 3). and
and
(x, z) _ (3, 1) or (2, 2).
is ample since
S -- C.
e + y = 1
e
C, or
for some
y = deg B, q = h1(S, OS) and
q = 1, we have
H def H(3)
and
L = xH(Y) + BS
We may assume
L2 = xz,
the first case we get
T
on a curve
2g - 2q = (x - 1)(z + 2q - 2) > 0, so
When
of
Y
S.
(15.5) In case a) we have and
K + L
Therefore either
S # IP2.
for some vector bundle
a) S = F(Y)
x E Z
is not nef, but
implies
d = xz = 12
Hence g 0.
y > 0
Hence
since
z k x
(x, z) = (2, 6) or (3, 4).
and we can easily obtain
60),
61) or 62).
(15.6) In case (15.4.b), we may assume (-1)-curve
E, since otherwise we have
is nef as in (14.3), so we have If
we have If
d = 4, we get
0).
LE a 2 Then
for any
L + 2K
0 S (L + 2K)L = 4 - d.
2K + L ti 0
by the index theorem, so
7).
d = 3, we have
0 2 (3K + L)2 = 9K2 - 3 K2 S 0, which implies
KL = -1
and
(3K + L)L = 0.
by the index theorem.
Hence
Therefore
(2K + L)2 s -1, contradicting the
125
§ 15: Polarized manifolds of sectional genus two
nefness of
Thus this case is ruled out.
2K + L.
d = 2, we have
If
Hence
theorem.
KL = 0
(2K + L)2 t -2, which is absurd.
The remaining case is
(mr,
bundle of
Hence
S
K2 = -1.
Then
ml = 2.
x(K1) < 0
L12 = 5, K1L1 = -1
q = 1, S1
L1 = xH + BS
and
z = 1, x = 5,
When Hence
2 2
0 = (L
2
+ 3K2)2.
mj = 3
Thus
is a
y = -2.
for
m2 = 2
= -3
Hence
=
K12 = 0.
Since
1P1-bundle over an elliptic curve
e, y
-1 = K1L1 = -z
q = 0, we have
K22 = 1, K L
0 g (L + 2K)2
(q, r) = (1, 1) or (0, 9).
and
= 1
e
and
as in (15.5). Let
Then
be as before.
Then
is ruled.
S
(L + 3K)2 = -2 < 0, we infer
Since
K12 = 9 - 8q - r, we have
When
and
m1 a 2,
Since
is the canonical
K.
On the other hand
by (14.13).
5 + 4K2, so
(S, L) --> (S1, L1)
be the adjoint spectrum.
K1 L1 = KL - m1 < 0, where
we have
K2 < 0
m1)
,
Let
d = 1.
be an admissible minimalization and let
-> (Sr, Lr)
-+
by the index
K2 g -1
and
and
Thus we have since
z = ex + 2y
5 = L12 = xz, so 5).
(L1 + 3K1)2 = -1.
L22 = 9, so
L2 ti -3K2
j a 3
and
and
(L2 + 3K2)L2 =
by the index theorem.
and we have
8).
Thus we have completed the proof of (15.2). (15.7) In the cases
For the case (15.2.2), we have the following.
more to say. Theorem. K '
0
and
3) ti 9) in (15.2), we have nothing
Let
L2 = 2.
(S, L)
be a polarized surface such that
Then one of the following conditions is
satisfied. 1) S
is the Jacobian of a curve of genus two and
class of a translation of the 1') S = C1 X C2
L
©-divisor.
for some elliptic curves
Cl, C2
and
is the
II. Sectional Genus and Adjoint Bundles
126
L = [F1 + F2], where
is a fiber of
F.
L = [Z + F], where
is a hyperelliptic surface and
2) S
S -+ C
F
is a fiber of the Albanese fibration
z
is a section of
a: S -b Alb(S)
a.
3) There is a finite double covering
branched
f: S -fIP2
along a smooth curve of degree six and
L = f 0(1).
is an Enriques surface and its K3-cover
4) S
Fa X FT
finite double covering of
member of
and
I4Ha + 4Ht1.
is a
branched along a smooth
is the pull-back of
L
H6 + Hr.
Proof.
By the classification theory of surfaces (see
(0.5.5)), S
is either an abelian surface, a hyperelliptic
surface, a K3-surface or an Enriques surface. abelian surface, we have If
S
is K3, we get
is an
1) or 1') by the classical theory.
is hyperelliptic, we obtain
S
S
If
h0(S, L) = 3
and
2) by [F22;(2.15)]. 4(S, L) = 1
If
by using
the Riemann-Roch theorem and the vanishing theorem.
Hence
3) is satisfied by (6.13). If
is an Enriques surface, let
S
Then
K3-cover.
with
n*0
h0(S, L)
h0(S, 2L + N) = 5
and
dim BsILI = 1, S
BsILI
2;2
0
Since
h0(S, 2L)
Hence
= 4
dim BsII a 0 and
2L = f*0(1).
H0(S, L)
Let and
4(^S`,
2;)
=
d(S, L) = 2.
would be ruled by (10.4.8).
by (4.15) and (3.5.2).
be linear basis of ly.
h0(S, 4L) = 17.
= 4, h0(S, 2L)
with
f:
N E Pic(S)
h0(S, L + N) = 2,
=
is at most finite, so
Moreover
for some
be the
By the Riemann-Roch theorem and the vanishing
2N = 0.
theorem we have
If
= 0s ® 0[N]
S
:r:
for = 2.
Therefore *
= n L.
Therefore
Thus we have a morphism (C0, C1)
and
H0(S, L + N)
(to, t1)
respective-
h0(S, 2L) = 5, we have a linear relation among
§ 15: Polarized manifolds of sectional genus two
127
E12 E H0(S, 2L). Therefore 0c1' C12' E02 is a hyperquadric in 1P3, and hence deg f = 2. W = f('`) 02'
The structure of such double coverings was studied in In particular
[F13].
[Fl3;§4]), so E
W
must be smooth since
W = {Z0Z1
for
Z0 1}
if the bases
2L)
Therefore
with
`1 = alai, 0 = a0a1
Z0 = a0a0,
covering transformation and
c
Thus
points, over which
acts on
t
f
and
c
W
1
by
The
a0.
Z.
t
ai = (-1)lai
= Z.
and
W with four fixed
f
is described as follows.
H0(S, 2L + N), but
Hence we have
Z1 =
is unramified.
The branch locus of Recall that
are
)
1
acts on
' _ -Zi, hence we may assume (-1)Ja
CJ its
n
ni
x IP1 = (01 0:01),(t0:T1
W = lP1
of
t
= 0 (cf.
Zi = n*Ci' Zi
and
chosen suitably.
1
a H0(S, 2L + N)
form a basis of
h0(S, 2L + N) = 5.
such that
H0(S, 2L + N).
i
and the
Counting the
dimensions we see that there is a linear relation among
the nCJ i's, and monomials in degree -
n n' Z0' Z1' Z0' Z1
is generated by
and we have a quadratic equation of
whose discriminant defines the ramification locus of Of course described.
c
77
_ -V .
Thus the structure of
In particular we have
(S, L)
f.
is well
4).
Thus we have completed the proof.
It is easy to find
examples of such polarized surfaces. (15.8) In the case (15.2.1), we have the following. Theorem.
type (15.2.1).
Let
Then
2,
C0' 1' 0 and 1 of
The graded algebra
4.
77
(S, L)
be a polarized surface of the
q = h1(S, OS) = 0.
the following conditions is satisfied.
Moreover, one of
128
II. Sectional Genus and Adjoint Bundles
1) Pg d=
h2(S, OS) = 0.
L # K
in
2) pg = 1
10
in
and
(S, L)
if
and
is a weighted complete intersection
IP(3, 3, 2, 2, 1).
(S, L)
is a weighted hypersurface of degree
IP(5, 2, 1, 1)
For a proof, see [BeLP;p.193]. S
h0(S, L) = 1
Pic(S).
of type (6, 6) in 3) pg = 2
Furthermore
In the case 1) above,
is called a numerical Godeaux surface.
We have several
examples and partial results on numerically Godeaux surfaces (cf. [Miyl], [Barl, 2]), but a complete classification of them is not yet known.
The surfaces of the above types 2) and 3) were studied by Catanese [Cat] and by Horikawa [Hor2-H] respectively. have
4(S, L) = 1
We
in the case 3),and (6.21) applies, since
the other cases in (6.20) are ruled out by (6.22) & (6.23). (15.9) In the case (15.2.1'), let elliptic fibration.
f: S -> C
be the
Then there are the following
possibilities (cf. [BeLP;p.192] and [Sern; Th. 2.1]). q = 1
and
pg = 0.
b) x(OS) = 0, g(C) = 1, q = 1
and
pg = 0.
a) x(OS) = 0,
c) x(OS) > 0,
C = IP1,
C = IPq = Pg = 0.
In the case
equivalent to
f
a) above, Serrano shows that
(D x E)/G --> DIG, where
G
is a finite
abelian group acting faithfully on smooth curves He further classified all such pairs
is
(S, L)
D
and
and found
several examples.
In fact, Serrano's method works also in the case and we can classify such pairs and construct examples.
b),
E.
§ 15: Polarized manifolds of sectional genus two However,
129
is not abelian in this case (cf. [F28]).
G
c), we have some results in [BeLP], but a
In the case
complete classification is not yet known. example of such
(S, L)
either.
(15.10) In the case (15.2.0), let Then
minimal reduction. curve
Z
(Mb, L1')
We have no
(L')2
passing a point on
Mb
and
1
>
be the
L
(Mb
LbZ > 1
for every
which is blown-up.
Hence
must be of one of the types 2), 3), 4), 6i) or 7).
(15.11) Now we study the case
n = dim M k 3.
First we
have the following.
be a polarized manifold with
Theorem.
Let
(M, L)
n = dim M k 3
and
g(M, L) = 2.
Then one of the following
conditions is satisfied. and
1 ) K -. (3 - n)L
d = Ln = 1.
f: M - Fn
2) There is a finite double covering along a smooth hypersurface of degree 2') (M, L) (M', L')
3) (M, L)
6, L = f 0(1), d = 2.
is a simple blowing-up of a polarized manifold of the above type
2) at a point. Moreover
n = 3.
is a scroll over a smooth surface.
4) There is a hyperquadric fibration smooth curve 5) (M, L)
branched
W
f: M -> W
onto a
as in (11.8;bl-Q).
is a scroll over a smooth curve of genus two.
The proof is outlined in (15.12) ti (15.13).
(15.12) We may assume that otherwise
is nef, since
5) is satisfied.
Suppose that = 2.
K + (n - 1)L
K + (n - 2)L
is nef.
Then
d
2g - 2
Moreover, by the fibration theorem (0.4.15), we have *
K + (n - 2)L = f A
for a morphism
f: M -
W
and an ample
H. Sectional Genus and Adjoint Bundles
130
line bundle
A
When
on
W.
and K = (2 - n)L.
Hence we have
Using
Using vanishing theorem as in (1.3) and
d = 1, we have
Thus
f.
A ". L
dim W > 0
deg A = 1
Hence
for any fiber
dim W > 1,
A2Ln-2 k 1
so
1).
The case
b2) we have
HF = 0(1)
K + (n - 2)L
and
is not nef.
b0) is easily rule out.
bl-V) as follows.
L = 2H + f B
Then In the
bl-Q) we have
3), and in the case
We rule out the case
4).
Set
H = K + 2L.
for some
B E Pic(W).
2 = (K + 2L)L2 = 4H3 + 4H2B, which is absurd. Finally we consider the case (11.8.a).
be the minimal reduction of
(M, L).
1), 2), 3), 4) or
the conditions
1) is ruled out since
arguments.
3), 4) and 5) there is a curve for every point (15.10).
infer
X
by a version of the index theorem (cf. [F21;(1.5)]).
(11.8) applies.
Hence
if
= 1.
(A - L)Ln-1 = 0, this implies
Since
(A - L)2Ln-2 2 0.
(15.13) Suppose that
Then
n-1
(6.14) ti (6.16), we can derive a contradiction by the
Thus we have
case
Ln-1X = 1
and
AML
since
is a Del Pezzo variety of degree one.
(X, LX)
argument in [F21;(1.4)]. and
dim W = 0
2) by (6.13).
dim W = 1, then of
Hence
= 0.
d(M, L) = 1 (see [F13;(1.11)] for details).
(6.4.3), we get
When
AMLn-1
d = 2, we have
p
Therefore
n = 3
on
Z
Let
(M', L')
This satisfies one of
5)
by the preceding
(L')n > 1.
with
In the cases
L'Z = 1
and
Z a p
This is impossible as in
M'.
(M', L')
is of the type
by the argument in [F21].
2), and we
Hence we have
2').
Thus we have completed the proof of (15.11). (15.14) In the cases
2), 2') and
5) in (15.11), we
§ 15: Polarized manifolds of sectional genus two
have nothing more to say.
In the cases
131
1), 3) and
have more precise results (cf. [F21], [F22]).
4) we
Here we
review them briefly.
(15.15) In case (15.11.1), we have
K = (3 - n)L
in
Pic(M)
H1(M, 0M) = 0
by [F21;(2.1)].
and
This is obvious
by Kodaira's vanishing theorem when n > 3.
When n = 3,
however, we need some deep results for the proof, such as Yau's solution of Calabi conjecture [Y], Miyaoka's inequality for Chern numbers [Miy2], or Kawamata's theory on Albanese mappings [Kawl].
1) If in addition
hypersurface of degree IP(5, 2, 1,
10
as in (15.8.3).
By using (2.6) we infer that
jective spice
IP(3, 3, 2, 2, 1,
In the above cases
This is true when
a=
4) If 111
Moreover
L
M
1
is a weighted
in the weighted pro-
---o 1) (cf. [F21;(2.4)]). 2), M
is simply connected. n = 3
Pic(M)) s 5.
a1(M) = Z/5Z
and the universal
is a hypersurface of degree five in
IP4.
= 0 (1).
t = 4, then
5) If
covering
#(torsion part of
of
as in
We have instead the following results.
s = 5, then
covering
1) and
(6, 6)
(M, L)
(M. L)
n > 3, but not always true when
d(M, L) 2 3. 3) r
is a weighted
in the weighted projective space
complete intersection of type
and
(M, L)
4(M, L) = 2, we have a ladder of
2) If (1.1).
1)
,
4(M, L) = 1,
a1(M) = 2;/4Z
and the universal
is a weighted complete intersection of type
(4,4) in the weighted projective space
IP(2,
2, 1, 1, 1, 1).
Thus, the results are higher dimensional versions of
132
II. Sectional Genus and Adjoint Bundles
The proof is similar to that of the case
those in (15.8). (15.7.4).
See [F21;(2.5),(2.6) & (2.8)] for details.
(15.16) The following problems are still unsolved.
1) Find examples and classify polarized pairs such that
n 3 4, K = (3 - n)L, Ln = 1
(M, L)
J(M, L) > 2.
and
2) Find examples and classify polarized threefolds L3 = 1, K = OM, 4(M, L) = 3
such that
(M, L)
and
g 3.
s
An example will be a polarized version of those in [Barl, 2].
is the scroll of
(15.17) In the case (15.11.3), (M, L) an ample vector bundle Set
A = det 8.
the types
on a surface
Then, by (13.12), we have
A2 = Ln + c2(6) a 2.
rational curve
n - 1
of rank
6
in
Z
Therefore
S.
2), 3), 4), 60), 61) or
S
is the
Hence
of genus two.
C
the cases 1') " 4) in (15.7) cannot occur.
S
for any
7) in (15.2).
Jacobian variety of a smooth curve
on
and
is of one of
(S, A)
1) In case (15.2.2), it turns out that
8n_1(C, o) 0 N
g(S, A) = 2
AZ 3 n - 1 k 2
Moreover
S.
Moreover
6 =
N
for some numerically trivial line bundle
and for some point
o
on
Here
C.
6n_1(C, o)
is
the Jacobian bundle which is described as follows. Let
Pk = C x
product of
C
...
and let
X C / Gk
be the
ak: Pk -> S = J(C)
induced by the Albanese mapping of known that
blowing-up at a point.
i-th projection
(C, o).
1k-2-bundle for
ak is a
C x
Let x C
k > 2
(P
k'
D)
be the morphism It is well and
n2
is the
o
of the
Di be the fiber over -* C
and set
This is an ample effective divisor on Moreover
k-th symmetric
D
Pk such that
Di)/6k.
Dk = 1.
is the scroll of a vector bundle on
S,
§ 15: Polarized manifolds of sectional genus two
which is defined to be
6k_1(C, o).
The proof of this fact
8 = 8n_i(C, o) 0 N
2) If
is of the type (15.2.4), we have
(S, A)
for any fiber
0(1) e 0(1)
F *
Therefore
n = 3
and
bundle
on
of rank two.
C
F(3) = S
and
8 = a
over
(0, 1) or (1, 0), and stable.
F
C
of
M
is the fiber product of
Moreover Both
L3 = 3.
AF = 2.
for some vector
(c1(3), c1(6)) _ are semi-
%
and
Y
c1(%) = -1.
such that
AF = 3
n: S --> C, so F = 0(1) e 0(1) e 0(1)
for some vector bundle
0 H(3)
2c1(%)H(3) + 3H(3)2 = 1, stable.
0 H(3)
n = 4
In the former case we have
0(2) e 0(1).
8 = a
a: S - C, since
of
is of the type (15.2.3), we have
(S, A)
for any fiber
*
C.
8F =
See [F22;(2.4)] for a proof.
3) If
or
is very
For details, see [F22;(2.1l)'(2.23)].
lengthy.
1P('S)
133
and
of rank three on
%
Moreover we have
c2(8) =
L4 = A2 - c2(6) = 2
and
19
is
See [F22;(2.6)].
When F = 0(2) a 0(1), we have an exact sequence 0 -+ 0[n*G + 2H(3)] --> 8 - 0[n*T + H(3)] - 0 G, T E Pic(C).
for some
Moreover we have (cf. [F22;(2.7)])
i) deg T = 0, deg C = -1, c2(6) = 1
and
L3 = 2, or
ii) deg T = 1, deg G = -2, c2(8) = 2
and
L3 = 1.
