Global Prym-Torelli theorem for double coverings of elliptic curves
aa r X i v : . [ m a t h . AG ] D ec GLOBAL PRYM-TORELLI THEOREM FOR DOUBLECOVERINGS OF ELLIPTIC CURVES
ATSUSHI IKEDA
Abstract.
The Prym variety for a branched double covering of a nonsingularprojective curve is defined as a polarized abelian variety. We prove that anydouble covering of an elliptic curve which has more than 4 branch points isrecovered from its Prym variety. Introduction
Let C and C ′ be nonsingular projective curves, and let φ : C → C ′ be a doublecovering branched at 2 n points. In [14] the Prym variety P ( φ ) for the doublecovering φ is defined as a polarized abelian variety of dimension d = g ′ − n ,where g ′ is the genus of C ′ . Let R = R g ′ , n be the moduli space of such coverings,and let A = A d be the moduli space of polarized abelian varieties of dimension d .Then the construction of the Prym variety defines the Prym map P : R → A , andthe Prym-Torelli problem asks whether the Prym map is injective. If g ′ = 0, thenit is injective by the classical Torelli theorem for hyperelliptic curves. We considerthe case g ′ > R ≤ dim A , where we note that dim R = 3 g ′ − n and dim A = ( g ′ − n )( g ′ + n )2 . The generically injectivity for the Prym map has beenproved in most cases. Theorem 1.1.
The Prym map is generically injective in the following cases; (1) (Friedman and Smith [7], Kanev [9]) n = 0 and dim R < dim A , (2) (Marcucci and Pirola [11]) g ′ > , n > and dim R < dim A − , (3) (Naranjo and Ortega [17]) g ′ > , n > and dim R = dim A − , (4) (Marcucci and Naranjo [10]) g ′ = 1 , n > and dim R ≤ dim A . The Prym varieties for unramified coverings have been intensively studied be-cause they are principally polarized abelian varieties. For ramified coverings, Na-garaj and Ramanan [15] proved the above Theorem 1.1 (2) for n = 2, and thenMarcucci and Pirola [11] proved it for any n >
0. When g ′ > R = dim A ,there are only two cases ( g ′ , n ) = (6 , , (3 , g ′ , n ) = (6 ,
0) then the Prym mapis generically finite of degree 27 ([6]), and if ( g ′ , n ) = (3 ,
2) then it is genericallyfinite of degree 3 ([15], [4]).Although the Prym map is not injective for many cases in Theorem 1.1 ([5], [16],[15], [18]), we prove the injectivity when g ′ = 1. The following is the main resultof this paper, which improves Theorem 1.1 (4). Mathematics Subject Classification.
Theorem 1.2 (Theorem 3.1) . If g ′ = 1 , n > and dim R ≤ dim A , then thePrym map is injective. To prove this theorem we use the Gauss map for the polarization divisor, whichis a standard approach to Torelli problems. Let L be an ample invertible sheafwhich represents the polarization of the Prym variety P = P ( φ ). For a member D ∈ |L| , we consider the Gauss mapΨ D : D \ D sing −→ P d − = Grass ( d − , H ( P, Ω P ) ∨ ) . It is not difficult to show that there exists a member D ∈ |L| such that thebranch divisor of Ψ D recovers the original covering φ : C → C ′ in a similar wayas Andreotti’s proof [1] of Torelli theorem for hyperelliptic curves. The essentialpart of our proof is to distinguish the special member D ∈ |L| . We study therestriction Ψ D | Bs |L| : Bs |L| \ D sing → P d − of the Gauss map to the base locus ofthe linear system |L| . Although Ψ D is difficult to compute, the restriction Ψ D | Bs |L| is rather simple for any member D ∈ |L| . By using the image of Ψ D | Bs |L| andthe branch divisor of Ψ D | Bs |L| , we can specify the member D ∈ |L| which has thedesired property.In