Annali di Matematica 181, 213–221 (2002) DOI (Digital Object Identifier) 10.1007/s102310100036
Gianfranco Casnati · Andrea Del Centina
A characterization of bielliptic curves and applications to their moduli spaces Received: March 25, 2000; in final form: March 10, 2001 Published online: May 29, 2002 – Springer-Verlag 2002 Abstract. We deal with the covers of degree 4 naturally associated to a bielliptic curve of genus g ≥ 6, giving a proof of the unirationality of the moduli space Mbe g of such curves, of the rationality of the Hurwitz scheme Hbe of bielliptic curves of even genus g, whereas, 4,g when g is odd, we construct a finite map C2g−2 → Mbe and compute its degree. g Mathematics Subject Classification (2000). 14H10, 14H45 Key words. curve – moduli – rationality
0. Introduction and notations Let C be a bielliptic curve of genus g ≥ 2k + ε ≥ 6, ε = 0, 1. This means that there exists a degree two morphism ϕ : C → E onto an elliptic curve E. Since g ≥ 6 by the Castelnuovo–Severi inequality (see for instance [1], p. 21) such a morphism is unique. The element i C ∈ Aut(C), which exchanges the two points of each fibre of ϕ, is called the elliptic involution on C and we have E ∼ = C/ i C . We notice that C is trivially tetragonal. In fact it suffices to compose ϕ with any map σ : E → PC1 of degree 2 in order to get a g41 on C. Again, by the Castelnuovo–Severi inequality, it follows that C cannot be hyperelliptic, neither trigonal and that every g41 on C is composed with the elliptic involution. Let : C → PC1 be the degree four cover corresponding to a g41 on C. It is known (see [7]) that factors through an embedding i : C → P into a PC2 -bundle whose fibres over PC1 are generated by the divisors in the g41 , followed by the natural projection π : P → PC1 . The main result of the present paper is the extention, developed in Sect. 2 (see formulas (2.2.1), (2.2.2), (2.2.3) and Theorem 2.3) and based on the techniques G. Casnati: Dipartimento di Matematica, Politecnico di Torino, c.so Duca degli Abruzzi 24, 10129 Torino, Italy, e-mail:
[email protected] A. Del Centina: Dipartimento di Matematica, Università degli Studi di Ferrara, via Machiavelli 35, 44100 Ferrara, Italy, e-mail:
[email protected] Both the authors have been partially supported by MURST in the framework of the national project “Geometria Algebrica, Algebra Commutativa ed Aspetti Computazionali”.
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of [7], of the description of the equations of i(C) inside P given in [8] and [9] to the cases 6 ≤ g ≤ 9. As a by-product of our study we get a proof of the unirationality of the moduli space Mbe g of bielliptic curves of genus g ≥ 6 (see Corollary 2.5), the rationality of the Hurwitz scheme Hbe 4,g of bielliptic curves of even genus g (see Corollary 3.3.2); whereas, when g is odd, we construct a finite map C2g−2 → Mbe g and compute its degree (see Proposition 3.4.2). The question whether Mbe g is rational when g ≥ 6, is open (for the cases g = 3, 4, 5 see [3], [2], [6]). Notations. As usual we denote by O X and ω X|C the structure sheaf and the canonical sheaf of the irreducible, smooth, projective variety X. If F is a locally free O X -sheaf and s ∈ H 0 X, F , we denote by D0 (s) the subscheme of X locally defined by the vanishing of s i.e. set-theoretically D0 (s) := {x ∈ X| s(x) = 0}. ˇ If E is any O X -sheaf then we denote its dual Hom O X E , O X by E. If g is an element of a certain group G then g denotes the subgroup of G generated by g. If X ⊆ PCn is any subscheme then X denotes the smallest subspace S ⊆ PCn containing X. We denote isomorphisms by ∼ = and birational equivalences by ≈. For all the other notations and definition we always refer to [10].