4) If S = Pa X
1P 1
n: S --b1P1 'J
def
and A = 2Ha + 3HV
n8[-Ha]
2HP a H
c2(8) = 3
For any fiber
AF = 2, so F = Ha a H
we have
By restricting
have
is of the type (15.2.60), we have
(S, A)
is locally free on 8
to a fiber of
Hence and
8
[H
L 3 = 9.
IP1
and
S - IP a
+ 2H
F
of
Therefore 8 = n*W 0 Ha.
we infer that '
a [Ha + H
and we
II. Sectional Genus and Adjoint Bundles
134
for any fiber 6
c9 0 H
ting
6
F
to the section
E
on
Hence
By restric
IP
n, we infer
of
and AE = 2.
and
c2(8) = 3
`
AF = 2
H1 = H - E.
n: S --> P1, where
of
for some vector bundle
HE = 0
since
is of the type (15.2.61), we have
(S, A)
If
= HP ® HA
8 = [2H - E] ® [2H - E],
Hence
L3 = 9.
The pairs
in these cases
(M, L)
Indeed, M
fact isomorphic to each other.
the fiber product of
and
® H1)
iP(2H
60) and
61) are in
is isomorphic to
W H
IP(H
over
IPA
and there are two scroll structures. 5) If
(S, A)
AE = 2
since
n = 3
is of the type (15.2.7), we have
E
for any (-1)-curve
on
It turns out
S.
that there are two possibilities: i)
(8
ti
[-K] ® [-K], c2(8) = 1
ii) c2(6) = 3
and
and
L3 = 3;
or
L3 = 1.
The latter case does really occur and we can describe the structure of
8
more precisely.
See [F22;(2.8)].
(15.18) In case (15.11.4), we employ the notation in (13.11).
Then
6n = 4nq + (n - 1)d + s k 4nq + (n - 1)d, so
q s 1. (15.19) When
i) d = 1,
q = 1, there are three possibilities:
ii) d = 2,
id) d = 3 = n.
or
If d = 1, we have s = n + 1, so b = 1 Moreover rank one.
deg Q k 0
for any quotient bundle
This case does occur for any
and Q
n k 3.
of
e = 0. 8
of
See
[F21;(3.8) & (3.9)] for details. If
d = 2, we have
case occurs for any
s = 2, so
n k 3.
b = 0
and
e = 1.
See [F21;(3.10) & (3.12)].
This
§ 15: Polarized manifolds of sectional genus two
135
If d = 3 = n, we have s = 0, so b = -1 Moreover every fiber of ample vector bundle.
is smooth, and
f: M --> W
Let
7L
is a member of
be a polar-
(M, L)
Assume that
ized manifold of the type (15.11.4).
b e
is an
See [F21;(3.13) & (3.14)].
Theorem (cf. [F21;(3.30)]).
M
8
q = 0, we have the following result.
(15.20) When
Then
e = 2.
and
I2H(6) + bH,I
on
1P(')
W = lP1.
for some
and one of the following conditions are satisfied.
1) d = 1, b = 5
and
[-HJ ® [-H]
6 = 0(-1, -1, 0, 0) (a=
®0®0). 2)
d = 2, b = 4
and
8
3)
d = 3, b = 3
and
8 = 0(0, 0, 0, 0).
3') d = 3, b = 3, 6
4)
d = 4, b = 2,
0(-1, 0, 0, 0).
0(-1, 0, 0, 1) and 6
BsILI _ 0. BsILI
is a point.
0(0, 0, 0, 1).
6 = 0(0, 0, 1, 1). 6) 6 = 0(0, 1, 1, 1). 7) d = 7, b = -1, 6 = 0(1, 1, 1, 1). 8) d = 8, b = -2, 6 = 0(1, 1, 1, 2). 8 ) d = 8, b = -2, 6 = 0(1, 1, 1, 1, 1). 9) d = 9, b = -3, 6 = 0(1, 1, 2, 2). 5)
d = 5, b = 1, d = 6, b = 0,
The proof is elementary but lengthy. Remark.
In all the above cases we have
so the results overlap with those in (10.7). the pair
(M, L)
in the above case
d(M, L) = 2,
Note also that
9) is isomorphic to the
pair of the type (15.17.4).
(15.21) Thus, by the results of this section, polarized
manifolds of sectional genus two are classified almost completely (see the chart in the next page).
II. Sectional Genus and Adjoint Bundles
136
Maeda [Mae] classified polarized surfaces of sectional genus three by the method in §14.
When
L
is very ample, we have more precise results.
This topic will be discussed in §17 in the next chapter. Classification of polarized manifolds of sectional genus two (n = dim M a 3, d = Ln, q = hl(M, 0M)) Chart.
1
2') d=1
2)
)
d=2 q=0
d= 1
q=0 n=3
double cover of
(15. 15)
3
4
)
5)
)
scr oll ove r S
scrol l over C q=2 (15. 18)
IPn (
15. 17)
q= 0
1)
2 )
3)
4)
5)
q=2 d=l
q= l d= 3 n= 3
q=l
q=0 d=9
q=0 n=3
n=3I
n=4
n=3
d=2
1)
2)
3)
4=1
4=2
J k3
3-i )
5-ii)
d=3
d=1
q=1 (15.19) 1
I
I
I
nk4
n=3
-I
I
3-ii) d=l
d=2
5-i)
q=0 (15.20)
-1
i)
I
I
I
I
ii)
n=3
n=4
d=l
d=2
iii) d=3
dIO
d=8
n=3 i=5
(15 16 1) .
i=4
t;g3
.
(15.15.4) (15.15.5) (15.16.2)
(15.22) Now we prove the theorem in (10.8.1). g(M, L) S 1,
If
bundle
8
(M, L)
is the scroll of a vector
over an elliptic curve
C
by (10.2).
Then
§ 15: Polarized manifolds of sectional genus two since
= cl(8)
h0(C, (8)
It is easy to show that such that
is ample (cf. [At]), so we have
6
rank(8) = n = d + h0(L) - d = 2.
Hence
h0(M, L) = Ln = 4.
has a sub-bundle
8
Let
deg(Y) k 2.
137
be the quotient bundle
2
This is identified with the restriction of section corresponding to Therefore
is spanned.
= 0
H1(Y (9 2v)
splits.
Hence
2.
L = H(8)
deg(2) a 2
d-i) in (10.8.1).
If
3" = 2, the
1P1
X
M = C x F1 Thus we
F1.
d-ii).
have
Now we consider the case
n = 2, and let
that
(M, L).
Then
be a minimal reduction of Hence
4), 6i), 7)
(M0, L0)
of a vector bundle h0(8)
=
h0(M0, L0)
deg(2) = 2
is of the type (15.2.9), it is a scroll over a curve
8
2
map
H0 (2)
as in the case
-
H1 (fl
C
of genus two.
Since
h0(M, L) = 4, there is an exact
such that
deg(g) _
g(C) = 2, both
Since
g = 1.
are the canonical sheaf
and
In case
b) in (10.8.1).
(#): 0 -> 3 --> 6 - 2 -* 0
sequence
satisfies one
(M0, L0)
9) in (15.2).
or
6i), we have clearly the condition If
Suppose first
g(M, L) = 2.
(M0, L0)
d0 = L02 2 4.
of the conditions
3"
Moreover, the natural
w.
vanishes, which is the Serre dual of
the extension class of p
L
Y -* 8 --> 2 - 0
0
would be a double covering of
p
to the
3 # 2, then
above sequence does not split, because otherwise and
8/T.
since
deg(f) = deg(2) = 2. If
and the sequence
Thus we have
of rank one
F
(#).
is a double covering of
Hence
(#)
F' X F1.
splits, but then
Thus this case is
ruled out. If
(M0, L0)
(M0, L0)
since
is of the type (15.2.4), we have d0 = 4.
Hence
M = F(Y)
(M, L)
for some vector
U. Sectional Genus and Adjoint Bundles
138
bundle
over an elliptic curve
3'
for some (1, 0).
such that
B E Pic(C)
C, and
(c1(3), deg(B)) = (0, 1) or
For any quotient bundle Therefore
deg B > 0.
S2(Y) ® B
Hence
is ample.
L = 2H(.T) + BM
2 3'
3
of
we have
is semistable and
h0(M, L)
9
aer
h0(C, 9) = C1(9) = 3
=
by (At], which contradicts the assumption. If
(M0, L0)
(M0, L0).
is of the type (15.2.7), we have
Moreover, any member
of
Z
ble, reduced and of arithmetic genus one.
the restriction of p
p
to
cannot be birational.
Z
I-KI
is irreduci-
Since
LZ = 2,
is a double covering.
In fact,
p
(M, L)
Hence
is a double covering
of a singular hyperquadric in this case.
Thus this case is
ruled out by assumption.
Combining the preceding arguments we complete the proof
when
n = 2.
When n k 3, we apply (15.11), and trace the
chart in (15.21). case
n = 2.
the condition
Since
Note that
q = 0
by the result in the
d = 4, we must reach (15.20), and have
c) in (10.8.1).
Thus we have completed the proof of (10.8.1).
Chapter 1. Classification Theories of Projective Varieties In this chapter we are mainly concerned with the cases in which L is very ample and char(.) = 0. We will survey the theories due to Castelnuovo, Ionescu, Sommese, et al. §16. Castelnuovo bounds P =
FN
which is not contained in any hyperplane, Castelnuovo has found an upper bound for the genus of C in terms of the degree. Here we provide a higher dimensional version of Castelnuovo bound using the hyperplane section method. For any curve
C
in
(16.1) Given positive integers 95c,d(µ) = (d - c - 1)µ - cµ(µ - 1)/2
where
q
and
c
and
we set
d,
y(c, d) _ 0c,d(q)
is the largest integer such that
cq
d - 1.
Then we have the following. Fact (Castelnuovo bound).
reduced curve of degree N + 1.
Then
d
in
Let
FN
h1(C, 0C) g y(N-1, d)
C
be an irreducible
such that unless
For a proof, see [GH;p.252] and [Rat]. C
h0(C, 0(1)) C
_
is strange.
Here, a curve
is said to be strange if all the tangent lines at smooth
points of
C
pass through a common point.
no curve is strange except lines.
Even if
If
char(g) = 0,
char(t) > 0,
smooth curves are not strange except conic curves in case char(3) = 2 (cf. [Ha4;p.312]). (16.2) Remark.
1) y(c, µc + 1) = cµ(µ - 1)/2 in
d
in the interval
µc + 1
y(c, d)
d s (µ + 1)c + 1.
is linear Hence
v = [(d - 2)/c], the largest integer
y(c, d) _ 0c'd(v)
for
with
We also have
cv g d - 2.
and
y(c, d) = Maxµ2O(Oc d(µ))'
IH: Classification Theories of Projective Varieties
140
For any fixed function in c
and
d
Hence
c.
for any fixed
µ,
is a decreasing
Oc,d(y)
is a decreasing function in
y(c, d)
On the other hand, y(c, d)
d.
increasing function in
d
if
c
is fixed.
2) In (16.1), it suffices to assume that is injective (or
H0(C, 0 C(1))
This is because
dim V = n
V
with
p
defined by
Let
h0(C, 0 C(1)) a N + 1).
BsILJ = 0
char(.) > 0, suppose in addition that
and
and the morphism
is birational onto its image.
for
L
d = Ln
d = 4(V, L).
When
Proof.
V
that
member
n = 1,
D
of
(16.1) applies since we may assume
When
is not strange.
n > 1, we take a general
and use induction on
ILI
n, using (16.2),
and further (0.2.9) if
d(D, LD) s d(V, L)
char(.) > 0.
This fact is very useful for classifying very ample line bundle (16.4) Corollary.
its canonical bundle.
M
Let
be a manifold and let
Suppose that
possibly not a morphism).
n = dim M
where
Let
Proof.
rational map
and M'
plKl.
Then
d
d
on
M'
char(.) = 0,
K
K
be
is nef
is birational (but
IKI
Kn > (n + 1)pg - n(n + 2),
pg = h0(M, K).
be a smooth model of the graph of the Then we have a birational morphism
r: M' --> M, an effective divisor
system
with
(V, L)
Here is another application.
L.
and the rational map defined by
Thus
When
is normal and
V
g(V, L) f y(d - d - 1, d)
Then
c.
be a line bundle on a variety
L
such that
ILK
is very ample.
H0(P, 0(1))
is a decreasing function in
y(c, d)
(16.3) Theorem.
is an
such that
gives a morphism
E
n IKI
on
M'
= E + A
p: M' - P =
1PN
and a linear and
BsA = 0.
with N = dimIKI
141
§ 16: Castelnuovo bounds
M' - p(M')
such that
W
def
is birational.
Set
L = [A],
d = Ln, g = g(M', L), 4 = 4(M', L), c = N - n = d - 4 - 1. v = [(d - 2)/c] s n.
Assume that
Then we claim
Indeed, this is true if
nd/2 > y(c, d).
d = (/c + 1 )c + 1 .
d = uc + 1
In the interval µc + 1 s d s (y + 1 )c + 1
both sides of the claimed inequality are linear in Hence the claim is always true, so
(16.2.1).
(R + E + nL)Ln-l 2 nd,
where
R
2g < nd. 1)L)Ln-1
=
is the ramification
v > n.
Now we have
d - 2 k (n + 1) c = (n + 1)(p9 - 1 - n), so
d k (n + 1)pg - n(n + 2) + 1.
Moreover
Kn = Ln
by (0.4.7)
since
K
Hence
Kn I (n + 1)pg - n(n + 2) + 1, as desired.
and
by
This contradicts the above estimate, so we
a.
conclude that
d
2g - 2 = (K' + (n -
On the other hand we have
divisor of
or
L
are nef and
E = x K - L
(16.5) Remark. 1) We may allow M
is effective.
to have canonical
singularities.
2) When
n = 2, (16.4) yields the famous estimate
c12 2 3pg - 7.
3) The birationality of (16.4).
For example, if
p
is essential in (16.3) and
p: V -> W
there is no upper bound for
is a double covering,
g(V, L).
(16.6) Now we consider the case where maximum
y(c, d).
g
attains the
The results will not be used in the later
sections of this book, but they are worth mentioning.
The
two-dimensional case was investigated by several authors in the study of surfaces of general type with (cf. [ASK], [Hor2], [Mir2]). Theorem.
c12
= 3pg - 7
Here we present the following.
Let the hypothesis be as in (16.3) and
III: Classification Theories of Projective Varieties
142
suppose in addition that
g(V, L) = y(c, d).
has a regular ladder and
L
Proof.
The case
n.
classical (see [GH; p.253]), so we assume
ladder and
LD
of
Then
ILI.
If
since
y(c, d)
is
Take a
has a regular
(D, LD)
is a decreasing function in
(V, L).
ample by (16.6).
L
is simply generated.
is ample, then
1),
is very
L
(V, L)
will be
There are three types of such
2) d = 2d.
1) d > 2d.
is a regular
D
Such a polarized variety
called a Castelnuovo variety.
In the case
L
by (16.3)
c = d - d - 1.
Therefore
Hence, by (2.5),
(16.7) If in addition
varieties:
n > 1.
g(D, LD) < y(c, d)
This contradicts the assumption. rung of
n = 1
is simply generated by the induction hypoth-
4(D, LD) < d, then
esis.
(V, L)
is simply generated.
We use induction on
general member D
Then
3) d < 24.
v = [(d - 2)/c] = 1
and
y(c, d) = A.
This case is treated in (3.5.3). For the case
2), we have the following.
(16.8) Theorem.
such that
(V, L)
BsILJ = 0, d = 2d
is simply generated, V
i) L
ities and ii)
Let
ILI
wV = (2 - n)L
in
and
be a polarized variety g > A.
has only Gorenstein singularPic(V), or
gives a finite double covering
polarized variety
(W, H)
Then either
such that
p: V
W onto a
4(W, H) = 0
and
p H = L. For a proof, see [F13;(1.4)].
Polarized varieties of the type (16.8.1) are our Castelnuovo varieties of the second kind.
They are
polarized higher-dimensional version of K3-surfaces and Fano
143
§ 16: Castelnuovo bounds
3-folds (cf. [Sail,2], [Isl,2], [Mukl,2]).
Polarized varieties of the type (16.8.1) are called hyperelliptic polarized varieties and were classified Important examples of such double
completely in [F13].
In fact, a minimal
coverings were studied by Horikawa.
surface of general type with
K2 = 2pg - 4
double covering over a variety with this case
K
admits such a However, in
d = 0.
is not always ample, and the branch locus may
have singularities, so the arguments are more difficult than in [F13].
Such surfaces often provide interesting examples
in deformation theory.
(16.9) For a Castelnuovo variety kind (the case
d < 2d), let
the hyperquadrics in dim X = n + 1
P = FN
4(X, H
and
X
)
X be the intersection of all containing = 0, where
This is proved by induction on
n.
V.
H
is a Castelnuovo variety.
and set
Then we have
H = 0(1).
The case
When
due to Castelnuovo (see [GH;pp.527-533]). a general hyperplane
of the third
(V, L)
D = V n H.
(D, LD)
Moreover, by (2.6), X n H
dim(X n H) = n
and
4(X n H, 0(1)) = 0
hypothesis, so
dim X = n + 1
and
is
n > 1, take
Then
intersection of hyperquadrics in H containing
C.
n = 1
D.
is the
Hence
by the induction
4(X, H) = 0.
Note that deg X = N + 1 - dim X = N - n = d - d - 1 V lies on X as a Well divisor. (16.10) Now we assume in addition that
We divide the cases according to the type of and (5.15), there are the following cases. 1) (X, H) = 2) (X, H)
(,Pn+l,
0(1))
is a hyperquadric in
p . n+2.
V X.
is smooth.
By (5.10)
III: Classification Theories of Projective Varieties
144
3) (X, H)
F1, or a generalized cone over
is a scroll over
such a scroll. 4) (X, H)
Is a Veronese surface
(IP2, 0(2)), or a
generalized cone over such a surface. (16.11) In case (16.10.1), V degree
Conversely, for any hypersurface of degree
d.
we have
is a hypersurface of
= d - 2, c = 1, a = [(d - 1)/c] = d - 1
1
d,
and
Thus the bound is in fact
g = (d - 1)(d - 2)/2 = y(c, d). attained.