Section 2, we summarize some basic properties of bielliptic curves and theirPrym varieties. In Section 3, we explain the strategy of the proof of Theorem 1.2 byusing the key Propositions in Section 6. In Section 4, we explicitly describe the baselocus of the linear system of polarization divisors. In Section 5, we show that therestricted Gauss map Ψ D | Bs |L| is the same map as the restriction of the Gauss mapfor the theta divisor on Jacobian variety of C . By giving a simple description forΨ D | Bs |L| , we prove some properties on the branch divisor of Ψ D | Bs |L| . In Section 6,we present key Propositions, which are consequences of the results in Section 5.In this paper, we work over an algebraically closed field k of characteristic = 2.2. Properties of bielliptic curves and Prym varieties
Let C be a nonsingular projective curve of genus g over k , and let σ be aninvolution on C . In this paper, we call the pair ( C, σ ) a bielliptic curve of genus g ,if g > E = C/σ is a nonsingular curve of genus 1. We denoteby φ : C → E the quotient morphism. First we note the following. Lemma 2.1 ([16] (3.3)) . Let ( C, σ ) be a bielliptic curve of genus g . If g > , then C is not a hyperelliptic curve. Let N : J ( C ) → J ( E ) be the norm map of φ , which is a homomorphism on theirJacobian varieties. Lemma 2.2 ([14]) . Let φ : C → E be the covering defined from a bielliptic curve ( C, σ ) . (1) φ ∗ : Pic ( E ) → Pic ( C ) is injective. (2) The kernel P of the norm map N : J ( C ) → J ( E ) is reduced and connected. RYM-TORELLI THEOREM FOR DOUBLE COVERINGS OF ELLIPTIC CURVES 3
By Lemma 2.2, the kernel P of the norm map N is an abelian variety of dimension n = g −
1. Let P ∨ be the dual abelian variety of P , and let λ P : P → P ∨ be thepolarization isogeny which is defined as the restriction of the principal polarizationon the Jacobian variety J ( C ). Then the polarized abelian variety ( P, λ P ) is calledthe Prym variety for the covering φ : C → E . We denote by K ( P ) ⊂ P the kernelof the polarization λ P : P → P ∨ . An ample invertible sheaf L on P represents thethe polarization λ P , if the polarization isogeny λ P is given by λ P : P ( k ) −→ P ∨ ( k ) = Pic ( P ); x t ∗ x L ⊗ L ∨ , where t x : P → P denotes the translation by x ∈ P ( k ). Lemma 2.3 ([14]) . Let ( P, λ P ) be the Prym variety defined from a bielliptic curve ( C, σ ) , and let L be an ample invertible sheaf which represents λ P . (1) K ( P ) = φ ∗ J ( E ) ⊂ J ( C ) , where J ( E ) denotes the set of points of order on J ( E ) . (2) deg λ P = 4 and h ( P, L ) = 2 . Proof of Main theorem
The main result of this paper is the following.
Theorem 3.1. If g > , then the isomorphism class of a bielliptic curve of genus g is determined by the isomorphism class of its Prym variety. Let (
P, λ P ) be the Prym variety of dimension n ≥ C, σ ) of genus g = n + 1. We will recover the data ( E, e + · · · + e n , η )from the polarized abelian variety ( P, λ P ), where E = C/σ is the quotient curve, e + · · · + e n is the branch divisor of the covering φ : C → E , and η ∈ Pic ( E )is the invertible sheaf with φ ∗ η ∼ = Ω C . We remark that η ⊗ ∼ = O E ( e + · · · + e n ),and η is the invertible sheaf which determines the double covering with the branchdivisor e + · · · + e n . Proof of Theorem 3.1.