1. General facts about tetragonal curves We begin by summarizing some results about covers of P1C . Recall that if C is a smooth curve, then a morphism : C → P1C is a cover of degree 4 if it is quasi-finite of degree 4. We refer to [7] for results about covers of degree 4. Let : C → P1C be a cover of degree 4. There exists a natural exact sequence #
0 → OP1 −→ ∗ OC → Eˇ → 0, C
where Eˇ is a locally free OP1 -sheaf of rank 3 called the Tschirhausen module C of . The sequence above splits (see [7]) and we obtain a decomposition ∗ OC ∼ = ˇ In particular, for each n ∈ Z, OP1 ⊕ E. C
(1.2)
ˇ h i C, ∗ OP1 (n) = h i P1C , OP1 (n) + h i P1C , E(n) . C
C
∼ ωC|C ⊗∗ O 1 (2) is defined and Moreover, the relative dualizing sheaf ωC|P1 = PC C invertible. By relative duality ∗ ωC|P1 ∼ = ∗ OC ˇ ∼ = OP1 ⊕ E (see [10], exercise C C III 6.10 b)). Theorem 1.3. Let C be a smooth, integral curve and : C → P1C a cover of degree 4.
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There exists an embedding i : C → P := P(E ) (Eˇ is the Tschirnhausen module of ) such that ωC|P1 ∼ = i ∗ OP (1) and = π ◦ i, π : P → P1C being the natural C projection. There exists a locally free OP1 -sheaf F of rank 2 such that det F ∼ = det E , and C fitting into an exact sequence of the form (1.3.1)
δ
0 → π ∗ det E (−4) → π ∗ F (−2) −→ OP → O X → 0.
Sequence (1.3.1) is unique up to isomorphism. The restriction to P y := π −1 (y) ∼ = PC2 of Sequence (1.3.1) is a minimal free resolution of the structure sheaf of C y := −1 (y). In particular the ideal of C y ⊆ P y is the complete intersection of two conics, hence C y is not contained in any line r ∈ Pˇ y .
Proof. See [7, Theorem 2.1]. Twisting Sequence (1.3.1) by OP (2) and applying π∗ we obtain (1.4)
0 → F → S 2 E → ∗ ω2C|P1 → 0. C
Since every locally free OP1 -sheaf splits into a direct sum of invertible sheaves C we can state the following: Definition 1.5. Let : C → P1C be a cover of degree 4, and fix decompositions 3 2 E∼ = i=1 OP1 (αi ), α1 ≤ α2 ≤ α3 , F ∼ = j=1 OP1 (β j ), β1 ≤ β2 . C C The triple α := (α1 , α2 , α3 ) and the pair β := (β1 , β2 ) are called the scrollar invariants of the cover . Remark 1.6. Let : C → P1C be a cover of degree 4 having α and β as scrollar invariants. If C is smooth and irreducible then α1 ≥ 1 (see [11, Proposition 1.2]). Conversely if α1 ≥ 1, then C is connected (Formula (1.2) with i = n = 0), whence, if C is smooth, then it is also irreducible. We have three monomorphisms OP1 (αi ) E into the three summands, C hence three fibrewise independent sections u ∈ H 0 P, OP (1) ⊗ π ∗ OP1 (−α1 ) ∼ = C 0 1 0 ∗ 0 1 ∼ H PC , E (−α1 ) , v ∈ H P, OP (1) ⊗ π OP1 (−α2 ) = H PC , E (−α2 ) , w ∈ C H 0 P, OP (1) ⊗ π ∗ OP1 (−α3 ) ∼ = H 0 P1C , E (−α3 ) . C Notice that δ ∈ HomOP π ∗ F (−2), OP ∼ = H 0 P, OP (2) ⊗ π ∗ Fˇ . Via the isomorphism H 0 P, OP (2) ⊗ π ∗ Fˇ ∼ = H 0 P, OP (2) ⊗ π ∗ OP1 (−β1 ) C ⊕ H 0 P, OP (2) ⊗ π ∗ OP1 (−β2 ) , C
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we can identify δ with a pair (a, b), where a(u, v, w) = a2α1 −β1 u 2 + 2aα1 +α2 −β1 uv + 2aα1 +α3 −β1 uw (1.7)
+ a2α2 −β1 v2 + 2aα2 +α3 −β1 vw + a2α3 −β1 w2 , b(u, v, w) = b2α1 −β2 u 2 + 2bα1 +α2 −β2 uv + 2bα1 +α3 −β2 uw + b2α2 −β2 v2 + 2bα2 +α3 −β2 vw + b2α3 −β2 w2 ,
ah , bh ∈ H 0 P1C , OP1 (h) . In particular C = D0 (a, b). C Conversely we have the following result:
Theorem 1.8. Let E , F be as above and define U := (a, b) ∈ H 0 P, OP (2) ⊗ π ∗ Fˇ | D0 (a, b) is a smooth curve . Then U is open. Proof. See [7, Theorem 4.5] and [5, Theorem 2.5].