(16.12) Suppose that (16.10.2) holds. P1 X IP1
two-dimensional hyperquadric over it, then
quadric
I = Sing(X)
for
divisor and let
I3`
Then
n E.
P
E
Q.
s = dim I
is smooth.
E
y e I n V, let
be the exceptional
be the proper transforms of V n E
and
This is impossible since
cone over
Hence we
where
Indeed, if
y, let
at
and IPn+l
X
if
V n 1 = 0.
be the blow-up of
V.
or a generalized cone
n - 1 - s > 2,
s = -1
and
We first claim
and
is a
is a generalized cone over a smooth hyper-
of dimension
0
X
can be viewed also as type 3).
X
X
assume that
If
I
is a linear n E
]Pn-1
X in
is a generalized
Thus the claim is proved.
By this claim
V
is a Cartier divisor on
a complete intersection in
P.
Moreover
X
X
and hence
has at most
isolated singularities.
Conversely, for any complete intersection of type
(2, a), we have d = 2a, d = 2a - 3, c = 2, y = [(d - 1)/c] a - 1
and
g = (a - 1)2,
so the bound
y(c, d)
is
attained.
(16.13) In case (16.10.3), we first study the case in
§ 16: Castelnuovo bounds
which
X
Then
is smooth.
8 = 0(a1,
145
43n+1)
def
V E
we have
a > 1
and
a, b
h0(V, L)
since
V
Hence
h0(X, Ha)
=
d = (aHa + bHP)Han = alai + b
161,
= 26j.
161
g(V, L) > 0.
since
= deg X = d - d - 1 = c,
161
n + 1 +
Then
X.
H. = H(8)
Set
where the
1PI,
for some integers
JaHa + bHfI
is a divisor on
over
[ajH,]
E)
6j's are positive integers. We have
is the scroll of
(X, 0(1))
and
g = a(a-1)61/2 + (a-1)(b-1) = (a-1)(d-c-1) - (a-1)(a-2)c/2. Thus the bound
is attained if and only if
y(c, d)
_ [(d-l)/c] or [(d-2)/c],
or equivalently,
K
a = n + 2
of
V
if and only if
L
This case provides examples in
b = 2 - 161.
and
is
1 - c t b s 1.
Note also that
There are many examples of this type. the canonical bundle
a - 1
which the bound (16.4) is attained.
Next we consider the case in which Sing(X) = Ridge(X)
have
smooth subvariety
det
with
S
R.
X
X = R * S
and
4(S, 0(1)) = 0
taking linear section (cf. (5.13), (5.15)). and suppose that
is singular.
(S, 0(1)) is a scroll of
for some
obtained by r = dim R
Set 0(61,
6n-r)
'
def
8
over
1P1.
such that
X
exceptional set of
n*OX(1) = Ha, a
and
be the proper transform of subvariety of
k
Suppose that R
is the vertex.
for some
a > 0.
V a R, then
(I, Ha)
and let
be
Then we have a birational morphism
the scroll of
n:
= 6 ® 0®(r+l)
Set
We
such that r = 0. If
D
D a
V
a=
n-1 (R)
F(0®(r+1))
a IF
on
is the Let
x R.
3`, namely, the unique
n(!`) = V.
Then
X
V I R, then
is a cone over a V
and
V E
S
10X(a)l
There are many examples of this type.
E n X' = S
contains linear
]P n-1
and
as in
If
146
III: Classification Theories of Projective Varieties
(16.12), where
denotes the proper transform on the
'
blowing-up at
and
R
n = 1 or E n V' = n-1(R)
case
is the exceptional divisor.
E
is a fiber of
1Pn-l
In either
S --+ F 1 .
Therefore
is a point.
n 11
is finite,
V
so it is an isomorphism by Zariski's Main Theorem. case we have
for some
on
JaHa + HftI
E
Hence
In this There
a > 0.
are many examples of this type also. Now suppose that
on X
By taking hyperplane sections
r > 0.
and using induction, we infer that either is isomorphic over any general point of
! --> V
V 0 R In
R.
either case the codimension of the exceptional set of is greater than one, so Moreover
E
1V
b = 0, then
section of
Hence
V.
11 E
have a non-constant morphism r = 1.
11D = F ]P1
i1D
L = [Ha ]V,
b = 1
when
'
Sn-r) ® 0®(r+l))
where
Ha
is
03t (1).
-* V
is a
D
However we
.
since
with
If
1.
D -* R, which
a > 0.
and this is a member of
V =
13` = F(0(S11
and
or
There are many examples of this type (with
Summary.
on
b = 0
JaHa + HP1, and
In particular
D -> R.
11
is smooth.
V
contains a fiber of
= V n D
contradicts
for
on
JaHa + bH D
since
11 = V
or
Hence
r = 1).
(aHa + bH,I
r = -1, 0, 1
Moreover
r = 1,
0 s b s 1 when r= 0, and 1- 161 sb;g 1 when r=-1. (16.14) Case (16.10.4) can be treated similarly. have line.
R n V = 0 Hence
dim R g 0.
bound
We
since the Veronese surface contains no
V E I0X(a)I
for some positive integer
a
and
Conversely any variety of this type attains the
y(c, d).
§ 17: Varieties of small degrees
147
There is no smooth polarized surface of this
Remark.
type (16.14) with K = L.
However, if we allow
an ordinary double point at
resolution, we can obtain a minimal surface
cl
2
= LV by choosing
K
type such that = 3pg - 7
for
of general
iV
suitably.
a
to have
be the minimal
`
and let
R
V
We have
V. These surfaces and their deformations
were studied by [Mir2]. §17.
Varieties of small degrees
We will classify projective varieties V c IPN by the value of d = deg V. This theory is due to Ionescu, who completed the classification for small degrees up to 8. (17.1) We will classify pairs
L
is
This is equivalent to classifying projective
very ample.
subvarieties
V
in
IPN
such that the restriction mapping
0: HO(IPN, 0(1)) _ HO(V, °V(1)) 0
such that
(V, L)
is injective if and only if
hyperplane in
IPN.
Note that
is bijective.
V
is not contained in any
is said to be linearly normal if
V
is surjective.
(17.2) We have 2 0.
The classification is obvious if In case
d = 3, we have
the former case latter case
(V, L)
(V, L)
is
d
2.
(c, 4) = (1, 1) or (2, 0). IP'
(IP1, 0(3)), X(1, 2),
denotes the scroll of the vector bundle F1.
.
is a hypercubic in
a generalized cone over one of them, where
over
c= N - n
since
0 s 4(V, L) s d - 1
E(1,
In
In the 1, 1)
or
1(611 ***1 6r)
0(S1) ® ... ® 0(b r)
These facts were pointed out in the paper [*],
which is, according to a rumor, due to Well. (17.3) In case
d = 4, we have
(c, 4) = (1, 2),
(2, 1)
111: Classification Theories of Projective Varieties
148
or
(3, 0).
is a hyperquartic in case
(V, L)
is a complete intersection of type
When
(cf. (6.8)).
A = 0,
(2, 2)
(1, 2).
It
A = 1
in case
is isomorphic to
(V, L)
(F100(4)), 1(1, 3), £(2, 2), Z(2, 1, 1), E(1, 1, 1, 1), (1P200(2))
This
or a generalized cone over one of them.
classification is due to Swinnerton-Dyer [Sw]. (17.4) As for the case
d = 5,
6
and
7, we refer the
reader to the well written paper [Iol] of Ionescu. was treated in [ioi-2] and [ioi-3].
d = 8
the strategy of his investigation.
The case
Here we outline
Many results hold for
cases of any characteristic, but we assume
char(.) = 0
for
the sake of simplicity.
When
c = N - n = 1, (V, L)
is obviously a hypersur-
We have some results and conjectures in case
face.
c = 2,
but this case is difficult to study in general. On the other hand, if is small.
c
is large, then
We have a complete classification in case
and a satisfactory theory in case
When
c
and
A
A = 0,
d = 1 (see Chapter I).
are in the intermediate range, the
problem is usually difficult. is bounded by (16.3).
(V, L)
A = d - 1 - c
The sectional genus For each fixed
g
of
(d, A, g), we
can classify smooth pairs having these invariants by the method in Chapter U, using adjoint bundles.
In fact, a
variant of this method described in the next section X18 is very useful. since
L
We can use the Apollonius method in addition
is very ample.
However we should be careful since
the very ampleness is not always preserved under various reduction steps in Chapter U, which often yields very subtle problems (see §18).
§ 17: Varieties of small degrees
149
It is sometimes useful to divide the cases according to the values of
q = h1(V, 0V).
We have
q s g
by Kodaira's
vanishing theorem.
Thus, when
char(.) = 0, Ionescu succeeded in obtaining
a classification theory of smooth projective varieties which is roughly summarized as follows. (17.5) The case
d = 5.
c = 1: a hypersurface of degree
5.
c = 2, 4 = 2: g s y(c, d) = 2. g = 0: impossible since this would imply g = 1:
(V, L)
4 = 0.
is an elliptic scroll.
g = 2 = n: simple blow-ups (seven times) of a polarized surface of the type (15.2.61). g = 2 < n: (V, L) c = 3, 4 = 1:
(V, L)
is of the type (15.20.5),(10.7.3.1). is a Del Pezzo manifold and hence a
linear section of
Gr(5, 2) c 1P9 (cf. (8.9)).
c = 4, 4 = 0: similar to the case (17.6) The case
d g 4.
d = 6.
c = 1: a hypersurface of degree c = 2, 4 = 3: g s y(c, d) = 4.
6.
A difficult case to study.
See [Iol] for details.
c = 3, 4 = 2: g s 2. g = 1: an elliptic scroll.
n = 2
since
4 = 2.
g = 2 = n: simple blow-ups (six times) of a polarized surface of the type (15.2.61). g = 2 < n: (V, L) c = 4, 4 = 1: (V, L)
is of the type (15.20.6),(10.7.3.2). is a Del Pezzo manifold.
See (8.1)
III: Classification Theories of Projective Varieties
150
and (8.11.6) for further results. c = 5, 4 = 0: similar to the case (17.7) When
d = 7, the situation is similar to, but
more complicated than, the case
d = 6.
The cases
(c, 4) =
are the most difficult to study.
(2, 4) and (3, 3) (17.8) Remark.
In the course of his study Ionescu
succeeded in classifying (L
d s 5.
is very ample, V
(V, L)
in the following cases
is smooth and
char(.) = 0
as above):
1) g(V, L) = 2 ([Iol;§3]). 2) g(V, L) = 3 ([Iol;S4]). 3) 4(V, L) = 2 ([Iol;Th.3.12]). 4) 4(V, L) = 3 ([Iol;Th.4.18]). 5) g(V, L) = 4 ([Iol-2;Th.1]). 6) g(V, L) = 5 ([Iol-2;Th.2]). 7) 4(V, L) = 4 ([Iol-2;Th.3]). 8) d(V, L) = 8 ([Iol-2]).
In [Io2], he gives also a rough classification in the case where
d 2 24 + 1.
There is no theoretical limit to such a classification theory of
(V, L)
with small invariants, but the technical
troubles increase rapidly beyond these ranges.
How can we generalize these results when singular and/or
V
is
char(t) > 0 ?
This would be an interesting problem. (17.9) The case
n = 2, i.e. the classification of
embedded surfaces, has a long history of investigation.
The
notion of sectional genus is also almost as old as the problem itself.
In fact, Ionescu's theory mentioned above
151
§18: Adjunction theories
is based on classical theories on surfaces as a starting n = dim V.
point for induction on
Recently this topic has begun to regain the attention of many mathematicians (cf. [BeLP], [B1L1".4], [Livl 4]).
Okonek [Okl-3] has introduced vector bundle techniques in such studies.
§18. Adjunction theories (18.1) As we have seen in §17, the sectional genus is an important notion in the study of projective varieties.
The main technical tool for such study is the adjunction theory.
Here we make a brief survey of it and its
applications.
We have a classification theory of L
and
for any fixed V
with ample
(V, L)
g = g(V, L), at least when
char(A) = 0
Why then do we want another theory to
is smooth.
classify pairs with very ample
L ?
There are in fact several reasons. 1) The methods in §13 and §14 yield a list of possible types of
(V, L)
with ample
but they do not say when
L
L
for any fixed
g = g(V, L),
can be very ample.
This is
often a non-trivial and very interesting problem. 2) Under the additional assumption that ample, the list of possible types of drastically cut off.
L
is very
is usually
(V, L)
Therefore, in order to classify such
pairs, it would be nicer to have a short-cut method which does not involve unnecessary and sometimes troublesome classification of pairs with general ample
L.
3) It is difficult to extend the method of Chapter II to
III: Classification Theories of Projective Varieties
152
the case in which
has bad singularities.
V
In positive
characteristic cases the situation is even more hopeless.
However, there is a good chance that the method described in this section works, at least partly, in such more general cases too.
(18.2) For simplicity we assume
char(s) = 0 n = 2
As in Chapter II, the cases
section.
in this n a 3
and
need separate treatments. We first consider the case Let
Theorem.
such that
L
n = 2.
be a smooth polarized surface
(S, L)
is very ample.
Then the following conditions
are equivalent to each other: 1) BsIK + LI
for the canonical bundle
# 0
K
of
S.
2) q = hl(S, 0S) = g(S, L). 3) (S, L)
a
is a scroll over a curve or
for some
(1P2, 0(a))
2. For the proof, see [V] and [So3;(1.5)&(1.5.2)].
except for the well-understood case above, a morphism
Thus,
defines
JK + LI
0, which will be called the adjunction map.
(18.3) We now divide the cases according to the nature of
¢.
Y
Let
When
be the image of
dim Y = 0, we have
classical Del Pezzo surface.
¢.
K = -L When
and
is a
(S, L)
dim Y = 1, 0
is a conic
bundle over a curve. In these cases the structure of
(S, L)
is easy to understand.
When
dim Y = 2, let
torization of
0.
Then
S --' S' --f Y S --, S'
be the Stein fac-
is the same map as
(14.3), which contracts all the (-1)-curves that
LE = 1.
E
on
in
0
S
Moreover there is an ample line bundle
such L'
§ 18: Adjunction theories
on
where
is ample and
H = K' + L'
such that
S'
153
is the canonical bundle of
K'
K + L = HS,
S'.
A natural method here is to study the pair and proceed inductively. ample.
However, H
(S', H)
is not always very
In order to make this method work, we need to know
the structure of
(S, L)
when H
is not very ample.
For
this purpose we have the result below. Let the hypothesis be as in (18.3),
(18.4) Theorem.
Then
dim Y = 2.
H
is very ample except in the following
cases. 1)
L = -2K
2) L' = -2K' and 3)
K2 = 2,
so
d = 8, g = 3.
(K')2 = 2, so
d = 7, g = 3.
and
L = -3K
K2 = 1,
and
4) (S, L) = (iP(8), 3H(8))
so
d = 9, g = 4.
for an ample vector bundle c1(8) = 1, so
an elliptic curve such that
6
on
d = 9, g = 4.
This was proved independently by Serrano, Ionescu [Io4]
and also by Sommese-Van de Ven [SV] after several partial results [V], [So3;3.1], [Einl].
Here we outline a proof
from [Io4] which is based on the following remarkable result due to Reider-Beauville (cf. [Rdr], [Bea2], [Sak3]). (18.5) Theorem.
surface
Let
D
be a nef divisor on a smooth
X with canonical bundle
1) BsJK + DI _ 0 divisor 2) K + D
Z
if
such that
Z
Then
D2 > 4, unless there is an effective DZ = 0, Z2 = -1
is very ample if
effective divisor
K.
DZ = 1, Z2 = 0.
D2 k 9, unless there is an
satisfying either
i ) DZ = 0
and
Z2 = -1 or -2,
ii) DZ = 1
and
Z2 = 0 or -1,
iii) DZ = 2
and
Z2
= 0, or
or
III: Classification Theories of Projective Varieties
154
and D
iv) Z2 = 1
is numerically equivalent to
(18.6) Outline of proof of (18.4).
When
not very ample.
have
R),
LZS = 1
so
ZS2 = -1
ZS2
and
with respect to
Z
L.
and
Hence we have a divisor
or iv) in (18.5.2).
iii)
is
Hence
= 0 or -1. 2
Thus the case
S'
ZS
is a line
(S, L) is not a scroll,
since
# 0
S
on
Z
In the case R), we
This contradicts
is a (-1)-curve.
ZS
0 < HZ = (K + L)ZS.
H
Suppose that
d = L2 3 9, we can apply (18.5) by
setting X = S', D = L'.
satisfying
3Z.
By a
ii) is ruled out.
similar (and slightly longer) argument we rule out also the case
In the case
iii).
Moreover
S = S'.
LZ = 3
occur.
implies
Hence
L2
=
L'2 = 9
and hence
is ample, irreducible and reduced.
Z
g(Z) s 1
implies
g(Z) = 0
iv), we have
since
KZ = -3
and
g(Z) = 1, so
L
is very ample.
However
g(S, L) = 1, which cannot
q = h1(S, OS) t g(Z) = 1.
It
is now easy to see that we have (18.4.3) (resp. (18.4.4)) if q = 0 (resp. q = 1).
When
d g 8, we use case-by-case arguments as in X17.
See [Io4;pp.355-356].
(18.7) Combining these tools we classify pairs with very ample line bundle may assume
BsJK + Ll = 0
Suppose that a: S -> S0
of
S
S
L
and
as follows.
L.
is not ruled.
Then the minimal model
is determined uniquely, and the problem
LO
is not
Such a classification can be done in
various cases, e.g. when number.
(S0, L0), where
However, we must be careful since
always very ample.
By (18.2), we
g(S, L) 3 2.
can be reduced to the classification of LO =
(S, L)
g = g(S, L)
is a fixed small
In general, however, the problem is not easy, since
§ 18: Adjunction theories
155
the classification theory of minimal surfaces is not so precise as we want.
When
S
is ruled, we study the problem by iterating
the process in (18.3).
dim Y < 2
If
ample, the structure of
(S, L)
wise we proceed to study
or if
H
is not very
is well understood.
Other-
Since the adjunction
(S', H).
process must terminate in finite steps for ruled surfaces,
we can establish a classification theory by a suitable induction, as in §14.
This method works for any fixed
g = g(S, L), provided
the classification theory of minimal surfaces is good enough for our purpose, and also for any fixed for any fixed
d = d(S, L)
and
However, there are many
d = L2, as in §17.
technical troubles.
(18.8) In the rest of this section we consider the case where
n k 3.
The simplest philosophy here is to reduce the
problem to lower dimensional cases by taking hyperplane sections.
This is indeed a powerful approach and has been
utilized by many mathematicians.
Here we review another
important theory, i.e. Sommese's theory on adjunction processes, which he developed in a series of papers, and which is analogous to our theory in §11. (18.9) In Sommese's theory
Moreover
larities.
is allowed to have certain mild singu-
V
Usually
is assumed to be
BsILI = 0, but is not necessarily very
spanned, which means ample.
L
V
is normal, Q-Gorenstein at every
point, and has only rational Gorenstein singularities except at finitely many points. Remark.
In many cases
'rational Gorenstein'
in the
III: Classification Theories of Projective Varieties
156
final assumption can be replaced by
'log-terminal'.
See
[F24].
(18.10) To begin with, we generalize (18.2) in the higher dimensional cases (compare (11.7)). Suppose that
Theorem. ample.
K
is smooth and
BsIK + (n - 1)LI = 0
Then
unless
V
L
is very
for the canonical bundle
is either a scroll over a curve or
(V, L)
g(V, L) = d(V, L) = 0.
A proof can be obtained by combining [So4;(3.1)] and Note that the exceptional cases are charac-
[So10;(3.6)).
g(V, L) = q def h1(V, 0 ). Here we provide a direct proof using induction on n. By (18.2), terized by the condition
the assertion is true when n = 2. x E BsIK + (n - 1)LI.
which contains We see
x.
D
Assume that
Take a general member is smooth since
L
D
n > 2 of
and
ILI
is very ample.
x E BsIK + (n - 1)LID = BsIKD + (n - 2)LDI,
so
is of the exceptional type by the induction
(D, LD)
hypothesis.
Hence
(18.11) Remark.
is also of the exceptional type.
(V, L)
It is an interesting but subtle
problem whether or not we can remove the assumption of very ampleness, assuming
BsILI = 0
instead.
a few counter-examples even in the case there are very few such examples.
very difficult to show that
Ln s 4
In fact there are
n = 2, but probably
In particular, it is not for any such counter
example.
Example 1. A2 = 1
and set
K + L = A
Take a Del Pezzo surface L = 2A.
is not spanned.
Then
L
(S, A)
such that
is ample and spanned, but
§18: Adjunction theories
157
Take an ample vector bundle
Example 2.
on an elliptic curve with
cl(8) = 1.
Let
of rank two
8
(P, H)
be the
scroll of
and set L = 2J_ Then L ii 2MPIS assd
spanned, but
K + L
is not spanned.
(18.12) In Chapter II, we classify
the nefness of
K + tL
for
K + tL
K + (n - 2)L
t > 0.
especially important in §13.
according to
(V, L)
is
However, the spannedness of
is. not easy to handle even if
t = n - 1.
The key
idea here, due to Sommese [Solo], is to study the semiThe main result in the case where
ampleness (cf. (0.4.2)). t = n - 1
is the following. Let
(18.13) Theorem.
K be a canonical
(V, L)
be as in (18.9) and let
Q-Cartier Well divisor of
V.
Then the
following conditions are equivalent. 1) h1(V, 0V) = g(V, L). 2) (V, L)
is a scroll over a smooth curve or
3) h0(m(K + (n - 1)L)) = 0
for every
J(V, L) = 0.
m > 0.
4) h0(OV(K + (n - 1)L)) = 0. 5) K'+ (n - 1)L
is not nef as a Q-bundle on
6) K + (n - 1)L
is not semiample as a Q-bundle.
V.
For the proof, see [SolO;(4.1),(4.2),(4.4)], where the following result plays an important role. (18.14) Theorem.
Let
normal variety and let divisor of
V.
ample divisor on
Let V.
A
V
be a locally
K be a canonical
Q-Gorenstein Q-Cartier Well
be an irreducible reduced effective Suppose that
V
has only log-terminal
singularities except at finitely many points and that is Cartier in a neighborhood of
A
for some positive
mK
III: Classification Theories of Projective Varieties
158
integer
m.
t E Q with
for some V
Suppose further that
unless
dim A = 1
(K + tL)A
t 2 2 - m-1.
Then
K + tL
A
is nef on
is a scroll over
(V, [A])
and
is nef on
A.
For a proof, see [F24;(5.7)], which slightly improves upon the result in [SolO;(2.1)] for the case t k 2.
m = 1
and
Because of this improvement we can weaken the
assumption on the singularity of
a little in (18.13)
V
(cf. Remark to (18.9)).
(18.15) To prove (18.13), we also use the following. Theorem ([SolO;(2.2)]).
K
(18.14) and suppose that
V, L
Let
Then
is Cartier.
semiample if it is nef and
A
and
be as in K + tL
is
is semiample.
[K + tL] A
Using these results, we prove (18.13) by induction on n = dim V.
(18.16) Thus, in general, we want results describing K + tL
the cases in which
is not nef (or not semiample).
If it is semiample, we study the fibration Im(K + tL)l
m > 0.
for
If
0
0
obtained by
is not birational, it gives
non-trivial information on the structure of
(V, L).
If
0
is birational (this is the general case), we study K + sL for smaller
s, after suitable birational contractions if
necessary. (18.17) Definition.
spectral value
a(V, L)
smallest real number
a
positive integers
and
u
Let of
(V, L) (V, L)
such that
be as in (18.9).
The
is defined to be the JuK + vLJ = 0
v, for which
uK
for all
is Cartier and
v < (n + 1 - o )u. This notion, introduced by Sommese, is very convenient
§ 18: Adjunction theories
159
He first proved
for formulating results systematically.
(V, L) (cf. [SolO;(1.2)]) and described the
for all
a k 0
structure of
with
(V, L)
a(V, L) g 3 (ibid. (4.4),(5.3)).
His results correspond roughly to ours in §11. a(V, L) E Q
whether or not
for all
He asks
This is an
(V, L).
interesting question, to which an affirmitive answer was given by Sakai [Sak2] when
is a normal surface.
V
But
A similar
this seems to be a difficult problem in general.
invariant can be defined for singularities (cf. (18.22)). Now we outline Sommese's results. (18.18) Fact.
Then 1)
Let
be as in (18.9) and
(V, L)
is of one of the following types.
(V, L)
a = 0.
(1Pn, 0(1)).
2) A hyperquadric in
n+1.
a = 1.
3) A scroll over a smooth curve.
a = 1.
4) (A generalized cone over) (F1, 0(e)), e k 3. and
a s 2.
a = 2 - 2/e
V has a quotient singularity.
5) (A generalized cone over) (F200(2)). 6) There is a surjective morphism variety
W with
= K + (n - I)L Warning.
dim W s 2
0: V -> W
such that
for some ample
a = 3/2.
onto a normal
Q-bundle
A
on
0 A
and
0*0V = 0W W.
a = 2.
The meaning of 'scroll' in [Solo] is
different from ours.
(18.19) In order to study the case
2 < a s 3, Sommese
introduced the following notion. Definition.
A normal polarized variety
said to be a reduction of of
V'
at a finite set
V', and if
L = L'V - E
(V, L)
F
if
V
(V', L')
is
is the blowing-up
contained in the smooth part of
for the exceptional divisor
E
III: Classification Theories of Projective Varieties
160
over
As in the smooth case, the minimal reduction of
F.
is well-defined.
(V, L)
(18.20) Fact.
Let
be as in (18.9) and suppose
(V, L)
Suppose in addition that
2 < a s 3.
codim(Sing(V)) 3 3
and further that
singularities when n s 3. reduction of
Let
V
is Gorenstein,
V
has only rational be the minimal
(V', L')
Then one of the following conditions
(V, L).
is satisfied.
0: V' - C
1) There is a surjective morphism curve
C
such that
fiber
F
of
onto a smooth
for any general
(F, LF) = (IP200(2))
a = 5/2.
0.
2) (V', L') = (Q3, 0(2)), where
T4.
is a hyperquadric in
3
a = 5/2.
3) (V', L') = (1P4, 0(2)).
a = 5/2.
4) (V', L')
a = 8/3.
(IF3, 0(3)).
5) There is a surjective morphism variety
W
with
dim W g 3
0: V' - W
such that
onto a normal
O*OV, = OW
*
0 A = K' + (n - 2)L'
where
K'
for some ample line bundle
is the canonical bundle of
For details and proofs see [Solo]. if and only if
K' + (n - 2)L'
V'.
and
A
on
W,
a = 3.
Note that
a
> 3
is nef and big.
(18.21) Using the above theories we can develop classification theories of polarized varieties of the type (18.9) according to the sectional genus. similar to those in Chapter U.
following result when Theorem.
such that
Let
In particular we have the
g = 1.
(V, L)
g(V, L) = 1
The results are of course
and
be a normal polarized variety BsJLJ = 0.
Then one of the
§18: Adjunction theories
161
following conditions is satisfied.
has only rational Gorenstein singularities and
1) V
is a Del Pezzo variety.
(V, L)
is a (generalized cone over a) scroll over
2) (V, L)
an elliptic curve.
d
vector bundle
k k 0
This means that there exist an ample on a smooth elliptic curve
and a birational morphism
(M, f *L)
f: M
such that
V
(0Cek).
d ®
is isomorphic to the scroll of
When n = 2, we can
For a proof, see [F24;(5.4)].
BsILI _ 0. Instead
obtain a similar result without assuming we assume that
C, an integer
is locally Gorenstein and that the
V
canonical bundle is negative (cf. [Bre], [HidW]).
Such a
surface is sometimes called a 'Del Pezzo surface', but it is not always a Del Pezzo variety in our sense in (6.3), since the condition (6.3;d) is satisfied only when
has no
V
irrational singularities. In the case
g a 2, we have the results [BeSiti4],
[FaS3], [LiS] etc. (18.22) Remark.
We can define a similar invariant for
singularities. Let
p
speaking, X Let
be a point on a normal variety
should be considered as a germ of a variety. S = {x E X
be a desingularization and set
n: M --> X
dim n-1(x) > 0),
We say that
E = a-1(S).
desingularization if the set
E
in
M
always exists by Hironaka's theory. M.
to be admissible if
is a divisor having
Let
An effective Q-divisor Supp(R) c E
and if
is a nice
a
Such a map
only simple normal crossing singularities.
cal bundle of
Strictly
X.
K
n
be the canoniR
on
K - R
M
is said
is n-nef,
III: Classification Theories of Projective Varieties
162
E
and
= -1 - InfR(v(R)), where
sible
such that
runs
i
n(Ei) a p, and then set
runs through all the admis-
R
Using (11.8.0) we can show that
Q-divisors.
n(C)
Ei's are components
v(R) = Mini(r), where
We set
ri a Q.
through all the suffix such that e
M
in
C
R = 2 riEi, where the
is a point. Thus of
for any curve
(K - R)C ? 0
i.e.
is
e
independent of the choice of the nice desingularization so it is denoted by of the singularity If
(X, p)
n,
and will be called the energy
e(X, p) (X, p).
Q-Gorenstein singularity, this cor-
is a
responds to the invariant introduced by Shokurov [Sho2].
Note also that log-terminal singularities can be defined as Q-Gorenstein singularities with negative energy. e(X, p) k - dim X
We conjecture that
equality holds if and only if
p
is a smooth point.
an interesting question whether or not (if
and that the
when
e Q
e
e
< 0
We also hope
> 0, there are many counter examples).
e
It is
that there is a classification theory of singularities such that
e
+ dim X, which corresponds to Sommese's spectral
value, is small.
The energy of a polarized variety by
e(V, L) = e(X, p) - 1, where
at the vertex
p
and
K
When
V
is defined
is the singularity
(X, p)
of the projective cone
(V, L) (cf. (19.12)).
(V, L)
X
associated with
has only mild singularities
is the canonical Q-bundle, we have another invariant
E'(V, L) = Sup{t E
and we have
e
a
e'
and
K - tL
IR
is nef),
a i e' + 1 + n, where
Sommese's spectral value and
If
is
The equalities
n = dim V.
hold in many cases, but not always.
a
V
has only
§ 18: Adjunction theories
163
log-terminal singularities and if Mori-Kawamata theory.
e'
< 0, then
by
E Q
E'
It would be an interesting (but
extremely difficult) problem to determine the set
Sn c IR
consisting of the values of energies of all the polarized varieties of a fixed dimension
Presumably
n.
Sn
will
look like, at least partly, as the energy spectrum of the n-body problem in quantum mechanics. r E
the set of
such that
IR
singular polarized surface
Alekseev [Al] studied for some possibly
K(S) ti rL
Even this special case
(S, L).
is very hard to study.
Given a pair tive divisor
D
(M, D)
on M
larities, we define E
If
= Sup{t E
M - D
IR
e(M, D) I
and an effec-
by
x(K(M) - tD) = dim M}.
is affine, this has a similar nature as the
M = 1n
and
D
Thus we conjecture that
is a hyperplane on it.
a classification theory of such pairs small.
M
having only normal crossing singu-
energies of other objects. unless
of a manifold
(M, D)
e
k -n
We will have with
E(M, D)
Moreover, the classification theory of these three
objects (polarized varieties, singularities and affine pairs) will be unified in future.
This unified theory, I
dream, might be related to (some non-linear version of) the quantum mechanics.
Chapter IV. Related Topics and Generalizations 519. Singular and quasi- polarized varieties (19.1) In the preceding theories on polarized varieties we usually needed some conditions on singularities.
This
may look like just a technical matter, but it often involves very delicate problems.
For example, let us consider the
following.
g(V, L) k 0
(19.2) Conjecture.
variety
for every polarized
(V, L).
This assertion is verified in the following cases. 1) (V, L)
has a ladder.
The proof is very easy.
Thus there is no problem if
L
is spanned by global sections.
is smooth and
2) V
char(s) = 0 (cf. (12.1)).
may have certain mild singularities (e.g.
In fact, V
log-terminal, see [F18], [F24;§31). 3) V
is a normal surface.
For a proof in this case, see [Sak2].
Despite these partial answers, the conjecture is still unproved in general.
(19.3) Here we present an approach for the study of singular polarized varieties.
As an application, we prove
Conjecture (19.2) in the case
n = dim V g 3.
A pair
Definition.
line bundle variety if in
V
and
L
L
on
V
(V, L)
of a variety
and a
will be called a quasi-polarized
is nef and big (e== LC a 0
Ln > 0).
V
Two such pairs
for any curve
C
(V1, L1) and (V2, L2)
165
§ 19: Singular and quasi-polarized varieties
are said to be birationally equivalent to each other if there is another variety morphisms
f
i
:
together with birational
V'
fl L1 = f2 L2.
V' --i Vi (i = 1, 2) such that
Under these notions, we can use various techniques in birational geometry. (19.4) Conjecture. g(V, L) a 0 variety
for any quasi-polarized
(V, L).
This assertion is logically stronger than (19.2), but sometimes easier to prove.
In the rest of this section, we assume
char(.) = 0.
We have
g(M, LM) g g(V, L)
M - V.
Hence, in order to prove (19.4), we may assume that
V
In the sequel we outline a proof of (19.4) in
is smooth.
the case
n = dim V g 3.
(19.5) Theorem.
Let
variety with n = dim V polarized variety to
for any desingularization
See [F24] for details. (V, L) 3.
be a quasi-polarized
Then there exists a quasi-
(V', L') which is birationally equivalent
(V, L), has only
Q-factorial terminal singularities,
and satisfies one of the following conditions: 1) K' + (n - 1)L'
divisor
K'
of
is nef for the canonical
Q-Cartier
V'.
2) d(V', L') = 0. 3) (V', L') Remark.
is a scroll over a smooth curve.
We omit the precise definition of
Q-factorial
terminal singularities, which can be found in [KMM] for example.
They are very mild singularities.
(19.6) Outline of proof of (19.5). V
is smooth and K + (n - 1)L
We may assume that
is not nef. By Mori-Kawamata
N: Related Topics and Generalizations
166
0: V -> W
theory (cf. (0.4.16)) there is a contraction an extremal ray
with
R
(K + (n - 1)L)R < 0.
unlike the ample case, it is possible that
When Hence set
LR = 0, we have
L = 0 A
is birational since
0
(X E W
dim T-1(x) > 0)
I
LR > 0,
If
2) or 3) is satisfied as in (11.7).
the condition
codim E = 1, it turns out that
L
terminal singularities, and we set codim E > 1, W
0+: V+ --* W
S
be the
E = 0-1(S).
has only
If
Q-factorial
(V11 L1) = (W, A).
has bad singularities.
we can take the flip
A E Pic(W).
Let
is big.
W
However,
LR = 0.
for some
and let
of
of
If
However, by [Mor5], and we set
0
(V+.
(V1. L1) =
(-P+)*A).
Next we examine the model
It has only mild
(V1, L1).
singularities so that Mori-Kawamata theory applies. we can replace (V1, L1)
(V1, L1)
by another model
(V2, L2)
if
1), 2), 3).
satisfies none of the conditions
In the above process the Picard number if
Hence,
p
codim E > 1, while it decreases by one if
is preserved codim E = 1.
By Shokurov's result (cf. [Shot] or [KMM;5-1-15]), flip processes cannot continue infinitely, so finite steps.
Since
terminate at some setting
(V
p
decreases after
is finite, the whole process must
p
Lk), and the proof is completed by
k'
(V', L') = (Vk' Lk).
Remark.
The above argument works in any dimension if
the Flip Conjecture I,
(19.7) Proof of g(V', L') 3 0.
II
in [KMM] is true.
(19.5) - (19.4).
It suffices to show
The argument used in (12.1) works here,
since we have the following result. (19.8) Theorem (cf. [F24;(1.1)]).