Let L be an ample invertible sheaf on P which representsthe the polarization λ P . We denote by K ( P ) the Kernel of λ P : P → P ∨ . ByLemma 2.3, we have ♯K ( P ) = 4 and h ( P, L ) = 2. We define the subset Π L in thelinear pencil |L| byΠ L = { D ∈ |L| | t x ( D ) = D ⊂ P for some x ∈ K ( P ) \ { }} , where t x is the translation by x ∈ P ( k ). By Lemma 4.8, Π L is a set of 6 membersfor any representative L of the polarization λ P . For a member D ∈ |L| \ Π L , weconsider the Gauss mapΨ D : D \ D sing −→ P n − = Grass ( n − , H ( P, Ω P ) ∨ ) , where Ψ D ( x ) is defined by the inclusion T x ( D ) ⊂ T x ( P ) ∼ = H ( P, Ω P ) ∨ of thetangent spaces at the point x ∈ D \ D sing ⊂ P . We set U D = Bs |L| \ D sing ,where Bs |L| ⊂ P denotes the set of base points of the pencil |L| . Let X ′ D =Ψ D ( U D ) ⊂ P n − be the Zariski closure of Ψ D ( U D ) ⊂ P n − , and let ν D : X D → ATSUSHI IKEDA X ′ D be the normalization. By Lemma 6.1, U D is a nonsingular variety, hencethere is a unique morphism ψ D : U D → X D such that Ψ D | U D = ν D ◦ ψ D . Weconsider the closed subset Z D = ψ D (Ram ( ψ D )) ⊂ X D , where Ram ( ψ D ) ⊂ U D denotes the ramification divisor of ψ D . By Proposition 6.2, Z D has a canonicaldecomposition Z D = S ni =1 Z D,i , and there is a unique hyperplane H D,i ⊂ P n − suchthat ν D ( Z D,i ) ⊂ H D,i for any 1 ≤ i ≤ n . Then the effective divisor ν ∗ D H D,i − Z D,i on X D has 2 irreducible components for general D ∈ |L|\ Π L , and these componentscoincide for special D ∈ |L| \ Π L . We define the subset Π ′L in the linear pencil |L| by Π ′L = { D ∈ |L| \ Π L | ν ∗ D H D,i − Z D,i is irreducible for 1 ≤ i ≤ n } . By Lemma 6.3, Π ′L is a set of 4 members for any representative L of the polarization λ P . For a member D ∈ Π ′L , we consider the dual variety ( X ′ D ) ∨ ⊂ ( P n − ) ∨ of X ′ D ⊂ P n − and the dual variety H ∨ D,i ⊂ ( P n − ) ∨ of H D,i ⊂ P n − . By Proposition 6.4, H ∨ D,i is a point on ( X ′ D ) ∨ , and we have an isomorphism( E, e + · · · + e n , η ) ∼ = (( X ′ D ) ∨ , H ∨ D, + · · · + H ∨ D, n , O ( P n − ) ∨ (1) | ( X ′ D ) ∨ ) . (cid:3) Pencil of polarization divisors
Let (
C, σ ) be a bielliptic curve of genus g = n + 1 >
3. For δ ∈ Pic n ( C ), we setthe divisor W δ ⊂ J ( C ) by W δ ( k ) = { L ∈ Pic ( C ) = J ( C )( k ) | h ( C, L ⊗ δ ) > } . We remark that the singular locus of W δ is given by W δ, sing ( k ) = { L ∈ Pic ( C ) | h ( C, L ⊗ δ ) > } , and dim W δ, sing = n − C is not a hyperelliptic curveby Lemma 2.1. Let λ C : J ( C ) → J ( C ) ∨ be the homomorphism defined by λ C : J ( C )( k ) → J ( C ) ∨ ( k ) = Pic ( J ( C )); x [ t ∗ x O C ( W δ ) ⊗ O C ( − W δ )] , which does not depend on the choice of δ ∈ Pic n ( C ). Let ι q : C → J ( C ) be themorphism defined by ι q : C ( k ) −→ Pic ( C ) = J ( C )( k ); q ′ [ O C ( q ′ − q )] . for q ∈ C ( k ). Lemma 4.1. x = ι ∗ q [ O J ( C ) ( W δ − W δ + x )] ∈ Pic ( C ) for any q ∈ C ( k ) and x ∈ Pic ( C ) .Proof. The statement means that ( − ◦ λ C is the inverse of the homomorphism ι ∗ q : J ( C ) ∨ → J ( C ) defined by the pull-buck ι ∗ q : Pic ( J ( C )) → Pic ( C ) of invertiblesheaves. It is well-known ([13, Lemma 6.9]). (cid:3) RYM-TORELLI THEOREM FOR DOUBLE COVERINGS OF ELLIPTIC CURVES 5