2. General representation of bielliptic curves In the following, C will always denote a fixed bielliptic curve of genus g = 2k + ε, where ε = 0, 1. We remark that if C is general, then Aut(C) = i C (see [4]). In [8] and [9] the following proposition is proved: Proposition 2.1. Let : C → P1C be a cover of degree 4, where C is bielliptic of genus g ≥ 6. Then β = (4, g − 1) and i) if g = 2k we have α = (2, k, k + 1) if g = 2k, ii) if g = 2k + 1 we have either α = (2, k + 1, k + 1) (the general case) or α = (2, k, k + 2) (the special case). In [8] it is also proved that the converse of Proposition 2.1 is true under the stronger condition g ≥ 10. The aim of this section is to find a converse of the above proposition which holds without any restriction. Remark 2.2. Assume that C has genus g ≥ 6 (which implies k ≥ 3). Let α and β be the scrollar invariants of any cover : C → P1C of degree 4 (as in Proposition 2.1). π As explained in the previous section, C = D0 (a, b) inside P := P(E ) −→ PC1 , where a and b are as in Equations (1.7). More precisely, if g = 2k (2.2.1) a(u, v,w) := a0 u 2 + 2ak−2 uv + 2ak−1 uw + a2k−4 v2 + 2a2k−3 vw + a2k−2 w2 , b(u, v, w) := 2b3−k uv + 2b4−k uw + b1 v2 + 2b2 vw + b3 w2 ; if g = 2k + 1 and is general (in the sense of Proposition 2.1), (2.2.2) a(u, v, w) := a0 u 2 + 2ak−1 uv + 2ak−1 uw + a2k−2 v2 + 2a2k−2 vw + a2k−2 w2 , 2 b(u, v, w) := 2b3−k uv + 2b3−k uw + b2 v2 + 2b2 vw + b 2w ;
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if g = 2k + 1 and is special (in the sense of Proposition 2.1), (2.2.3) a(u, v, w) := a0 u 2 + 2ak−2 uv + 2ak uw + a2k−4 v2 + 2a2k−2 vw + a2k w2 , b(u, v, w) := 2b4−k uw + b0 v2 + 2b2 vw + b4 w2 . In the equations above, ah. , b.h ∈ H 0 P1C , OP1 (h) . C
be as above. Choose (a, b) ∈ H 0 (P, Theorem 2.3. Let g ≥ 6 and π : P → OP (2) ⊗ π ∗ Fˇ ) as in Remark 2.2 and define C := D0 (a, b), := π|C . Assume that is a cover and that C is a smooth curve. Then C is a bielliptic curve of genus g if and only if P1C
(2.3.1)
b3−k = b4−k = 0, if g = 2k, b3−k = b3−k = 0, if g = 2k + 1 ( is general), b4−k = 0, if g = 2k + 1 ( is special),
Proof. We study only the case g = 2k; the other ones being similar. Since α1 = 2 then C is irreducible (see Remark 1.6). Let A := D0 (a), B := D0 (b), Z := D0 (v, w), U := D0 (u) ⊆ P. If E0 := OP1 (k) ⊕ OP1 (k + 1) and C C π0 P0 := P(E0 ) −→ P1C then U ∼ = P0 , in particular U is smooth. If b3−k = b4−k = 0 then a0 = 0, otherwise Z ⊆ C and C would be reducible. Then one easily checks that Z ∩ C = ∅, thus the projection from Z, f : P \ Z → U, induces a double cover from ϕ : C → E := B ∩ U. If e ∈ E is singular then B would be singular along the line r := ϕ−1 (e) . Since r ∩ C = ∅ then C = A ∩ B should be singular too. We then conclude that E is smooth. On the other hand, π0|E has degree 2 and it is branched exactly along D0 (b1 b3 − b22 ) ⊆ P1C , hence E is elliptic and so we obtain that C is bielliptic. Conversely let C be bielliptic. The elliptic involution i C allows us to build
ϕ−1 (e) ⊆ P, T := e∈E
which is a conic-bundle whose fibres are all degenerate. Let OP (T ) ∼ = OP (2) ⊗ π ∗ OP1 (−n). Z := Sing(T ) is a curve which cannot intersect X, thus X · Z = 0. C Notice that X ∈ |2U|, hence we get U · Z = 0. It follows that each general divisor in |U| intersects T along a curve isomorphic to E. On the other hand, adjunction formula on P yields 0 = 2 pa (E) − 2 = 2(g − n − 1), thus n = g − 1 and T ∈ |B|. Since B is the unique element of |B| containing C then T = B. The discriminant ∆ of B must then be zero. If b3−k = 0 then k = 3, thus b3−k would be a non-zero constant. With a suitable linear transformation on the variables u, v, w we can assume b4−k = 0, hence ∆ = b3−k b23 . Thus necessarily b3 = 0, then B should be reducible and the same should be true for C. We conclude that b3−k = 0, then ∆ = b1 b24−k . If b4−k = 0 then b1 = 0 and again we would get the reducibility of C.
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Fix E , hence g ≥ 6, let U := (a, b) ∈ H 0 P, OP (2) ⊗ π ∗ Fˇ | b satisfies conditions (2.3.1) , and let U0 ⊆ U be the subset containing sections (a, b) such that D0 (a, b) is a smooth curve. Proposition 2.4. U0 is an open and dense subset of U. Proof. In any case, Theorem 1.8 says that U0 is open inside U. Hence we have to prove only that U0 = ∅. For each g ≥ 3 it is easy to build bielliptic curves of genus g. Indeed it suffices to choose any elliptic curve E and consider the cover ϕ : C → E defined by an invertible O E -sheaf L of degree g − 1 and a general section s ∈ H 0 E, L2 . This proves the statement if g is even. If g is odd, say g = 2k + 1, let σ : E → P1C be any cover of degree 2. Then in [8] and [9] it is proved that := σ ◦ ϕ is special (general, respectively) if and only if L ∼ = σ ∗ OP1 (k) (L ∼ = σ ∗ OP1 (k), respectively). This C C completes the proof when g is odd. As a by-product of Propositions 2.3 and 2.4, we obtain the existence of a dominant rational map U Mbe g . Thus the following corollary holds:
Corollary 2.5. Mbe g is unirational. 3. The action of SL 2 (Aut(E) × Aut(F )) on U0 Now we study the action on U0 of the algebraic group H := SL 2 (Aut(E ) × Aut(F )) .