Let
(V, L)
be a
167
§ 19: Singular and quasi-polarized varieties
quasi-polarized variety with
i(V, L) g 0.
4(V, L) = g(V, L) = 0.
and
Let
Proof.
be the normalization of the graph of
G
the rational mapping defined by tional morphism
n*ILI
n: G --' V, a morphism
= E + p*IHI
E
H = OP(1).
for
F
Let
S = deg W.
and
(0.4.9), A = L - D
4(V, L) g 0
implies
Ln = Ln-wHw = S = Hn. for all
Ln-jHj = 8
dim p(E) < n - 1. curve
Hence
Remark.
Then
by (0.4.7). Ln a b
Hence k
Ln-wHw =
h0(V, L) - w,
d(V, L) = 0 = A(W, H), w = n Ln 2 Ln-1H a
...
and
. Hn, we get
In particular Hn-lE = 0.
Hence
E # 0, we apply (11.8.0) to find a
EC < 0
and
p(C)
But
is a point.
case.
Moreover
g(V, L) _
The above argument works well even if
(19.9) Remark. g(V, L)
BsILI = 0.
Thus we have completed the proof.
char(.) > 0, or if
V
is a non-KAhler Moishezon variety.
Quasi-polarized varieties with small
seem to be classified in a similar way to the ample First of all we conjecture
normal and
= 2
If
E = 0, so
g(W, H) = 0.
n
p.
LC = (E + H)C = EC < 0, contradicting the nefness of
then
L
W = p(G), w = dim W
is effective on any general fiber
Since j.
such that
C
such that
Now, by Kodaira's Lemma
Therefore, by (0.4.7),
SLn-wF a S. so
DF
Ln-wF 2 An-wF > 0
F, we have
Set
G
is an ample c-bundle for some effective
Since
D.
on
p: G - p = FdimILI
be a general fiber of
0 g d(W, H) ¢ w + 8 - h0(V, L).
Q-divisor
Then we have a bira-
ILI.
and an effective Cartier divisor
L.
BsILI = 0
Then
g(V, L) = 0.
4(V, L) = 0
if
V
is
This and (19.4) are true when
g 2 (cf. [F24;52]), or if
n g 3
by (19.5) (the case
was previously studied by Sakai).
n
IV: Related Topics and Generalizations
168
(19.10) Until now we have assumed that
What can be said if
variety.
V
V
is a
is allowed to be
non-reduced or reducible ? This is a natural, important and interesting question, but we do not have simple answer.
1) There is no lower bound for
4(V, L).
To produce examples, we add a large number of embedded points into
h0(V, L)
Then
V.
Hence
remains the same.
becomes large, while
4(V, L)
Ln
becomes arbitrarily
There are also many other ways of producing such
negative. examples.
2) There is no lower bound for
g(V, L).
We can construct examples similarly to the above. 3) When
V
is reducible, it is not obvious how we
should define invariants like
d(V, L), g(V, L).
Thus, there are numerous difficulties, but there are also encouraging positive results. 4) Let
M
manifold
be a surjective morphism from a
f: M --* C
onto a smooth curve
line bundle on
The triple
M.
Del Pezzo fibration if any general fiber
F
of
Let
(f, M, L)
(F, LF) f.
C.
L
be an
f-ample
will be called a
is a Del Pezzo manifold for
A singular fiber in such a
fibration can be viewed as a sort of
'Del Pezzo scheme'.
In [F26] all the possible types of such singular fibers are classified.
because
M
The result is surprizingly simple, mainly is assumed to be smooth.
If we allow
M
to
have mild singularities, we get many more examples of such Del Pezzo schemes.
§19: Singular and quasi-polarized varieties
169
(19.11) It should be mentioned that singular polarized varieties (or even schemes) appear in a natural way in the theory of Mori-Kawamata.
0: M - W
Let
tremal ray (x E W
Suppose that
R.
dim $-1(x) > 0)
I
codim E = 1
we assume and
be the contraction morphism of an ex-
E = '-1(S).
and S
and
Set
S =
For simplicity ER < 0
Then
is a point.
Thus, the classifi-
is a polarized variety.
[-E])
(E,
is birational.
0
cation of such contractions is very closely related to that See [Mor3], [Bell], [Be12],
of polarized varieties. [F25;§3].
(19.12) Here we discuss another important relationship between the theory of polarized varieties and the theory of singularities.
Given a normal polarized variety be the scroll of of
6 = L ® OV
over
defines the unique member
iS
v: P -> V.
a section of
HD = 0
(V, L), let
The subbundle
V.
D0
(P, H)
L
JH - LPI, which is
of
and the normal bundle
0
[DO]D = -L
is negative.
So, by Grauert's criterion, there
0
is a birational contraction point
on
v
0: P - C When
C = C(V, L).
very ample and the image of
V
is just the projective cone over this case
C - O(D_)
is another section of OV
of
8.
L
of
is simply generated ((-4
is projectively normal), C V
with vertex
is the affine cone over x
to a normal
D0
v.
In
V, where
D_
corresponding to the subbundle
With these observations in mind, by conical
singularity of type isomorphic to
C
at
(V, L) v.
we mean a singularity which is
Many properties of such a
singularity can be translated to those of
(V, L).
For
IV: Related Topics and Generalizations
170
example we have:
is Cohen Macaulay if and only if
1) v
Macaulay and
Hq(V, tL) = 0
0 < q < n = dim V.
2) v
q, t E Z
such that
It is Gorenstein if and only if a E Z
for some
WV = 0V[aL]
for any
is locally
V
in addition.
is a rational singularity if and only if
Hq(V, tL) = 0
for any
q > 0
and
t k 0, provided
V
has
only rational singularities. 3) If
is simply generated, then the multiplicity of
L
the singularity
v
dimension of Remark.
d(V, L), and the embedding
is equal to
v
h0(V, L).
is
Taking conical singularities after choosing a
(V, L), we can produce various examples of singu-
suitable
larities with the desired properties. (19.13) In view of (19.12.3), we define d + e - h - 1
multiplicity proved
for a
d-dimensional local ring
of embedding dimension
e
4(R, m) ? 0
A(R, m) =
[Sal l'6] further studied the cases in which
Ooishi [001,2] defined the notions of
of
Abhyankar [Ab]
is Cohen Macaulay.
(R, m)
if
h.
(R, m)
d
Sally
is small.
4-genus and sectional
genus for noetherian rings and studied their properties. (19.14) The smoothability of conical singularities is
closely related to the existence of a polarized manifold (M, A)
such that
If such tions of A to
and
(D, AD) = (V, L)
exists, let
(M, A)
µ: IP(A (D OM) --* M
0M
respectively.
M0 + It D.
Let
A
for some M0
and M
D E
JAI.
be the sec-
corresponding to the subbundles
Then
M_
is linearly equivalent
be the pencil containing them.
Let
§20: Ample vector bundles
A'
be its image on the cone
of
A'
is isomorphic to
171
C(M, A).
Any general member M0 + µ D
M, while the image of
C(V, L), at least set-theoretically.
is isomorphic to
This
is also true in the scheme-theoretical sense under some additional assumptions.
Thus, C(V, L)
is smoothable in a
certain sense (see [F9;(3.11)]).
On the other hand, if the pair smoothable to a pair ation, the rung
D
(M, D)
is a deformation of
polarized manifold.
(V, L)
is
under a flat polarized deform-
very roughly speaking, C(V, L) small deformation of
(C(V, L), q5(D_))
D_ = V.
Hence,
is not smoothable if any cannot be a rung of another
This was the basic idea of [So2].
See
also [F9], [F14], [Sol], [Bad2'5], [Lvl,2]. §20.
Ample vector bundles
A pair (V, 8) consisting of a variety V and an ample vector bundle 8 on V can be viewed as a generalization of the notion of a polarized variety. We review a few theories on such pairs. Throughout this section we assume n = dim V a 2, r = and V is smooth.
rank 8 a 2, char(s) = 0
First, following the idea of [YZ), we give vector bundle versions of the results in §11 on adjoint bundles. (20.1) Theorem (compare [YZ;Theorem 1]). Set and assume r . n. ®n+10(1)),
Then
K + A
is ample unless
A = det 8 (V, 8) =
where K is the canonical bundle of
(lPn
V.
For the proof we need the following lemmas.
For any extremal ray
(20.2) Lemma (cf. [Mor3;(l.4)]).
V, there is a rational curve
R
on
R
such that
-KZ s n + 1.
Z
in
V
representing
IV: Related Topics and Generalizations
172
(20.3) Lemma.
Let
R
be an extremal ray of E
Suppose that there is a subset
there is a rational curve -KZ g n + 1 -
such that any
V
is contained in
R
extremal curve representing
of
representing
Z
V.
Then
E.
such that
R
E.
For a proof, see [Io3;(0.4)]. (20.4) Proof of (20.1). nef.
If not, there is an extremal rational curve
that
(K + A)Z < 0
and
-KZ g n + 1
since the pull-back of
r > n
K + A
First we prove that
such
Z
by (20.2).
is
But
AZ k
to the normalization of
8
is the direct sum of line bundles of positive degrees.
Z
Thus
we get a contradiction, as desired.
Now, by (0.4.15), we have a fibration
on W such that
H
an ample line bundle
this is not ample, there is a curve of
0.
We may assume that
(20.3).
Therefore
contradicts
and
0 H = K + A.
if
contained in a fiber
Z
is an extremal curve as in
Z
-KZ s n - 1
0 = HZ = (K + A)Z
is not birational and
0: V -* W
0
if
dim W < n.
Hence
AZ k r > n.
and
dim W > 0,
If
This
is birational.
can be
Z
viewed as an extremal rational curve of a general fiber of
0, so
KZ = KFZ k - dim F - 1 k -n
and
-KZ t n + 1 and Let
(P, H)
canonical bundle
Moreover
K + A = 0. AZ
of
P
is
6
r = n + 1
over
-(n + 1)H
bundle formula in (0.1.3) for scrolls. negative.
since
0: P -> X
V.
Then the
by the canonical
In particular it is
Hence there is an extremal ray
contraction morphism
This
by (20.2).
r.
be the scroll of KP
F
Thus we conclude that
again yields a contradiction. dim W = 0
0
R
of
P
whose
is different from the map
§20: Ample vector bundles
By (0.4.16.2), no curve is contracted by both
n: P --+ V.
maps
and
a.
Assume that
Hence
0-1(x).
n(Z)
x
Take a point
(KP + qH)R k 0
Then
q a n + 1, but then
in
Z
is birational.
dim 0-1(x) = q > 0.
that
173
such
by (11.5).
is a point for some curve
This case is ruled out by the above
observation.
We have
dim X < dim P.
contracted to a point by n
F
:
0, so
0, hence
F
Since the canonical bundle of have of
(F, HF) = (1Pn, 0(1))
again, we obtain
nF
V = 1Pn
V.
Hence
F
dim X = dim F = n.
KP F = -(n + 1)HF, we
is
Using the finiteness
by (1.3).
Now it is easy to see that line in
Similarly
dim X a n.
n g dim F = 2n - dim X, so
is
a
is a finite morphism for any general fiber
F --' V
of
No curve in a fiber of
by [Laz2]. 6(-1)
is trivial on any
is trivial by [OSS;3.2.1], so
t(-l)
8 . ®n+10(1), as desired.
Using similar arguments we can also prove the following results.
See [YZ] for details.
(20.5) Theorem (cf. [YZ;Theorem 2]). is nef unless
(V, 8)
(1Pn
If
®n0(1)).
(20.6) Theorem (cf. [YZ;Corollary 1]).
non-zero effective divisor on crossing singularities.
V
1Pn
and
D
be a
V having only simple normal
If the locally free sheaf
is a hyperplane on
(20.7) Theorem (cf. [YZ;Theorem 3]).
K + A
D
Let
of vector fields with logarithmic poles along then
r = n, K + A
D
0(V, D) is ample,
V.
If
r = n - 1,
is nef except in the following cases.
1) There is a line bundle
L
on
V
such that
(V, L)
is a
IV: Related Topics and Generalizations
174
scroll over a smooth curve vector bundle 2) (V, 8)
on
for some
C.
®n-10(1))
(1Pn,
3) (V, 8)
4) V
3
8 = 3® ® L
and
C
n-20(1)
(1Pn,
is a hyperquadric
(D 0(2)). 1Pn+1
in
Qn
and
®n 10(1).
8
Furthermore, by using the results and techniques in [Mor2], [Laz2;S4] and [CS], we can prove the following result, as conjectured in [Muk3]. (20.8) Theorem (cf. [F27]).
If
r = n, K + A
is ample
except in the following cases. 1) (V, 8)
ti
(D n0(1)).
(]P',
2) There is a line bundle scroll over a smooth curve vector bundle 3) (V, 8) =
7
on
C
V
such that
and
8 ti 3® ® L
is a
for some
® 0(2)).
is the tangent bundle of 5) (V, 8) = (Qn, en,(,)). 4) V = Fn and
(V, L)
C.
®n-10(1)
(]Pf,
on
L
8
V.
(20.9) Now we review vector bundle versions of the classification theory by sectional genus. Clearly
(V, A)
is a polarized variety for
Its sectional genus is called the The pairs with
(V, 8).
A = det S.
c1-sectional genus of
g(V, A) g 2
are classified in
[F22] as follows.
(20.9.0) g(V, A) = 0
if and only if
V = 1P2
and
8 =
0(1) ® 0(1).
(20.9.1) g(V, A) = 1
exactly in the following cases:
1) (V, 8) = (]P200(2) e 0(1)). 2) V =
T2
and
8
is the tangent bundle of
V.
§20: Ample vector bundles
3) (V, 8) =
175
0(1) ® 0(1) ® 0(1)).
(1P2
2
Of course
4) (V, 8) = (Q2, 0(1) ® 0(1)).
X 1F1.
= 1P1
5) (V, 8) = (1P3, 0(1) (B 0(1)).
(20.9.2) If
g(V, A) = 2, then
Thus
n = 2.
8
is of
one of the types classified in (15.17).
The above results are based on the classification of the polarized manifold (20.10) Let
H
(V, A).
(P, H)
is ample by definition.
called the cation of
be the scroll of
The sectional genus
0(1)-sectional genus of (V, 8)
over
8
with small
(V, 8).
V. Then
g(P, H)
is
The classifi-
0(l)-sectional genus is in
fact a part of the classification of polarized manifolds.
Indeed, for each type in the classification list of polarized manifolds, we simply check whether or not it admits a structure of a scroll of a vector bundle.
The result is as
follows (cf. [F22;S3]). (20.10.0) g(P, H) = 0
if and only if
V = pn
and
8 =
0(1) ® 0(1).
(20.10.1) g(P, H) = 1
exactly in the following cases:
1) (V, 8) = (1P2, 0(2) ® 0(1)).
2) (V, 8) = (1P2, 0(1) ® 0(1) ® 0(1)). 3) V = F2 4) (V, 8)
and (Q2
6
is the tangent bundle of
V.
0(1) ® 0(1)).
(20.10.2) g(P, H) = 2
exactly in the following cases:
1) (V, 8) = (Q3, 0(1) ® 0(1)). 2) (V, 8)
is the type of (20.9.2).
(20.11) We have assumed
n 3 2
since, when
V
is a
N: Related Topics and Generalizations
176
curve, both
g(V, A)
and
g(P, H)
are the genus of
V.
At present, we do not have a good vector bundle version of the theory of
d-genus.
§21. Computer-aided enumeration of ruled polarized surfaces of a fixed sectional genus (21.1) Below is the core subroutine of our computer program "Classification of Rational Polarized Surfaces" written in N88-Japanese-BASIC.
This subroutine is designed
to enumerate, for any fixed values of
L2, KL
and
the possible deformation types of polarized surfaces such that
L + 2K
S # 1P2, S
is rational and either
K2 = 8
K2, all (S, L)
or
is nef.
Using this subroutine and by the observation in §14, we can produce programs for various purposes. 1160 1170 1180 1190 1200 1210 1220 1230 1240 1250 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410
'LL, KL, KK is fixed IF KK<8 THEN 1190 M=O:B=-KL:C=LL:GOSUB 1340:RETURN B(0)=-KK-KK-KL:C(0)=4*(KK+KL)+LL I=1:M=2:B=B(0):C=C(0) IF C>0 THEN 1240 IF C=0 THEN GOSUB 1510 IF I>1 THEN I=I-1:G0TO 1290 ELSE RETURN M(I)=M:B(I)=B:C(I)=C IF KK+I>0 THEN 1300 IF KK+I=O THEN 1280 I=I+1:B(I)=B:C(I)=C:M(I)=M:GOTO 1260 IF B>0 THEN D=B*B:GOTO 1320 M=M(I)+l:B=B(I)-KK-I+1:C=C(I)-B(I)-B:GOTO 1210 IF B<=0 THEN 1230 ELSE D=B*B IF D
1420 Y=(Z-X)V2 1430 IF (M OR Y) THEN GOSUB 1540 ELSE RETURN
177
§21: Computer-aided enumeration of ruled polarized surfaces
1440 1450 1460 1470 1480 1490 1500 1510 1520 1530 1540
IF E=1 THEN RETURN ELSE J=Y IF (X AND 1)>O THEN 1470 Y=Y-X12:IF Y>0 OR (Y=O AND M>O) THEN E=1:GOSUB 1540 IF (J<=X OR (J AND 1)>O OR X+X<J) THEN RETURN E=1:Y=X-JV2:X=J IF (M OR Y) THEN GOSUB 1540 RETURN IF B
(21.2) We will explain precisely how the above program Basically we employ the notation in §14.
works.
1170-80: If
K2 = 8, then
S
is a
IP1-bundle over
IP1.
This case is treated in the subroutine beginning at 1340. 1190-200: Since m = (M
possible fixed
m1)
,
r'
We will enumerate all the
(S, L).
of
in the lexicographical order.
m
Namely, for any
ml, we consider first the case of smallest
mj-1,
possible
K2 < 8, we study the adjoint spectrum
mj, and then proceed to cases of larger
A. = L. + miKi, bi = -KiAi Note that
convention.
and
bi = bi-l
ci = A i and
2
where
ci = ci-1
mj.