Let P be the kernel of the Norm map N : J ( C ) → J ( E ), and let D δ ⊂ P thefiber of the restriction of the norm map N | W δ : W δ → J ( E ) at 0 ∈ J ( E ). Wedenote by L δ = O P ( D δ ) = O J ( C ) ( W δ ) | P the restriction of O J ( C ) ( W δ ) to P . Since W δ is the theta divisor of J ( C ), the ample invertible sheaf L δ represents the thepolarization λ P . Lemma 4.2. D δ + φ ∗ s ⊂ P is a member of the linear system |L δ | for any s ∈ Pic ( E ) .Proof. By Lemma 4.1, φ ∗ s = ι ∗ q [ O J ( C ) ( W δ − W δ + φ ∗ s )] ∈ Pic ( C )for s ∈ Pic ( E ) and q ∈ C ( k ). We set s ′ ∈ Pic ( J ( E )) by s = ι ∗ φ ( q ) s ′ , where ι φ ( q ) : E ∼ → J ( E ) is the isomorphism determined by ι φ ( q ) ( φ ( q )) = 0. Then we have N ∗ s ′ = [ O J ( C ) ( W δ − W δ + φ ∗ s )] ∈ Pic ( J ( C )) , because φ ∗ s = ι ∗ q N ∗ s ′ and ι ∗ q : Pic ( J ( C )) → Pic ( C ) is an isomorphism. Since( N ∗ s ′ ) | P = 0 ∈ Pic ( P ), we have O P ( D δ + φ ∗ s ) ∼ = O J ( C ) ( W δ + φ ∗ s ) | P ∼ = O J ( C ) ( W δ ) | P = L δ . (cid:3) We denote by C ( i ) the i -th symmetric products of C . For δ ∈ Pic n ( C ), we definethe morphism β iδ : C ( n − i ) × E ( i ) → J ( C ) by β iδ : C ( n − i ) ( k ) × E ( i ) ( k ) −→ J ( C )( k ) = Pic ( C );( q + · · · + q n − i , p + · · · + p i ) C (cid:16) n − i X j =1 q j (cid:17) ⊗ φ ∗ O E (cid:16) i X j =1 p j (cid:17) ⊗ δ ∨ . We remark that W δ = Image ( β δ ), and we set B iδ = ( Image ( β iδ ) (1 ≤ i ≤ n ) , ∅ (2 i > n ) . Lemma 4.3. B δ \ W δ, sing = ∅ and B δ ⊂ W δ, sing .Proof. Let B be the image of the morphism β : C ( n − × E → C ( n ) defined by β : C ( n − ( k ) × E ( k ) → C ( n ) ( k ); ( q + · · · + q n − , φ ( q )) q + · · · + q n − + q + σ ( q ) . Since C is not a hyperelliptic curve, we have dim ( β δ ) − ( W δ, sing ) = n − < n − B , hence B δ = β δ ( B ) * W δ, sing .To prove the second statement, we assume that n ≥
4, because B δ = ∅ for n = 3.Let F ⊂ C ( n − × E (2) be the fiber of the composition C ( n − × E (2) β δ −→ J ( C ) N −→ J ( E )at η − N ( δ ) ∈ J ( E )( k ), where η ∈ Pic n ( E ) denotes the unique invertible sheaf on E with φ ∗ η ∼ = Ω C . We set U = ( C ( n − × E (2) ) \ ( F ∪ ( C ( n − × ∆ E )), where ∆ E ⊂ E (2) ATSUSHI IKEDA denotes the image of the diagonal in E × E . For y = ( q + · · · + q n − , φ ( q ) + φ ( r )) ∈ U ( k ), there are points q ′ , r ′ ∈ C ( k ) such that ( O E ( φ ( q ′ )) ∼ = η ⊗ O E ( − φ ( q ) − · · · − φ ( q n − ) − φ ( q ) − φ ( r )) , O E ( φ ( r ′ )) ∼ = η ⊗ O E ( − φ ( q ) − · · · − φ ( q n − ) − φ ( q ) − φ ( r )) . Then we have Ω C ( − q − · · · − q n − − q − σ ( q ) − r − σ ( r )) ∼ = φ ∗ η ⊗ O C ( − q − · · · − q n − − q − σ ( q ) − r − σ ( r )) ∼ = O C ( σ ( q ) + · · · + σ ( q n − ) + q ′ + σ ( q ′ ) + r + σ ( r )) ∼ = O C ( σ ( q ) + · · · + σ ( q n − ) + q + σ ( q ) + r ′ + σ ( r ′ )) . We remark that φ ( q ) = φ ( q ′ ) and φ ( q ) = φ ( r ), because y / ∈ F and y / ∈ C ( n − × ∆ E .Hence we have h ( C, Ω C ( − q − · · · − q n − − q − σ ( q ) − r − σ ( r ))) > β δ ( y ) ∈ W δ, sing . Since U ⊂ ( β δ ) − ( W δ, sing ) is a dense subset of C ( n − × E (2) , we have C ( n − × E (2) = ( β δ ) − ( W δ, sing ). (cid:3) Lemma 4.4.
For s ∈ Pic ( E ) , D δ = D δ + φ ∗ s ⊂ P if and only if s = 0 or s = η − N ( δ ) .Proof. For L ∈ P ( k ) ⊂ Pic ( C ), we have 0 = φ ∗ N ( L ) = L + σ ∗ L ∈ Pic ( C ).Hence we have D δ = D δ + φ ∗ ( η − N ( δ )) , because L ∈ D δ ( k ) ⇐⇒ h ( C, L ⊗ δ ) > ⇐⇒ h ( C, σ ∗ L ⊗ σ ∗ δ ) > ⇐⇒ h ( C, L ∨ ⊗ σ ∗ δ ) > ⇐⇒ h ( C, Ω C ⊗ L ⊗ σ ∗ δ ∨ ) > ⇐⇒ L ∈ D [Ω C ] − σ ∗ δ ( k ) = D δ + φ ∗ ( η − N ( δ )) ( k ) . We assume that D δ = D δ + φ ∗ s for s = 0 ∈ Pic ( E ). Let α δ + φ ∗ s : C ( n − × C → J ( C )be the morphism defined by α δ + φ ∗ s : C ( n − ( k ) × C ( k ) −→ Pic ( C ) = J ( C )( k );( q + · · · + q n − , q ) C ( q + · · · + q n − + 2 q ) ⊗ δ ∨ ⊗ φ ∗ s ∨ . Then the set D δ \ ( W δ, sing ∪ B δ ∪ Image ( α δ + φ ∗ s )) is not empty, becausedim D δ ∩ ( W δ, sing ∪ B δ ∪ Image ( α δ + φ ∗ s )) < n − D δ . For L ∈ D δ ( k ) \ ( W δ, sing ∪ B δ ∪ Image ( α δ + φ ∗ s )), there is r + · · · + r n ∈ C ( n ) ( k ) suchthat L ⊗ δ ⊗ φ ∗ s ∼ = O C ( r + · · · + r n ), because L ∈ D δ ( k ) = D δ + φ ∗ s ( k ) ⊂ W δ + φ ∗ s ( k ).Since L ∈ W δ ( k ) \ W δ, sing ( k ), we have h ( C, Ω C ⊗ L ∨ ⊗ δ ∨ ) = h ( C, L ⊗ δ ) = 1. Let q + · · · + q n ∈ C ( n ) ( k ) and q ′ + · · · + q ′ n ∈ C ( n ) ( k ) be the effective divisors definedby L ⊗ δ ∼ = O C ( q + · · · + q n ) , Ω C ⊗ L ∨ ⊗ δ ∨ ∼ = O C ( q ′ + · · · + q ′ n ) . Let φ ( u i ) ∈ E ( k ) be the point determined by s = [ O E ( φ ( r i ) − φ ( u i ))]. Then L ⊗ δ ⊗ O C ( σ ( r i )) ∼ = O C ( r + · · · + r n − r i + u i + σ ( u i )) . If σ ( r i ) / ∈ { q ′ , . . . , q ′ n } , then h ( C, L ⊗ δ ⊗ O C ( σ ( r i ))) = h ( C, Ω C ⊗ L ∨ ⊗ δ ∨ ⊗ O C ( − σ ( r i ))) + 1 = 1 , RYM-TORELLI THEOREM FOR DOUBLE COVERINGS OF ELLIPTIC CURVES 7 hence q + · · · + q n + σ ( r i ) = r + · · · + r n − r i + u i + σ ( u i ) . Since s = 0, we have σ ( r i ) = r j for some j = i , and L ∈ B δ ( k ). It is a contradictionto L / ∈ B δ ( k ), hence σ ( r i ) ∈ { q ′ , . . . , q ′ n } for any 1 ≤ i ≤ n . Here the condition L / ∈ Image ( α δ + φ ∗ s ) implies that ♯ { r , . . . , r n } = n and L ∨ ⊗ σ ∗ δ ⊗ φ ∗ s ∼ = O C ( σ ( r ) + · · · + σ ( r n )) = O C ( q ′ + · · · + q ′ n ) ∼ = Ω C ⊗ L ∨ ⊗ δ ∨ . Hence we have φ ∗ s = [Ω C ] − δ − σ ∗ δ = φ ∗ ( η − N ( δ )), and s = η − N ( δ ) byLemma 2.2. (cid:3) Let B δ ⊂ J ( C ) be the subset B δ = \ s ∈ Pic ( E ) W δ + φ ∗ s . Lemma 4.5. B δ \ W δ, sing = B δ \ W δ, sing .Proof. If L ∈ B δ ( k ), then L ⊗ δ ∼ = O C ( q + · · · + q n − + q + σ ( q )) for some q , . . . , q n − , q ∈ C ( k ). For s ∈ Pic ( E ), there is a point q ′ ∈ C ( k ) such that s = [ O E ( φ ( q ′ ) − φ ( q ))]. Since L ⊗ δ ⊗ φ ∗ s ∼ = O C ( q + · · · + q n − + q ′ + σ ( q ′ )), wehave h ( C, L ⊗ δ ⊗ φ ∗ s ) > L ∈ W δ + φ ∗ s ( k ). Hence the inclusion B δ ⊂ B δ holds.For L ∈ B δ ( k ) \ W δ, sing ( k ), there is a unique r + · · · + r n ∈ C ( n ) ( k ) such thatΩ C ⊗ L ∨ ⊗ δ ∨ ∼ = O C ( r + · · · + r n ), because h ( C, Ω C ⊗ L ∨ ⊗ δ ∨ ) = h ( C, L ⊗ δ ) = 1.Let Σ ⊂ Pic ( E ) be the finite subset defined byΣ = { s ∈ Pic ( E ) | L ⊗ φ ∗ s ∈ β δ (∆ ( n ) ) } , where ∆ = { σ ( r ) , . . . , σ ( r n ) } ⊂ C and ∆ ( n ) ⊂ C ( n ) . For s ∈ Pic ( E ) \ (Σ ∪ { } ),there is a divisor q + · · · + q n ∈ C ( n ) ( k ) such that L ⊗ δ ⊗ φ ∗ s ∼ = O C ( q + · · · + q n ),because L ∈ B δ ( k ) ⊂ W δ + φ ∗ s ( k ). Since s / ∈ Σ, we may assume that q n / ∈ ∆. Thecondition σ ( q n ) / ∈ { r , . . . , r n } implies that h ( C, Ω C ⊗ L ∨ ⊗ δ ∨ ⊗ O C ( − σ ( q n ))) = 0and h ( C, L ⊗ δ ⊗ O C ( σ ( q n ))) = 1. Let φ ( q ′ ) ∈ E ( k ) be the point determined by s = [ O E ( φ ( q n ) − φ ( q ′ ))]. Then L ⊗ δ ⊗ O C ( σ ( q n )) ∼ = O C ( q + · · · + q n − + q ′ + σ ( q ′ )) . Since s = 0, we have σ ( q n ) ∈ { q , . . . , q n − } and L ∈ B δ ( k ). (cid:3) Lemma 4.6.
The map
Pic ( E ) −→ |L δ | ; s D δ + φ ∗ s is a double covering, and the base locus Bs |L δ | of the linear system |L δ | is B δ ∩ P ,which is of dimension n − .Proof. The map is well-defined by Lemma 4.2. Since dim |L δ | = 1, it is a doublecovering by Lemma 4.4. Hence we haveBs |L δ | = \ s ∈ Pic ( E ) D δ + φ ∗ s = B δ ∩ P. ATSUSHI IKEDA
By Lemma 4.3, B δ is irreducible of dimension n −
1. Since the restriction of theNorm map N | B δ : B δ → J ( E ) is surjective, we have dim B δ ∩ P = n −
2, hencedim B δ ∩ P = n − (cid:3) Lemma 4.7.