Definition 3.1. Two covers : C → P1C and : C → P1C are said to be isomorphic if there exists a commutative diagram
C −→ ϑ
P1C α
C −→ P1C , for some suitable α ∈ SL 2 and isomorphism ϑ : C → C . Let (a, b), (a , b ) ∈ U0 sections giving rise to isomorphic covers : C → P1C and : C → P1C . Since the Tschirnhausen modules of and α ◦ coincide with E we then obtain an automorphism β : E → E . Finally the diagram 0 → F → S 2 E → ∗ ω2C|P1 → 0 C 2 S β 0 → F → S 2 E → ∗ ω2C |P1 → 0 C
yields an automorphism γ : F → F . Thus if (a, b) and (a , b ) define isomorphic covers there is h ∈ H such that h(a, b) = (a , b ). We can summarize the above description in the following:
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Lemma 3.2. Let U0 and H be as above. H acts on U0 and the orbit with respect to this action of the section (a, b) ∈ H 0 P, OP (2) ⊗ π ∗ Fˇ corresponding to a cover contains all the sections (a , b ) ∈ H 0 P, OP (2) ⊗ π ∗ Fˇ defining covers isomorphic to . 3.3. The Hurwitz scheme of bielliptic curves of even genus with a fixed g41 . Assume that g = 2k for some k ≥ 3 and let E := OP1 (2) ⊕ OP1 (k) ⊕ OP1 (k + 1). C C C Up to H we can assume that b1 = b0 x 1 (see Formula (2.2.1)). This shows that the H-orbits of V0 := {(a, b) ∈ U0 | b = b0 x 1 } , are dense in U0 . Moreover, let SL2,x1 be the stabilizer of [1, 0] ∈ P1C inside SL2 , and define K := SL2,x1 (Aut(E ) × Aut(F )) ⊆ H. If h ∈ H, (a, b) ∈ V0 and h(a, b) ∈ V0 then it is not difficult to check that h ∈ K ; thus: Theorem 3.3.1. U0 /H ≈ V0 /K is rational. Proof. The above description shows that V0 is a relative section of U0 in the sense of Sect. 2.8 of [12]. Since H is a semidirect product of triangular groups then it is connected and solvable, thus the rationality of V0 /K follows from Theorem 2.11 in [12]. Let H4,g be the Hurwitz scheme of tetragonal curves of genus g. We consider D0 (a, b) ⊆ P × U0 . D := (a,b)∈U0
D → U0 is a flat family since the Hilbert polynomial of D0 (a, b) inside P is constant for each (a, b) ∈ U0 . It follows the existence of a rational map u : U0 H4,g whose fibres are exactly the H-orbits. We set Hbe 4,g := im(u) ⊆ H4,g : it is irreducible and it can be regarded as the Hurwitz scheme of bielliptic curves of genus g. Corollary 3.3.2. If g is even then Hbe 4,g is rational. Proof. By the above description it follows that Hbe 4,g ≈ U0 /H which is rational by Theorem 3.3.1. be Remark 3.3.3. We have a natural forgetful morphism Hbe 4,g → Mg whose fibre over the point [C] is isomorphic to C/ i C .