Set
m0 = 2
by
if
mi = mi-l'
1210-20: If If
c = 0, the mapping
(14.3; 0) or 1). 1230: A. if
c = ci > 0, then we blow down further.
i = 1.
mi-l
0
defined by
mi.
Hence we are done
i > 1, we should consider the case in which
is one bigger.
1240-70: If
1280: When
Ki2
=
K2 + i < 0, we blow down further.
Ki2 = 0, bi
should be positive by (14.13).
1290: Here we study the case in which than as it was.
Let
bi' = bi - Ki-12 and since
is of the type
This case is treated in 1510-30.
is not nef for larger
If
A i
bi
'
mi is one bigger
denote the new values. Then we have
ci' = ci - 2bi + Ki-12 = ci - bi - bi'
-Ki-1(Li-1 + miKi-1) and
ci = (Li-1 + miKi-1)' 2
IV: Related Topics and Generalizations
178
1300: Here
Ki2 > 0, so
bi g 0, we have with
(Li + tKi)2 g 0
Hence
t s m.
should change
bi
for some real number
is not nef for larger
A i
When
should be positive.
mi
t
and we
mi-1'
1310: Since
K1. 2 > 0, we should have
Otherwise we change
the index theorem. 1320-30: If
Ki2 = 8,
S
is a
bi2
ciKi2
by
mi.
Otherwise
]P1-bundle.
we continue to blow down. 1340: Set
x2e + 2xy = c
Then
r = 0).
Ar = Lr + mrKr = xH + yF (set
A0 = L
ex + 2y + 2x = b.
and
this subroutine we study this Diophantine equation. that
m > 0
r > 0
if
Note also that and
c > 0.
Hence
c = xz.
while
x(b - 2x) = c.
equation and then consider 1350: Since Hence
m = 0
In
Note
r = 0 (cf. 1180).
if
z = b - 2x
Then
z = ex + 2y.
Set
if
We first solve this
e, y.
c > 0, both
x
and
are positive.
z
b = z + 2x k 3.
1360-80: We first look for the smallest solution 1390: No solution is found such that remains to consider the case y a 0
and
x > 1, we have
b = 2x + 2 = c + 2, so Therefore we may assume x = 3
and
y = 0.
z s 3 z > 1.
x = 1
If
so it
4x s c,
Since
x > 1.
and
x.
z = 2, then
should have been a solution. Then
z = ex + 2y = 3.
This is possible only when
e = 1,
b = c = 9.
Thus this step 1390 is justified. 1400-20: Here equation. odd.
e = 0
Hence
If
x
is the smallest solution of the
z = ex + 2y e = 1
is possible.
and
is odd, then
z - x = 2y k 0.
e
If
and z
x
must be
is even,
§21: Computer-aided enumeration of ruled polarized surfaces
means
1430: m = 0
m > 0, Ar
If
ample.
1440: If
K2 = 8,
is just nef and
is odd, x
z
y > 0
so
1450-60:,If both
since
L
is
is possibly
y
0.
is the unique solution.
and
z
179
x
are even, then
possible provided the corresponding
e = 1
is
is positive (or
y
non-negative when r > 0). 1470-1500: We now consider whether or not there is another solution for b
only if
and
c
x.
By (14.11), such a solution exists If so, there are possibly
are even.
four cases: (e, x, y) = (0, a, P), (0,
with
fi,
or
a)
x = a
solution
(1,
fi,
(1, a, P - a/2),
with
a - P/2)
a < P.
The solutions
is treated in the preceding steps.
(0, P, a)
is essentially the same as (0, a,
Hence we should consider the solution This is actually the case if
P
(1,
in a situation as in (14.3;0) or 1).
Moreover, if
is a Del Pezzo surface.
fi,
is even and
1510-30: Here we consider the case
b = -KiLi a 0.
The fi).
a - X4/2).
8 f 2a.
c = 0.
Thus we are
In this case we have
b = 0, then
A. = 0
and
This is possible only if
2
Ki
S.
> 0.
If these conditions are satisfied, we have a solution (e, x, y) = (0, 0, b/2).
Recall that we may assume
e = 0
(cf. (14.17)).
1540: In the subroutine beginning here, the data of the
new type of polarized surfaces thus found is handled appropriately.
(21.3) We have produced similar programs for the classification of irrational ruled surfaces too.
The
algorithm is rather simpler than that in the rational case. (21.4) Here we exhibit several samples of results of
IV: Related Topics and Generalizations
180
First we
our computer experiments by the preceding program. present a list of
(S, L)
such that
is not a scroll, g(S, L) = 4 S
g
meaning of
q, mj, e, x, y
1) 0: 2) 0:
1 1
3) 0: 1 4) 0: 5) 0: 6) 0:
1 1
1
too.
-1: -1: -1: -1: -1: -2:
0 0 0 0 0 0
1
-2: 0 -2: 0 -2: 0 -2: 0 -2: 0 -2: 0 -2: 0 -2: 0 -3: 0 -3: 0 -3: 1 -3: 0 -3: 0 -4: 0
21) 0: 2 22) 0: 2 23) 0: 2
-1: 0 -1: 0 -1: 0
7) 0: 8) 0: 9) 0: 10) 0:
11) 0: 12) 0: 13) 0: 14) 0:
1 1 1
1 1
1 1 1
15) 0: 1 16) 0: 1 17) 0: 1 18) 0: 1 19) 0: 1
20) 0:
24) 0: 2 25) 0: 2 26) 0: 2 27) 0: 2 28) 0: 2 29) 0: 2 30) 0: 2 31) 0: 2 32) 0: 2 33) 0: 2 34) 0: 2 35) 0: 2 36) 0: 2 37) 0: 3 38) 0: 3 39) 0: 3 40) 0: 3 41) 0: 3 42) 0: 3 43) 0: 3
-1: 0 -1: 0 -2: 0 -2: 0 -2: 1 -2: 0 -2: 0 -2: 0 -2: 0 -3: 0 -3: 1 -3: 0 -4: 0 -1: 0 -1: 0 -1: 0 -1: 0 -2: 0 -2: 0 -2: 0
x 38 26 22 22 21 22 16
g < 4.
.. ,
m1)
19 13 11 11 10 11
19 13 11 11 10 11
19 13 11 11 10 11
19 13 11 11 10 11
19 13 11 11 10 11
19 13 11 10 10 11
19 12 11 10 10 11
18 12 10
8
8
8
8
7
7
7
8 7
7 7
7
7
8 7
14
16: 15:
20
20: 10 10 10 10 10 10 10
14 12 14 12
14: 13: 14: 12: 11: 14: 10: 4: 9: 8: 7:
11
14 10 9
8 8 6
7
7
7
7
7
7
6
6
6
6
6
6
7
7
7
6 5
6
7 6
7
6 5
7 5
5 7 5
5
7 5 4 4 4 3
5 7 5
7 5 5 7 5
4 4 4 3
4
4 4 3 3
4 4 3 3
4
4 4 3
7 5
4 4 3
6
9
10 10 6 9
6 6
6
7
4 5 5
5 5
5
7
6
4 4 4 3 3
4 4 3 3 2
28 20 20
28: 14 14 14 14 14 14 14 13 20: 10 10 10 10 10 10 9 9 20: 10 10 10 10 10 10 10 7
18 17 16 12 11 12 10 10
18: 17: 16: 12: 5: 12: 11: 10: 9: 8: 3: 8: 5:
9 8
7 6 5
20 16 14 14 12
10 8
The
are as in §14.
y: (mr
38: 26: 23: 22: 21: 22:
is nef or
Similar lists are obtained
See [F21], [Mae] for
for larger
K2, e,
L + 2K
and either
is a F1-bundle over a curve.
q:L2
is ruled, (S, L)
S
6
7 8 9 9
4 5 6 3
2
4 5 4 4 5 2 3 4 3 3 2
3 3 4 4 4 2 2 2 3 3 2
6
5
5 5 6 5
4 4 4 3 3 2
4 4 3 3 3 2
7 5 5 4 5 4 4 3 3 3 2
20: 10 10 10 10 10 10 10
9
7 8 3 4 5 3 3 4 4 2 3 3 2 4
16: 15: 14: 12: 10: 9:
9
9
9
8 8 6
8 8
8 8
8 8
8
6 5 6 5 5
6
6
5
5
6
6
5 5
5
8 6 5 6 5
5
5
4 4 3 3 2
4
4 4 3
5
6 5 5
4 4 3 3 2 8
4
4 3 3 2
4 3 3 2
8 7
8
7
5
8 7 7 6 5
5
7 6 5
4
4
4
4
7 7 6
6
7
3 2 8 7 7 6 5
4
5 6
8 8
2 2 2 2 3 2
2
7
8 8 8
9
3
5 6
8 8 8
9
2 2
8 7
8
6
5
7
7
5
6 6
6
6
5
2
5
6 6 5
4
4
3 4
3 3
2 2 2 3 3 3 4 2 2 3 2
2 2
2
2 2 2
2
2
§21: Computer-aided enumeration of ruled polarized surfaces
44) 0: 3 45) 0: 3 46) 0: 4 47) 0: 4 48) 0: 4 49) 0: 4 50) 0: 4 51) 0: 4 52) 0: 5 53) 0: 5 54) 0: 5 55) 0: 6 56) 0: 7 57) 0: 8 58) 0: 9
-2: 0 -3: 0 -1: 0 -1: 0 -1: 0 -2: 0 -2: 0 -3: 0 -1: 0 -1: 0 -2: 1 -1: 0 0: 0 0: 0
59) 60) 61) 62) 63) 64) 65) 66) 67)
0:12 0:18 0:20 0:20 1: 1 1: 1 1: 1 1: 1 1: 1
68) 69) 70) 71) 72)
1: 1: 1: 1: 1:
1
73) 74) 75) 76) 77) 78) 79) 80) 81) 82) 83) 84) 85) 86) 87) 88) 89) 90) 91) 92) 93) 94) 95) 96) 97)
1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1: 1:
2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4
3: 8: 8: 8: -1: -1: -2: -2: -2: -2: -2: -3: -3: -3: -1: -1: -1: -2: -2: -2: -2: -2: -3: -3: -1: -1: -1: -1: -2: -2: -2: -2: -3: -1: -1: -1: -2: -2: -3: -1: -1: -1: 0:
98) 99) 100) 101)
1 1 1 1
1: 5 1: 5 1: 6 1: 7
1: 0
0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 1 0 1 1 1 0 1 1 1 1 0 1 0 1 0 1 0 0 1
8 6
14 12 10 8 6
4 10 8 5
6 8
4 6 4 3 2 2
8: 6: 14: 12: 11: 8: 7: 6: 10: 8: 2: 6: 8: 5: 6:
4 3 7 6
4: 3:
2
5
4 3 2 5 4 2 3 4 2 3
6
25
1:
7
21 -10:
4
15 19 13 11 13
5 3
1:
-9: 1:
-4: -6:
4 4 2
1: 2:
3 3
27 -13:
5 6
7
19 17 15 11 10 10
7 6 5
4 3 2 5 4 2 3 4 2 3 2
4 3 7 6
5
5
4 3 2 5 4 2 3 4 2 3
4 3 2 5 4 2 3 4 2 3
2
2
1:
-7: -7:
2 2
3 3 4 2 2 3
2 2
3
7 3 4 3 3 4
2 2 3 3
1: 2:
2
2
2
3
3
2
19 14 14 13 11
-9:
4
1:
5 5 6
8 8 7
1:
9
7 6
5
13 10 10 6 6
1: 1:
-4: -3:
-6: -5: -5: -3: -2: -1: -6: 1:
-4: 1:
-2:
4
2:
9
-4:
7
1: 1:
5 7
-3:
3
2
2
3 3 3 2
2
2 3 2
2
2 2 2
2
3
4 4 2 2 2 2 3 2
4 3 7 6
5: 4:
37 -18:
9
4 3
4 3 7
3 2 7
6 5
6 5
4 3
4 3 2
2 5
4 2
3 4 2 3 2
5
4 2 3 4 2 3
3 2 7 6 5 4 3 2 5 3 2 2 4 2 3
3 2 6
4 5
2 3 2 4 3 2 2 3 2
181
3 2 3 4 4 2 2 2 2 3 2 2
3
2
2
2 2 2
2
2
IV: Related Topics and Generalizations
182
102) 103) 104) 105) 106) 107) 108)
1: 8 1: 8 1: 9 1:12 1:12 2: 4 2: 4
0: 0 0: 1 0:
1
0: 0: -8: -8:
0 1 0 1
4 4 3 2 2 2 2
1:
-1: 0: 3: 2: 1: 0:
(21.5) Now we can calculate the number of deformation types of ruled polarized surfaces with
g(S, L) = 4
Suppose that the minimal reduction
follows.
of one of the types in the above list.
(S0, L0)
(S0, L0) ?
The number of (-1)-curves of
S --' S0
d0 = (L0)2, so there are
deformation types of such
polarized surfaces
d0
(S, L), including
However, occasionally, (S0, L0) by
(S, L).
x = 2
for
is smaller than
(S0, L0)
itself.
is not determined uniquely
This happens to be the case if and only if (S0, L0).
in the case of types
In the above list (21.4), this occurs 61), 62), 105), 106), 107), 108).
Thus, for example, types 61) and 62) together yield types of
is
How many types of
do we have whose minimal reduction is
(S, L)
as
(S, L), instead of
we just add up
d0
2 x 20 = 40
types.
21
Otherwise
to conclude that there are 346 types.
The existence of these types is uncertain.
If the
following conjecture is true, all the types of ruled polarized surfaces enumerated by this method do exist (for larger
g
too).
Conjecture.
surface
S
effective S
Any ruled surface can be deformed to a
such that the cone in
H2(S; IR)
of pseudo-
1R-divisors is generated by all the (-1)-curves on
and curves of positive self-intersection numbers.
(21.6) The table below shows the number of possible types
(S0, L0)
with given
(g, q, d0)
as in (21.4).
Here
§21: Computer-aided enumeration of ruled polarized surfaces
183
5 s g s 7. g
5
q
0
1
32 18
9
72 41 35 18 8 7 4 3 2
10
1
1
4 3
11 12 13 14 15 16 17 18
0
0
2
2
1
1
3
0 0 0
0 0 0 2
1 1
3 0 0
0
1
1
d0 1
2 3
4 5 6 7 8
6
0
1
2
3:
0 :221 0 :146 1 94
58 62 42 36 19
0 0 0 0
0 0 0 2
2:
19 15
:
0
72
:
0: 35 0: 19 0: 17 2: 10
4 4 4 3 1
7
0
1
2
3
:525 :384 :257 :146
103 116 111
1
51
1
0 0 0 0
0
:100
9
1
10
0 0 0 0 0
: : :
65 41
33 34 23
34
21
0 0 0 0
:
16
9
1
7 8
4
5
5
2 2 1
1 1 1
0 0 0 0 0 0
2
2
2
1
0
1 1
1
9 2 1
2
0 0
20
0 0 0
0 0 0
1
2
21
1
0
24 27 28
2
0 2 0 2
0
1
5
0 0
0 2
0 0 2
1 1
32
0 0
2
A similar table is obtained for larger
g
too.
(21.7) By the method (21.5), we can calculate the number
t(g)
of types of ruled polarized surfaces of
sectional genus
g
2
3
4
5
0
18
1
9
69 40
199 142
567 329 12
q 2 3 4 5
t(g)
g g 10
The result for
g.
5
7
6
is as follows.
8
9
10
1836 865
4699 1923
12273 3631
30965 7760
70975 14876
19
31
5
9
60 16
64 23
107 30
5
9
13
15985
38821
86006
5
27
Problem.
109
346
908
2725
6962
Study the asymptotic behaviour of
t(g)
when
184
References
The subscripts in the reference codes are sometimes omitted or abbreviated in the text. For example, the paper [KaMM] is often referred to as [KMM]. [Ab] S. S. Abhyankar, Local rings of high embedding dimension, Amer. J. Math. 89 (1967), 1073-1077. [Al] V. A. Alekseev, Fractional indices of log del Pezzo surfaces, Math. USSR Izvestija 33 (1990). [An] M. Andreatta, The stable adjunction mapping, Math. Ann. 275 (1986), 305-315. [AnS]
and A. J. Sommese, Generically ample divisors on normal Gorenstein surfaces, in Singularities, Contemporary Math. 90 (1989), 1-20.
[Ar] M. Artin, Algebraization of formal moduli II, Existence of modifications,
Ann. of Math., 91 (1970), 88-135.
[ASK] T. Ashikaga and K. Konno, Algebraic surfaces of general type with
c12 = 3p9 - 7,
preprint.
[At] M. F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc., (3) 7 (1957), 414-452. [Badl] L. Badescu, A remark on the Grothendieck-Lefschetz theorem about the Picard group, Nagoya Math. J. 71 (1978), 169-179. on ample divisors, Nagoya Math. J., 86 (1982), 155-171.
[Bad2]
,
On ample divisors II, in Proceedings of the Week of Algebraic Geometry, Bucharest 1980, pp. 12-32, Teubner Texte z. Math., Band 40, 1981.
[Bad3]
,
[Bad4]
,
The projective plane blown-up at a point as an
ample divisor,
Atti. Accad. Ligure, 38 (1981), 3-7.
Hyperplane sections and deformations, in Proc. Conf. "Recent Trends in Mathematics", pp. 13-19,
[BadS]
,
Teubner Texte z. Math., 50 (1982).
References
185
[Barl] R. Barlow, Some new surfaces with
pg = 0,
Duke Math. J., 51 (1984), 889-904. [Bart]
with
A simply connected surface of general type pg 0, Invent. math., 79 (1985), 293-301. ,
[BarPV] W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Ergebn. d. Math. u. ihr. Grnzg. 3 Folge, Band 4, Springer, 1984.
,
[Beal] A. Beauville, Surfaces algebriques complexes, Asterisque 54 (1978); English translation: London Math.
Soc. Lecture Notes [Beat]
,
68 (1983).