Let L be an ample invertible sheaf which represents the polarization λ P on P , then there is δ ∈ Pic n ( C ) such that N ( δ ) = η and L ∼ = L δ .Proof. For any δ ′ ∈ Pic n ( C ), we have L⊗L ∨ δ ′ ∈ Pic ( P ), because L δ ′ gives the samepolarization as λ P . Then L ∼ = t ∗ x L δ ′ ∼ = L δ ′ + x for some x ∈ P ( k ). Let s ∈ Pic ( E )be a point with 2 s = η − N ( δ ′ + x ). For δ = δ ′ + x + φ ∗ s , we have N ( δ ) = η and L ∼ = L δ (cid:3) For an ample invertible sheaf L which represents the polarization λ P , we set asubset in the linear system |L| byΠ L = { D ∈ |L| | t x ( D ) = D for some x ∈ K ( P ) \ { }} , where K ( P ) is the kernel of the polarization λ P . Lemma 4.8. ♯ Π L = 6 .Proof. By Lemma 4.7, there is δ ∈ Pic n ( C ) such that N ( δ ) = η and L ∼ = L δ .For any D ∈ |L δ | , by Lemma 4.6, there is s ∈ Pic ( E ) such that D = D δ + φ ∗ s . If D δ + φ ∗ s ∈ Π L δ , then by Lemma 2.3, there is t ∈ J ( E ) \{ } such that t φ ∗ t ( D δ + φ ∗ s ) = D δ + φ ∗ s . Since t φ ∗ t ( D δ + φ ∗ s ) = D δ + φ ∗ ( s − t ) and t = 0, by Lemma 4.4, we have δ + φ ∗ ( s − t ) = δ + φ ∗ s + φ ∗ ( η − N ( δ + φ ∗ s )) = δ − φ ∗ s, hence t = 2 s by Lemma 2.2. It means thatΠ L δ = { D δ + φ ∗ s ∈ |L δ | | s ∈ J ( E ) \ J ( E ) } . Since ♯ ( J ( E ) \ J ( E ) ) = 12 and D δ + φ ∗ s = D δ − φ ∗ s , we have ♯ Π L = 6. (cid:3) Gauss maps
Gauss map for Jacobian and Gauss map for Prym.
LetΨ J ( C ) ,δ : W δ \ W δ, sing −→ P ( H ( C, Ω C ) ∨ ) = Grass ( n, H ( C, Ω C ) ∨ )be the Gauss map for the subvariety W δ ⊂ J ( C ). For L ∈ W δ ( k ) \ W δ, sing ( k ),the tangent space T L ( W δ ) of W δ at L defines the image Ψ J ( C ) ,δ ( L ) by the naturalidentifications T L ( W δ ) ⊂ T L ( J ( C )) ∼ = (Ω J ( C ) ( L )) ∨ ∼ = H ( J ( C ) , Ω J ( C ) ) ∨ ∼ = H ( C, Ω C ) ∨ . Lemma 5.1.
For L ∈ W δ ( k ) \ W δ, sing ( k ) , the image Ψ J ( C ) ,δ ( L ) of the Gauss mapis identified with the canonical divisor q + · · · + q n + q ′ + · · · + q ′ n ∈ | Ω C | = Grass (1 , H ( C, Ω C )) ∼ = P ( H ( C, Ω C ) ∨ ) , where the effective divisors q + · · · + q n and q ′ + · · · + q ′ n are uniquely determinedby L ⊗ δ ∼ = O C ( q + · · · + q n ) and Ω C ⊗ L ∨ ⊗ δ ∨ ∼ = O C ( q ′ + · · · + q ′ n ) .Proof. It is a special case of Proposition (4.2) in [3, Chapter IV]. (cid:3)
RYM-TORELLI THEOREM FOR DOUBLE COVERINGS OF ELLIPTIC CURVES 9
Lemma 5.2.
Let K ∈ | Ω C | be an effective canonical divisor. If q + σ ( q ) ≤ K forsome q ∈ C ( k ) , then K = P ni =1 ( q i + σ ( q i )) for some q , . . . , q n ∈ C ( k ) .Proof. When K = m X i =1 ( q i + σ ( q i )) + q + n − m − X j =1 r j for 1 ≤ m ≤ n −
1, we show that σ ( q ) ∈ { r , . . . , r n − m − } . First we assume that1 ≤ m ≤ n −
2. Since C is not a hyperelliptic curve, by Clifford’s theorem, wehave m +2 > h ( C, O C ( m X i =1 ( q i + σ ( q i ))+ q + σ ( q ))) ≥ h ( E, O E ( m X i =1 φ ( q i )+ φ ( q ))) = m +1and m + 1 > h ( C, O C ( m X i =1 ( q i + σ ( q i )))) ≥ h ( E, O E ( m X i =1 φ ( q i ))) = m, hence h ( C, O C ( P mi =1 ( q i + σ ( q i )) + q + σ ( q ))) = m + 1 and h ( C, O C ( P mi =1 ( q i + σ ( q i )))) = m . Since σ ( q ) is not a base point of |O C ( P mi =1 ( q i + σ ( q i )) + q + σ ( q )) | = φ ∗ |O E ( P mi =1 φ ( q i ) + φ ( q )) | , we have m = h ( C, O C ( m X i =1 ( q i + σ ( q i )) + q )) < h ( C, O C ( m X i =1 ( q i + σ ( q i )) + q + σ ( q ))) = m + 1 , hence h ( C, O C ( n − m − X j =1 r j − σ ( q ))) = h ( C, O C ( n − m − X j =1 r j )) = n − m − . It implies that σ ( q ) ≤ P n − m − j =1 r j . We consider the case m = n −
1. Let η ∈ Pic n ( E ) be the invertible sheaf with φ ∗ η ∼ = Ω C . There is a point q ′ ∈ C ( k ) suchthat P n − j =1 φ ( q i ) + φ ( q ′ ) ∈ | η | . Then O C ( q + r ) ∼ = O C ( q ′ + σ ( q ′ )). Since C is not ahyperelliptic curve, we have q + r = q ′ + σ ( q ′ ) and σ ( q ) = r . (cid:3) By the injective homomorphism H ( E, η ) −→ H ( C, φ ∗ η ) ∼ = H ( C, Ω C ) , we have the closed immersion ι : P ( H ( E, η ) ∨ ) −→ P ( H ( C, Ω C ) ∨ ) . Lemma 5.3.