3.4. A finite map C2g−2 → Mbe g for g odd. Assume g = 2k + 1, where k ≥ 3. Let us consider the factorization = σ ◦ ϕ, where ϕ : C → E is induced by the bielliptic involution i C and σ : E → P1C is a cover of degree 2. As explained in
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the proof of Proposition 2.5, the cover ϕ is induced by an O E -sheaf L of degree g − 1 and by a section s ∈ H 0 E, L2 . The cover is special if and only if L∼ = σ ∗ OP1 (k), general otherwise. C Let E := OP1 (2) ⊕ OP1 (k) ⊕ OP1 (k + 2) (the special case: see ProposiC C C tion 2.1 ii)). Let V := {(a, b) ∈ U| ak−2 = ak = a2k−4 = b2 = 0} , and V0 := V ∩ U0 . If we denote by TF and TE the tori of diagonal matrices of Aut(F ) and Aut(E ), respectively, and we set K := SL 2 × TF × TE ⊆ H then, as in the previous example, V is a (H, K )-section of U. Theorem 3.4.1. U0 /H ≈ V0 /K is rational. Proof. The statement follows from a Theorem of Katsylo (see [12, Sect. 2.9]). Now there exists the forgetful map p : V0 → Mbe g sending the section (a, b) ∈ V0 1 be representing : C → PC , to [C] ∈ Mg ⊆ Mg . By Lemma 3.2 above such a map is H-equivariant, thus we get a rational map p : C2g−2 Mbe g . Proposition 3.4.2. p is dominant and finite of degree deg( p) = k 2 . Proof. We prove that p is surjective. Indeed let [C] ∈ Mbe g and let ϕ : C → E be the corresponding coverwhich is induced by an O E -sheaf L of degree g − 1 = 2k and by a section s ∈ H 0 E, L2 . There is an effective divisor D ∈ Div(E) such that L ∼ = O E (D), thus there is a point P ∈ E such that 2k P ∈ |D| (see [10, Exercise IV 4.6]). In particular, we can consider the linear system |2P| and the corresponding morphism σ : E → P1C . Then := σ ◦ ϕ : C → P1C is special, thus it corresponds to a section (a, b) ∈ V0 , hence p(a, b) = [C]. Moreover, all special covers arise in this way. If [C] is general then ϕ, hence O E (2k P), is uniquely determined. In particular, we have (2k)2 possible different choices for P which, four by four, give rise to isomorphic sheaves O E (2P) (which define σ : E → PC1 ). In particular, we have k 2 non-isomorphic covers over [C], i.e. p is generically finite of degree k 2 . Corollary 2.5 together with the results of [2], [3], [6] pose the question of the possible rationality of Mbe g when g ≥ 6. This question is still open and, in the light of the above results, it seems to be rather difficult. References 1. Accola, R.: Topics in the theory of Riemann surfaces. L.N.M. 1595. Springer 1994 2. Bardelli, F., Del Centina, A.: The moduli space of genus four double covers of elliptic curves is rational. Pacific J. Math. 144, 219–227 (1990) 3. Bardelli, F., Del Centina, A.: Bielliptic curves of genus three: canonical models and moduli space. Indag. Math. (N.S.) 10, 183–190 (1999) 4. Cornalba, M.: On the locus of curves with automorphisms. Ann. Mat. Pura Appl. CIL, 135–151 (1987)
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5. Casnati, G.: Covers of algebraic varieties III. The discriminant of a cover of degree 4 and the trigonal construction. Trans. Amer. Math. Soc. 350, 1359–1378 (1998) 6. Casnati, G., Del Centina, A.: The rationality of the moduli space of bielliptic curves of genus five. Bull. London Math. Soc. 28, 356–362 (1996) 7. Casnati, G., Ekedahl, T.: Covers of algebraic varieties I. A general structure theorem, covers of degree 3, 4 and Enriques surfaces. J. Algebraic Geom. 5, 439–460 (1996) 8. Del Centina, A., Gimigliano, A.: Scrollar invariants and resolutions of certain d-gonal curves. Ann. Univ. Ferrara – Sez. VII – Sc. Mat. XXXIX, 187–201 (1993) 9. Del Centina, A., Gimigliano, A.: Corrigenda to “Scrollar invariants and resolutions of certain d-gonal curves”. Ann. Univ. Ferrara – Sez. VII – Sc. Mat. XLI, 147 (1995) 10. Hartshorne, R.: Algebraic geometry. New York: Springer 1977 11. Lazarsfeld, R.: A Barth–type theorem for branched coverings of projective space. Math. Ann. 249, 53–58 (1980) 12. Popov, V.L., Vinberg, E.B.: Invariant theory. In: Parshin, A.N., Shafarevich, I.R. (eds.): Algebraic Geometry IV. Encyclopedia of Mathematical Sciences 55 (1991)