Letter on Reider's criterion.
d-folds whose canonical bundle is not numerically effective, according to Mori and Kawamata, Annali Mat. Pura e Appl., 147 (1987), 151-172
(Bell] M. Beltrametti, On
[Be12]
,
Contractions of non-numerically effective
extremal rays in dimension 4, in Proc. Conf. on Algebraic Geometry, Berlin 1985, pp.24-37, Teubner Text z. Math. 92, 1987. [BeBS]
,
A. Biancofiore and A. J. Sommese, Projective
n-folds of log-general type, I, Trans. AMS, 314 (1989), 825-849. [BeFS]
,
P. Francia and A. J. Sommese, On Reider's
method and higher order embeddings, Duke Math. J. 58 (1989), 425-439. [BeLP]
A. Lanteri and M. Palleschi, Algebraic surfaces
,
containing an ample divisor of arithmetic genus two, Arkiv for mat., 25 (1987), 189-210. and M. Palleschi, A footnote to a paper by Geometrae Ded., 22 (1987), 149-162. Ionescu,
[BePl]
M. Schneider and A. J. Sommese, Threefolds of degree 9 and 10 in 1P5, preprint, Bayreuth, 1989.
[Be SS]
[BeSl]
,
and A. J. Sommese, On generically polarized
Gorenstein surfaces of sectional genus two, J. rein. angew. Math., 386 (1988), 172-186.
References
186 [BeS2]
&
,
On normal Gorenstein polarized
varieties of sectional genus 3 and 4, Indiana Univ. Math. J., 37 (1988), 667-686. [BeS3]
&
,
On the relative adjunction mapping,
,
New properties of special varieties
preprint, 1989. [BeS4]
&
arising from adjunction theory, preprint, 1989. [BiLl] A. Biancofiore and E. L. Livorni, Algebraic non ruled surfaces with sectional genus equal to seven, Ann. Univ. Ferrara, Sez. VII Sc. Mat., 32 (1986), 1-14. [BiL2]
,
Algebraic ruled surfaces with low sectional Ricerche di Mat., 36 (1987), 17-32.
genus,
On the iteration of the adjunction process in the study of rational surfaces,
[BiL3]
,
Indiana Univ. Math. J., 36 (1987), 167-188.
On the genus of a hyperplane section of a geometrically ruled surface,
[BiL4]
,
Annali di Mat. pura ed appl., 147 (1987), 173-185. [BiFL] A. Biancofiore, M. L. Fania and A. Lanteri, Polarized surfaces with hyperelliptic sections, preprint. [BIG] S. Bloch and D. Gieseker, The positivity of the Chern classes of an ample vector bundle, Invent. Math., 12 (1971), 112-117. [B0M] E. Bombieri and D. Mumford, Enriques classification of surfaces in char
p;
II,
in Complex Analysis and
Algebraic Geometry, pp. 23-42, Cambridge Univ. Press, 1977.
[Bre] L. Brenton, On singular complex surfaces with negative canonical bundle, with applications to singular C2
compactification of and 3-dimensional rational singularities, Math. Ann. 248 (1980), 117-124. [BrL] S. Brivio and A. Lanteri, On complex projective surfaces with trigonal hyperplane sections, manuscripta math.
65 (1989), 83-92.
References
187
[Cas] G. Castelnuovo, Numero delle involuzione rationali giacenti sopra uno curve di date genere, Rendi dello R. Acad. Lincei, Ser. 4 (1889), 5. [Cat] F. Catanese, Surfaces with
K2 = pg = 1
and their
period mapping, in Proceedings on Algebraic Geometry at Copenhagen 1978, pp. 1-29, Lecture Notes in Math. 732, Springer, 1979.
[CS] K. Chou and E. Sato, Smooth projective varieties dominated by G/P, preprint, Kyushu Univ., 1989. [D]
P. Deligne, Theorie de Hodge II,
Pubi. Math. I. H. E. S., 40 (1971), 5-57. [DG] (= [EGA]) J. Dieudonne and A. Grothendieck, Elements de la geometrie algebrique, Pubi. Math. I. H. E. S., 4, 8, 11, 17, 20, 24, 28, 32 [Einl] L. Ein, Surfaces with a hyperelliptic hyperplane section,
Duke Math. J.
50 (1983), 685-694.
Nondegenerate surfaces of degree J. rein. angew. Math. 351 (1984), 1-11.
[Ein2]
,
n+3
in
F.,
[EiH] D. Eisenbud and J. Harris, On varieties of minimal degree, in Algebraic Geometry, Bowdoin 1985, part I, pp. 3-13, Proc. Symp. in Pure Math. 46, AMS, 1987. [Ekl] T. Ekedahl, Foliations and inseparable morphisms, in Algebraic Geometry, Bowdoin 1985, part II, pp. 139-149, 46, AMS, 1987.
Proc. Symp. in Pure Math. [Ek2]
Canonical models of surfaces of general type in positive characteristic, Publ. Math. I. H. E. S., 67 (1988), 97-144. ,
[Fa] M. L. Fania, Extension of modifications of ample J. Math. Soc. divisors on fourfolds, & ibid. part II, Japan 36 (1984), 107-120 [FaL]
&
38 (1986), 285-294.
and E. L. Livorni, Polarized surfaces of A-genus 3, preprint.
References
188
E. Sato and A. J. Sommese, On the structure of 4-folds with a hyperplane section which is a ]P1-bundle over a surface that fibers over a curve, Nagoya Math. J., 108 (1987), 1-14.
[FaSS]
,
and A. J. Sommese, On the minimality of hyperplane sections of Gorenstein threefolds, in Proc. in
[FaSi]
honor of W. Stoll, Notre Dame, pp.89-113, Vieweg, 1984. [FaS2]
&
,
Varieties whose hyperplane section are
IPk-bundles,
Annali d. Scuola Norm. Sup. Pisa, 15 (1988), 193-218. & on the projective classification of smooth n-folds with n even, preprint.
[FaS3]
,
[Fnl] G. Fano, Sopra alcune varieta algebriche a tre dimensione aventi tutti i generi nulli,
in Atti Cong.
internaz. di mat. Bologna, 4, pp. 115-119, 1931. [Fn2]
Sulle varieta a tre dimensioni a curve-sezioni Mem. R. Acad. Italia, 8 (1937), 23-64.
,
canoniche, [Fn3]
Nuove ricerche sulle varieta algebriche tre dimensioni a curve-sezioni canoniche, Comm. Dont. Ac. Sci. 11 (1947), 635-720. ,
[Fl] T. Fujita, On the structure of polarized varieties with d-genera zero, J. Fac. Sci. Univ. of Tokyo, 22 (1975), 103-115. [F2]
[F3]
Defining equations for certain types of polarized varieties, in Complex Analysis and Algebraic Geometry (edited by Baily and Shioda), pp. 165-173, Iwanami and Cambridge Univ. Press, 1977. ,
,
An extendability criterion for vector bundles on Proc. Japan Acad. 54 (1978), 298-299.
ample divisors, [F4]
On the hyperplane section principle of Lefschetz, J. Math. Soc. Japan 32 (1980), 153-169. ,
189
References
[F5]
On the structure of polarized manifolds with total deficiency one, part I, II & I, J. Math. Soc. Japan, 32 (1980), 709-725 & 33 (1981), 415-434 & 36 ,
(1984), 75-89. [F6]
On topological characterizations of complex projective spaces and affine linear spaces, ,
Proc. Japan Acad., 56 (1980), 231-234. [F7]
Vector bundles on ample divisors J. Math. Soc. Japan 33 (1981), 405-414.
[F8]
On L-dimension of coherent sheaves & Corrections, J. Fac. Sci. Univ. of Tokyo, 28 (1981), 215-236 & 29 (1982), 719-720.
[F9]
Impossibility criterion of being an ample divisor, J. Math. Soc. Japan, 34 (1982), 355-363.
[F10]
,
,
,
Projective threefolds with small secant
,
Sci. Papers of Col. Gen. Ed., Univ. of Tokyo, 32 (1982), 33-46. varieties,
[F11]
,
On polarized varieties of small
Tohoku Mathematical J. [F12]
d-genera,
34 (1982), 319-341.
Theorems of Bertini type for certain types of polarized manifolds, J. Math. Soc. Japan 34 (1982), ,
709-718. [F13]
[F14]
On hyperelliptic polarized varieties, Tohoku Math. J. 35 (1983), 1-44. ,
,
Rational retractions onto ample divisors,
Sci. Papers of the College of Arts and Sciences, Univ. of Tokyo, 33 (1983), 33-39. [F15]
[F16]
Semipositive line bundles, J. Fac. Sci. Univ. of Tokyo 30 (1983), 353-378. ,
,
Polarized manifolds of
J-genus two, part
I
J. Math. Soc. Japan 36 (1984), 709-730. [F17]
,
Projective varieties of d-genus one,
Algebraic and Topological Theories - to the memory of Dr. Takehiko MIYATA, pp. 149-175, Kinokuniya, 1985.
References
190
[F18]
On polarized manifolds whose adjoint bundles are not semipositive, in Algebraic Geometry Sendai 1985, pp. 167-178, Advanced Studies in Pure Math. 10, ,
Kinokuniya, 1987. [F19]
,
A generalization of Lefschetz theorem,
Proc. Japan Acad. 63 (1987), 233-234. [F20]
Triple covers by smooth manifolds, J. Fac. Sci. Univ. of Tokyo 35 (1988), 169-175. ,
, Classification of polarized manifolds of
[F21]
in Algebraic Geometry and Commutative Algebra, in Honor of Masayoshi NAGATA, pp. 73-98, Kinokuniya, 1987. sectional genus two,
[F22]
genera, [F23]
Ample vector bundles of small c1-sectional J. Math. Kyoto Univ. 29 (1989), 1-16.
Polarized manifolds of degree three and J-genus J. Math. Soc. Japan 41 (1989), 311-331. ,
two, [F24]
Remarks on quasi-polarized varieties, Nagoya Math. J. 115 (1989), 105-123.
[F25]
On singular Del Pezzo varieties, in Algebraic Geometry, Proceedings L'Aquila 1988, pp. 117-128, Lecture Notes in Math. 1417, Springer, 1990.
,
,
[F26]
[F27]
,
On Del Pezzo fibrations over curves, preprint.
,
On adjoint bundles of ample vector bundles,
preprint. [F28]
On certain polarized elliptic surfaces of sectional genus two, in preparation. ,
[FuR] T. Fujita and J. Roberts, Varieties with small secant 103 varieties: the extremal case, Amer. J. of Math. (1981), 953-976. [Ful] W. Fulton, Intersection Theory, Ergebn. d. Math. u. ihr. Grenzg., 3. Folge, Band 2, Springer, 1984. [Gor] R. Goren, Characterization and algebraic deformations of projective space, J. Math. Kyoto Univ., 8 (1968), 41-47.
References
191
[GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Interscience, 1978. [Grol] A. Grothendieck, Sur la classification des fibres holomorphes sur la sphere de Riemann, Amer. J. Math. 79 (1957), 121-138. et al., Revetements Stales et Groupe Fondamental (SGA 1), Lecture Notes in Math. 224, Springer, 1971.
[Gro2]
Cohomologie Locale des Faisceaux Coherents et Theoremes de Lefschetz Locaux et Globaux (SGA 2),
[Gro3]
,
North-Holland, 1968. [Hal] R. Hartshorne, Ample vector bundles,
Publ. Math. I. H. E. S., 29 (1966), 63-94. [Ha2]
Residue and Duality, Lecture Notes in Math. 20, Springer, 1966.
[Ha3]
, Ample Subvarieties of Algebraic Varieties, Lecture Notes in Math. 156, Springer, 1970.
[Ha4]
,
,
Algebraic Geometry, Graduate Texts in Math. 52,
Springer, 1977.
[HidW] F. Hidaka and K. Watanabe, Normal Gorenstein surfaces with ample anti-canonical divisor, Tokyo J. Math. 4 (1981), 319-330.
[Hirn] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math., 79 (1964), 109-326. [Hirz] F. Hirzebruch, Topological Methods in Algebraic Geometry, Grundl. der math. Wissens. 131, Springer, 1956 (3rd enlarged edition 1966) [HiK]
and K. Kodaira, On the complex projective spaces, J. Math. Pure Appl., 36 (1957), 201-216.
[Horl] E. Horikawa, On deformations of quintic surfaces, Invent. Math.
31 (1975), 43-85.
References
192
Algebraic surfaces of general type with small c12, part I, II, IQ, IV, V. Ann. Math. 104 (1976), 358-387 Invent. Math. 37 (1976), 121-155 & ibid. 47 (1978) & 209-248 & ibid. 50 (1979), 103-128 & J. Fac. Soc.
[Hor2]
,
Univ. of Tokyo
28 (1981), 745-755.
On algebraic surfaces with pencils of curves of genus 2. in Complex Analysis and Algebraic Geometry pp. 79-90, Cambridge Univ. Press, 1977.
[Hor3]
,
[Iii] S. Iitaka, On D-dimensions of algebraic varieties, J. Math. Soc. Japan, 23 (1971), 356-373. [Ii2]
Algebraic Geometry, Grad. Texts in Math. 76, Springer, 1982. ,
[Ioll P. Ionescu, Embedded projective varieties of small invariants, in Proc. on Algebraic Geometry at Bucharest 1982, pp. 142-186, Lecture Notes in Math. 1056, 1984. [Iol-2]
,
ibid., part II,
Rev. Roum. Math. Pures Appl. [Iol-3]
,
31 (1986), 539-544.
ibid., part II,
INCREST Preprint Series in Math., No. 59/1988. [Io2]
On varieties whose degree is small with respect to codimension, Math. Ann. 271 (1985), 339-348.
[Io3]
Generalized adjunction and applications, Math. Proc. Cambridge Phil. Soc., 99 (1986), 457-472.
,
,
[Io4]
Ample and very ample divisors on surfaces,
,
Rev. Roum. Math. Pures Appl.
33 (1988), 349-358.
[Isl] V. A. Iskovskih, Fano 3-folds, Nauk SSSR Izvestija [Is2]
I (Russian), Izv. Akad.
41 (1977); English translation: Math. USSR 11 (1977), 485-527.
Fano 3-folds, II (English translation),
,
Math. USSR Izvestija
12 (1978), 469-506.
[Kawl] Y. Kawamata, Characterization of Abelian varieties, Compositio Math., 43 (1981), 253-276.
A generalization of Kodaira-Ramanujam's vanishing theorem, Math. Ann., 261 (1982), 43-46.
[Kaw2]
,
References
193
[KaMM] Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the Minimal Model Problem, in Algebraic Geometry Sendai 1985, pp. 283-360, Advanced Studies in Pure Math., 10, Kinokuniya, 1987.
[K1] S. L. Kleiman, Toward a numerical theory of ampleness Ann. of Math., 84 (1966), 293-344. [Kob0] S. Kobayashi and T. Ochiai, Characterization of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ., 13 (1973), 31-47. [Kod1] K. Kodaira, On a differential-geometric method in the theory of algebraic stacks,
Proc. Nat. Acad. Sci. USA, 39 (1953), 1268-1273. [Kod2]
,
On stability of compact submanifolds of Amer. J. Math., 85 (1963), 79-94.
complex manifolds,
[KO1M] J. Kollar and T. Matsusaka, Riemann-Roch type inequalities, Amer. J. Math., 105 (1983), 229-252. [Lan] A. Lanteri, On polarized surfaces of d-genus two,
Annali di Mat. pura ed appl., 151 (1988), 317-329. [LanP] A. Lanteri and M. Palleschi, Projective manifolds containing many rational curves, Indiana Univ. Math. J., 36 (1987), 857-865. [Lazl] R. Lazarsfeld, A Barth-type theorem for branched coverings of projective space, Math. Ann. 249 (1980), 153-162.
Some applications of the theory of positive vector bundles, in Complete Intersections, pp. 29-61, Lecture Notes in Math. 1092, Springer, 1984.
[Laz2]
,
[LazV] R. Lazarsfeld and A. Van de Ven, Topics in the Geometry of Projective Space: Recent Work of F. L. Zak, DMV Seminar, Band 4, Birkhauser, 1984. [LibW] A. S. Libgober and J. W. Wood, Uniqueness of the complex structure on Kahler manifolds of certain homotopy types, preprint, 1988.
References
194
[Livl] E. L. Livorni, Classification of algebraic surfaces with sectional genus less than or equal to six. I: Rational surfaces, Pacific J. Math. 113 (1984), 93-114. [Liv2]
,
II: Ruled surfaces with dim ¢K+L(X) = 1
Canad. J. Math., 38 (1986), 1110-1121. [Liv3]
,
Math. Scand.
IQ: Ruled surfaces with dim OK+L(X)
2,
59 (1986), 9-29.
Classification of algebraic non-ruled surfaces with sectional genus less than or equal to six,
[Liv4]
,
Nagoya Math. J., 100 (1985), 1-9.
Some results about a smooth hyperplane section of a 3-fold of non negative Kodaira dimension,
[Liv5]
,
Arch. Math., 48 (1987), 530-532.
[LiS] E. L. Livorni and A. J. Sommese, Threefolds of nonnegative Kodaira dimension with sectional genus less than or equal to 15,
Annali d. Scuola Norm. Sup. Pisa, 13 (1986), 537-558. [Lvi] S. L'vovsky, On the extension of varieties defined by quadratic equations, Mat. Sbornik 135 (177) (1988), 322-324; English Translation: Math. USSR Sbornik 63 (1989), 305-317. [Lv2]
,
Criterion for the nonrepresentability of a
variety as a hyperplane section, Ser. I: Mat. Meh.
Vestnik Moskov. Univ.
1985, n- 3, 25-28.
Translation: Moscow Uni. Math. Bulletin [Lv3]
,
English 40 (1985).
Extensions of projective varieties and
deformations, preprint, 1990.
[Mae] H. Maeda, On polarized surfaces of sectional genus three, Sci. Pap. Col. of Arts & Sci. Univ. of Tokyo, 37 (1987), 103-112.
[Mat] T. Matsusaka, Polarized varieties with a given Hilbert Amer. J. Math., 94 (1972), 1027-1077. polynomial,
References
195
[Ma M] T. Matsusaka and D. Mumford, Two fundamental theorems
on deformations of polarized varieties, Amer. J. Math., 86 (1964), 668-684. [Mil] J. Milnor, Morse Theory, Annals of Math. Studies 51, Princeton Univ. Press, 1963. [Mirl] R. Miranda, Triple covers in algebraic geometry, Amer. J. Math., 107 (1985), 1123-1158. [Mir2]
,
On canonical surfaces of general type with Math. Zeits. 198 (1988), 83-93.