For L ∈ B δ ( k ) \ W δ, sing ( k ) , Ψ J ( C ) ,δ ( L ) ∈ ι ( P ( H ( E, η ) ∨ )) ⊂ P ( H ( C, Ω C ) ∨ ) . Proof.
For L ∈ B δ ( k ) \ W δ, sing ( k ), by Lemma 5.1, the image Ψ J ( C ) ,δ ( L ) of the Gaussmap is given by q + · · · + q n + q ′ + · · · + q ′ n ∈ | Ω C | ∼ = P ( H ( C, Ω C ) ∨ ) , where the effective divisors q + · · · + q n and q ′ + · · · + q ′ n are uniquely determinedby L ⊗ δ ∼ = O C ( q + · · · + q n ) and Ω C ⊗ L ∨ ⊗ δ ∨ ∼ = O C ( q ′ + · · · + q ′ n ). Since L ∈ B δ ( k ), by Lemma 4.5, σ ( q i ) = q j for some i = j . By Lemma 5.2, we haveΨ J ( C ) ,δ ( L ) ∈ ι ( P ( H ( E, η ) ∨ )). (cid:3) Let Ψ
P,δ : D δ \ D δ, sing → P ( H ( P, Ω P ) ∨ ) = Grass ( n − , H ( P, Ω P ) ∨ )be the Gauss map for the subvariety D δ ⊂ P . For L ∈ D δ ( k ) \ D δ, sing ( k ), the tangentspace T L ( D δ ) of D δ at L defines the image Ψ P,δ ( L ) by the natural identifications T L ( D δ ) ⊂ T L ( P ) ∼ = (Ω P ( L )) ∨ ∼ = H ( P, Ω P ) ∨ . For L ∈ P ( k ), the tangent space T L ( P ) of P at L is naturally identified with theorthogonal subspace V P = ( φ ∗ H ( E, Ω E )) ⊥ ⊂ H ( C, Ω C ) ∨ ∼ = T L ( J ( C ))to φ ∗ H ( E, Ω E ) ⊂ H ( C, Ω C ), and it corresponds to the ramification divisorRam ( φ ) ∈ | Ω C | of the covering φ : C → E . We define the finite set Σ δ byΣ δ = { β δ ( r + · · · + r n ) ∈ J ( C ) | r + · · · + r n ≤ Ram ( φ ) } . Lemma 5.4. D δ, sing = ( W δ, sing ∪ Σ δ ) ∩ D δ .Proof. If L ∈ D δ ( k ) ∩ W δ, sing ( k ), then L ∈ D δ, sing ( k ). If L ∈ D δ ( k ) \ W δ, sing ( k ),then by Lemma 5.1, L ∈ D δ, sing ( k ) ⇐⇒ T L ( W δ ) = T L ( P ) ⊂ T L ( J ( C )) ⇐⇒ L ∈ Σ δ . (cid:3) Lemma 5.5. ( B δ ∩ P ) \ W δ, sing = ( B δ ∩ P ) \ D δ, sing .Proof. By Lemma 4.5, ( B δ \ W δ, sing ) ∩ Σ δ = ( B δ \ W δ, sing ) ∩ Σ δ , and it is emptybecause Ram ( φ ) is reduced. Hence by Lemma 5.4, W δ, sing ∩ B δ ∩ P = W δ, sing ∩ B δ ∩ D δ = D δ, sing ∩ B δ = D δ, sing ∩ B δ ∩ P . (cid:3)
We denote by π : P ( H ( C, Ω C ) ∨ ) \ { V P } −→ P ( H ( P, Ω P ) ∨ );[ V ⊂ H ( C, Ω C ) ∨ ] [ V ∩ V P ⊂ V P ∼ = H ( P, Ω P ) ∨ ]the projection, where V P = ( φ ∗ H ( E, Ω E )) ⊥ ⊂ H ( C, Ω C ) ∨ is the image of thedual of the restriction H ( C, Ω C ) ∼ = H ( J ( C ) , Ω J ( C ) ) ։ H ( P, Ω P ) . Lemma 5.6. Ψ P,δ ( L ) = π ◦ Ψ J ( C ) ,δ ( L ) for L ∈ D δ ( k ) \ D δ, sing ( k ) . RYM-TORELLI THEOREM FOR DOUBLE COVERINGS OF ELLIPTIC CURVES 11
Proof.