K 2 = 3x - 10,
[Miyl] Y. Miyaoka, Tricanonical maps of numerical Godeaux surfaces,
Invent. math., 34 (1976), 99-111.
The Chern classes and Kodaira dimension of a minimal variety, in Algebraic Geometry, Sendai 1985,
[Miy2]
,
pp. 449-476, Advenced Studies in Pure Math. 10, Kinokuniya, 1987. [Moil] B. G. Moishezon, On n-dimensional compact varieties with n algebraically independent meromorphic functions I, II, E. Izv. Akad. Nauk SSSR Ser. Math., 30 (1966), 133-174, 345-386 and 621-656. AMS translations Ser. 2, 63, 51-177.
The Castelnuovo-Enriques contraction theorem for arbitrary dimension (Russian), Izv. Akad. Nauk. SSSR Ser. Math., 33 (1969), 974-1025.
[Moi2]
,
[Morl] S. Mori, On a generalization of complete intersections, J. Math. Kyoto Univ. 15 (1975), 619-646. [Mor2]
,
bundles, [Mor3]
,
Projective manifolds with ample tangent Ann. Math. 110 (1979), 593-606.
Threefolds whose canonical bundles are not Ann. Math. 116 (1982), 133-176.
numerically effective,
Classification of higher-dimensional varieties in Algebraic Geometry Bowdoin 1985, part I, pp. 269-331 Proc. Symp. in Pure Math. 46, AMS, 1987.
[Mor4]
,
[Mor5]
,
Flip theorem and the existence of minimal
models for 3-folds, J. of AMS, 1 (1988), 117-253.
References
196
[Mukl] S. Mukai, Curves, K3 surfaces and Fano 3-folds of in Algebraic Geometry and Commutative genus s 10 Algebra, in Honor of M. Nagata, pp. 357-377, ,
Kinokuniya, 1988. [Muk2]
Fano manifolds of coindex 3, preprint.
,
Problems on characterization of the complex projective space, in Birational Geometry of Algebraic Varieties - Open Problems, The 23rd International Symp. Division Math., The Taniguchi Foundation, Katata, 1988.
[Muk3]
,
[Muml] D. Mumford, Pathologies
I,
Amer. J. Math. 89 (1967), 94-103.
Enriques' classification of surfaces in char p: I, in Global Analysis, pp. 325-339, Princeton Univ. Press, 1969.
[Mum2]
,
[Mum3]
,
Varieties defined by quadratic equations, in
Questions on Algebraic varieties (CIME, Ciclo, Varenna 1969), pp. 29-100, Edizioni Cremonese, 1970. III
[Nag] M. Nagata, On rational surfaces
Mem. Coll. Sci. Univ. Kyoto
I,
32 (1960), 351-370.
[Nk] Y. Nakai, A criterion of an ample sheaf on a projective scheme, Amer. J. Math., 85 (1963), 14-26. [Nkm] I. Nakamura, Threefolds homeomorphic to quadrics
Q3
in
IP4,
IF3
or hyper-
preprint, Hokkaido Univ.
[Nkn] S. Nakano, On the inverse of monoidal transformation, Publ. RIMS Kyoto Univ., 6 (1971), 483-502. [Oki] C. Okonek, Moduli reflexiver Garben and Flachen von kleiner Grad in IF4, Math. Z. 184 (1983), 549-572. [Ok2]
,
Uber 2-codimensionale Untermannigfaltigkeiten in IF4 and IFS 7 187 (1984), 209-219.
von Grad Math. Z. [Ok3]
Flachen von Grad 8 in Math. Z. 191 (1986), 207-223. ,
IF4,
References
197
Vector Progress in
[OSS] C. Okonek, M. Schneider and H. Spindler,
Bundles on Complex Projective Spaces, Birkhauser, 1980. Math. 3 ,
[Ool] A. Ooishi, Genera and arithmetic genera of commutative rings, Hiroshima Math. J., 17 (1987), 47-66. [Oo2]
, 4-genera and sectional genera of commutative rings, Hiroshima Math. J., 17 (1987), 361-372.
[Ra] C. P. Ramanujam, Remarks on the Kodaira vanishing theorem & a supplement, J. Indian Math. Soc. (N. S.) 36 (1972), 41-51
&
38 (1974), 121-124.
[Rat] J. Rathmann, The uniform position principle for curves 276 (1987), 565-579. in characteristic p, Math. Ann. [Rdl] M. Reid, Surfaces with p9= 0, K2 = 1, J. Fac. Sci. Univ. of Tokyo, 25 (1978), 75-92. [Rd2]
Projective morphisms according to Kawamata, Preprint, Warwick. ,
[Rd3]
Surfaces of small degree,
,
Math. Ann. [Rd4]
275 (1986), 71-80.
Young person's guide to canonical singularities in Algebraic Geometry Bowdoin 1985, part I, pp.345-414, ,
Proc. of Symp. in Pure Math. 46, A. M. S., 1987. [Rdr] I. Reider, Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. Math., 127 (1988), 309-316. [Rob] J. Roberts, Projective Embeddings of Algebraic Varieties, Monograf. d. Inst. Mat. 19, Univ. Nac. Autonoma de Mexico, 1988. [Sail] B. Saint-Donat, On Petri's analysis of the linear system of quadrics through a canonical curve, Math. Ann. 206 (1973), 157-175. Projective models of K3-surfaces, Amer. J. Math. 96 (1974), 602-639.
[Sai2]
,
[Sakl] F. Sakai, Ample Cartier divisors on normal surfaces,
J. rein. angew. Math., 366 (1986), 121-128.
References
198
[Sak2]
,
on polarized normal surfaces,
Manuscripta Math., 59 (1987), 109-127. Reider-Serrano's method on normal surfaces, in Algebraic Geometry, Proceedings in L'Aquila 1988,
[Sak3]
,
pp. 301-319, Lecture Notes in Math. 1417, Springer, 1990.
[Sall] J. D. Sally, On the associated graded ring of a local Cohen-Macaulay ring,
J. Math. Kyoto Univ., 17 (1977), 19-21. [Sal2]
,
Cohen-Macaulay local rings of maximal embed-
ding dimension, J. of Algebra, 56 (1979), 168-183. [Sa13]
,
Stretched Gorenstein rings,
J. London Math. Soc. (2), 20 (1979), 19-26. [Sa14]
,
Tangent cones at Gorenstein singularities,
Compositio Math., 40 (1980), 167-175.
Good embedding dimensions for Gorenstein singularities, Math. Ann., 249 (1980), 95-106.
[Sal5]
,
The Poincare series of stretched CohenMacaulay rings, Can J. Math., 32 (1980), 1261-1265.
[Sa16]
,
[Satl] E. Sato, A variety which contains a P1-fiber space as an ample divisor, in Algebraic Geometry and Commutative Algebra, in Honor of Masayoshi NAGATA, pp. 665-691, Kinokuniya, 1987. [Sat2]
,
On projectively uniruled varieties, preprint.
and H. Spindler, On the structure of 4-folds with a hyperplane section which is a 1P1-bundle over a ruled surface, in Proc. Complex Analysis and Algebraic Geometry Conf. 1985, pp. 145-149, Lecture Notes in
[Bats]
Math. 1194, Springer, 1986.
[Sei] A. Seidenberg, The hyperplane sections of normal Trans. A. M. S., 69 (1950), 357-386. varieties, [Ser] J-P. Serre, Faisceaux algebriques coherents, Ann. Math., 61 (1955), 197-278.
199
References
[Sern] F. Serrano, Elliptic surfaces with an ample divisor of genus two, preprint, 1989. [Sha] I. R. Shafarevich et al., Algebraic Surfaces, Proc. Steklov Inst. Math.
75 (1965), AMS Translations, 1967.
[Shol] V. V. Shokurov, Theorems on non-vanishing, Math. USSR Izv., 26 (1986), 591-604.
Problems, in Birational Geometry of Algebraic Varieties - Open Problems, The 23rd Intern. Symp. Div. of Math., The Taniguchi Foundation, 1988, Katata.
[Sho2]
,
[Sol].A. J. Sommese, On manifolds that cannot be ample Math. Ann., 221 (1976), 55-72. divisors, [So2]
,
Non-smoothable varieties, 54 (1979), 140-146.
Comment. Math. Helv. [So3]
,
Hyperplane sections of projective surfaces I.
The adjunction mapping,
Duke Math. J., 46 (1979), 377-401. [So4]
On hyperplane sections, in Algebraic Geometry Proc. Chicago Circle Conf. 1980, pp. 323-271, Lecture Notes in Math. 862, Springer, 1982.
[So5]
On the minimality of hyperplane sections of projective threefolds, J. rein. angew. Math., 329 (1981), 16-41.
[So6]
,
,
,
Ample divisors on 3-folds,
in Algebraic Threefolds, pp. 229-240, Lecture Notes in Math. 947, Springer, 1982. [So7]
,
Configurations of -2 rational curves on hyper-
plane sections of projective threefolds, in Classification of Algebraic and Analytic Manifolds,
pp. 465-497, Progress in Math. 39, Birkhauser, 1983. [So8]
[So9]
Ample divisors on normal Gorenstein surfaces, Abh. Math. Sem. Univ. Hamburg, 55 (1985), 151-170. ,
,
Ample divisors on Gorenstein varieties,
in Proc. of Complex Geometry Conf. Nancy 1985, Revue de l'Inst. E. Cartan, 10, 1986.
200
References
On the adjunction theoretic structure of projective varieties, in Proc. Complex Analysis and Algebraic Geometry Conf. 1985, pp. 175-213, Lecture
[Solo]
,
Notes in Math. 1194, Springer, 1986.
On the nonemptiness of the adjoint linear system of a hyperplane section of a threefold, J. rein. angew. Math. 402 (1989), 211-220.
[Soil]
[S0V]
,
and A. Van de Ven, On the adjunction mapping Math. Ann. 278 (1987), 593-603.
[Sw] H. P. F. Swinnerton-Dyer, An enumeration of all varieties of degree 4, Amer. J. Math. 95 (1973), 403-418. [Ta] H. Tango, On a criterion of a projective space,
Kyoto Univ. of Education
Bull.
Ser. B, 47 (1975), 1-11.
[Tol] H. Tokunaga, Triple coverings of algebraic surfaces according to the Cardano formula, to appear in J. Math. Kyoto Univ. [To2]
On the construction of triple coverings of certain type of algebraic surfaces, preprint.
[To3]
Non-Galois triple coverings and double coverings, preprint.
[U]
K. Ueno, Classification Theory of Algebraic Varieties
,
,
and Compact Complex Spaces, Lecture Notes in Math. 439, Springer, 1975. [Umz] Y. Umezu, Quartic surfaces of elliptic ruled type, Trans. AMS, 283 (1984), 127-143.
[Ur] T. Urabe, Classification of non-normal quartic surfaces, Tokyo J. Math. 9 (1986), 265-295. [V] A. Van de Ven, On the 2-connectedness of very ample divisors on a surface, Duke Math. J. 46 (1979), 403-407. [Vil] E. Viehweg, Klassifikationstheorie algebraischer Varietdten der Dimension drei, Compositio Math. 41 (1980), 361-400. [Vi2]
,
Vanishing theorems,
J. reine angew. Math., 335 (1982), 1-8.
References [W]
201
P. M. H. Wilson, Towards birational classification of algebraic varieties, Bull. London Math. Soc., 19 (1987), 1-48.
[Y]
S. T. Yau, On the Ricci curvature of a compact Kdhler manifolds and the complex Monge-Ampere equations I, Comm. Pure Appl. Math. 31 (1978), 339-411.
[YZ] Y.-G. Ye and Q. Zhang, On ample vector bundles whose adjunction bundles are not numerically effective,
preprint, Duke Univ., 1989, to appear in Duke Math. J. [Zr] O. Zariski, Introduction to the problem of minimal models in the theory of algebraic surfaces, Pubi. Math. Soc. Japan 4 (1958). [Zkl] F. L. Zak, Projections of algebraic varieties, Math. USSR Sbornik, 44 (1983), 535-544. [Zk2]
,
Severi varieties,
Math. USSR Sbornik [*]
54 (1985), 113-127.
* * *, Correspondence,
Amer. J. Math. 79 (1957), 951-952.
After the completion of the manuscript, the following book has been published, which contains many papers on topics discussed in this book. Algebraic Geometry, Proceedings of the International Conference held in L'Aquila 1988, Lecture Notes in Math. 1417, Springer, 1990.
202
Subject Index adjoint bundle: 39,40,86,93,96,97,104,152,153,156,157,158, 165,171,173,174 adjoint spectrum: 116,121,125,177,180 adjunction theory: §18 admissible minimalization: 116 ample vector bundle: 41, 113,132 adjoint bundle of 171,173,174 c -sectional genus of 174 175 0i1)-sectional genus of :
:
:
ampleness criterion for blow down: 61 Nakai's 14 6 Serre's Bertini theorem for irreducibility, reducedness: 32 for k-Macaulayness: 8 for normality: 9 for smoothness: 36,62 : :
bi-elliptic surface:= hyperelliptic surface big: 16,164 blowing-down: 4 extension of 60 :
blowing-up: 3 canonical bundle formula for Chow ring of 12 simple 103 canonical bundle formula for blowing-up: 4 for double covering: 49 for scrolls: 2
:
4
:
:
Castelnuovo bound: 139,140 Castelnuovo variety: 142,143 characterization of In (and
Qn): 22,23,24,52,59,62,93,104, 159,171,173,174,175
Chern class: 11 Chow ring: 10 of scroll: 12 Clifford's theorem: 31
Cohen Macaulay singularity: 22,36,38,43,44,49,50,56,170 deformaton type: 108,110,118,182 (generalized) cone: 42,79,159,161,169,170 degree: 4 Del Pezzo fibration: 168
203
Subject Index
Del Pezzo surface: 62,115,161 of degree one: 123,134,153,156 of degree two: 153 Del Pezzo variety (or manifold): 44,83 45,46,82,107,161 characterization of classification of 72 of degree 1: 50 2: 49 3: 47 4: 47 5: 69,77 6: 66,76 7: 65 8: 65 in positive characteristic case: 73 d-genus: 26 :
:
desingularization: 9 double covering: 48,49,51,53,85,89,91,126,129,142,143 elliptic ruled variety: 123,133,134,153,157 elliptic scroll: 47,84,89,107,137,517,161 embedded component: 7,23,168 energy: 161 Enriques surface:. 20,126 exceptional divisor: 3 extension theorem (of Lefschetz type) of blowing-dowm: 60 of line bundle: 55,76 of linear system: 57 of morphism: 57,58 of scroll structure: 60 of vector bundle: 71 extremal ray (extremal curve): 18,93,166,171,172 Fibration theorem: 17,115,172 flatness criterion: 6 flip: 166 higher direct image: 4 Hilbert polynomial: 13,24,25,45,95,99,108,110 hyperelliptic fibration: 52,86 hyperelliptic polarized variety: 143,90 hyperelliptic surface (:= bielliptic surface): 21,126 hyperquadric fibration: 87,100,112,129,134,135,159,160 index theorem: 15,98,110,167 intersection number: 11,13 Jacobian vector bundle: 132
204
Subject Index
K3-surface: 20,126 Kleiman's criterion of nefness: 15 Kodaira's Lemma for bigness: 16 ladder: 28 regular existence of
:
28,30,141 30,32,36,37,38,43,44 :
Lefschetz theorem: 55 Leray spectral sequence: 5 lift to characteristic zero: 73 log-terminal singularity: 96,156,157,162,164 k-Macaulay singularity: 7,8,44,57,58,59 Cohen Macaulay singularity: 22,36,38,43,44,49,50,56,170 minimal reduction: 103,11,121,129,130,159,160 Nakai's criterion of ampleness: 14 nef: 14,15,16,17,18 and big: 16,164 normal singularity: 8,9,17,30,43,78,80,81,82,164,169 polarized manifold of d-genus 0: 41,107 1: 47-55,72 2: 92,150 150 fixe d of sectional genus 0: 30,41,107 1: 72,84,107 2: 86,122,129,150 3: 136,150 :
fixed: 108'
polarized varieties of 4-genus
0: §5,42,79,142,143,167 1: §6,§9,82,83
proper transform: 4 Q-bundle: 14
Q-divisor: 14 quasipolarized variety: 164 rational scroll: 41,80,107,145,§17 relatively (very) ample: 3 relative rational map: 3 ridge: 42,79,145 Riemann-Roch-Hirzebruch formula: 13 rung: 25 regular
:
26,27
,114ti ,150
205
Subject Index
scroll: 2,102,160,172 over curves: 96,107,108,109,122,129,152,156,157, 158,159,165,174 surfaces: 98,101,113,129,132,159 12 Chowring of :
e llip ti c ext ens i on o f
47 , 84 89 , 107 , 137 ,§ 17 161 s truc t ure: 60 41 , 80 , 107 145 ,§ 17
:
rati ona l
,
:
,
,
sectional genus: 25,26
c-
:
Oil)-
174 :
175
Segre class: 12 semiample: 14,157,158 Serre's condition Serre duality: 6
Sk: 8
Serre's normality criterion: 8 vanishing theorem: 6 Severi variety (in the sense of Zak): 105 simple blow-up: 103 simply generated: 27,30,35,38,44,86,142,170 spectral sequence: 5 spectral value (in the sense of Sommese): 158,159,160,162 Stein factorization: 1 strict transform: 4 Todd class: 12
triple cover: 90 upper-semicontinuity theorem: 6 vanishing theorem Kawamata-Viehweg's 16 Kodaira's 16 Mumford's 17 Serre's 6 Sommese's 56 :
:
:
:
:
varieties of minimal degrees: 43 varieties of small degrees: §17 varieties with many rational curves: 104 Veronese surface: 39,41,80,96,97,106,112,146,159 (weighted) hypersurface (or complete intersection): 28,47, 48,50,52,53,59,89,128,131,144