For L ∈ D δ ( k ) \ D δ, sing ( k ), the tangent spaces at L satisfies T L ( P ) ∩ T L ( W δ ) = T L ( D δ ) ⊂ T L ( J ( C )) , because P ∩ W δ = D δ ⊂ J ( C ). Since T L ( P ) ⊂ T L ( J ( C )) is identified with V P ⊂ H ( C, Ω C ) ∨ by T L ( J ( C )) ∼ = H ( C, Ω C ) ∨ , we have Ψ P,δ ( L ) = π ◦ Ψ J ( C ) ,δ ( L ). (cid:3) By Lemma 5.3, we have the morphismΨ BJ ( C ) ,δ : B δ \ W δ, sing −→ P ( H ( E, η ) ∨ )satisfying ι ◦ Ψ BJ ( C ) ,δ = Ψ J ( C ) ,δ . Lemma 5.7.
The restriction Ψ P,δ | ( B δ ∩ P ) \ D δ, sing of the Gauss map Ψ P,δ is identifiedwith the restriction Ψ BJ ( C ) ,δ | ( B δ ∩ P ) \ D δ, sing of Ψ BJ ( C ) ,δ by the isomorphism π ◦ ι : P ( H ( E, η ) ∨ ) ∼ −→ P ( H ( P, Ω P ) ∨ ) . Proof.
Since the composition H ( E, η ) ֒ → H ( C, Ω C ) ∼ = H ( J ( C ) , Ω J ( C ) ) ։ H ( P, Ω P ) , is an isomorphism, it is a consequence of Lemma 5.3 and Lemma 5.6. (cid:3) Description for the restricted Gauss maps.
Let γ δ : E ( n − × E → J ( E )be the morphism defined by γ δ : E ( n − ( k ) × E ( k ) −→ Pic ( E );( p + · · · + p n − , p ) E ( p + · · · + p n − + 2 p ) ⊗ ( N ( δ )) ∨ . Let X δ ⊂ E ( n − × E be the fiber of γ δ at 0 ∈ J ( E ), and let Y δ ⊂ C ( n − × E bethe fiber of the composition C ( n − × E φ ( n − × id E −→ E ( n − × E γ δ −→ J ( E )at 0 ∈ J ( E ). We denote by ψ δ : Y δ → X δ the induced morphism by φ ( n − × id E .Let ν δ : X δ → | η | ∼ = P ( H ( E, η ) ∨ ) be the morphism defined by ν δ : X δ ( k ) → | η | ; ( p + · · · + p n − , p ) p + · · · + p n − + p + t η − N ( δ ) ( p ) , where t η − N ( δ ) ( p ) ∈ E ( k ) is the point determined by[ O E ( t η − N ( δ ) ( p ))] = [ O E ( p )] + η − N ( δ ) ∈ Pic ( E ) . We remark that β δ ( Y δ ) = B δ ∩ P ⊂ J ( C ), and we set Y ◦ δ = ( β δ ) − (( B δ ∩ P ) \ D δ, sing ) = ( β δ ) − (( B δ ∩ P ) \ D δ, sing ) . Lemma 5.8.
The diagram Y ◦ δ β δ −→ ( B δ ∩ P ) \ D δ, sing ψ δ ↓ ↓ Ψ BJ ( C ) ,δ X δ −→ ν δ P ( H ( E, η ) ∨ ) is commutative. Proof.
Let L ∈ Pic ( C ) be the invertible sheaf which represents the point β δ ( y ) ∈ J ( C ) for y = ( q + · · · + q n − , φ ( q )) ∈ Y ◦ δ ( k ). Then q + · · · + q n − + q + σ ( q ) ∈ C ( n ) ( k )is the unique effective divisor with L ∼ = O C ( q + · · · + q n − + q + σ ( q )) ⊗ δ ∨ . Since ν δ ◦ ψ δ ( y ) = φ ( q ) + · · · + φ ( q n − ) + φ ( q ) + t η − N ( δ ) ( φ ( q )) ∈ | η | , we have q + σ ( q ) + · · · + q n − + σ ( q n − ) + q + σ ( q ) + r + σ ( r ) ∈ | Ω C | , where r ∈ C ( k ) is given by φ ( r ) = t η − N ( δ ) ( φ ( q )). Then σ ( q ) + · · · + σ ( q n − ) + r + σ ( r ) ∈ C ( n ) ( k ) is the unique effective divisor with O C ( σ ( q ) + · · · + σ ( q n − ) + r + σ ( r )) ∼ = Ω C ⊗ L ∨ ⊗ δ ∨ , and by Lemma 5.1, Φ BJ ( C ) ,δ ( L ) is equal to ν δ ◦ ψ δ ( y ), (cid:3) Lemma 5.9. X δ and Y δ are nonsingular projective varieties.Proof. We fix a point p ∈ E ( k ). Let γ δ,p : E ( n − → J ( E ) be the morphismdefined by γ δ,p : E ( n − ( k ) −→ Pic ( E ) = J ( E )( k ); p + · · · + p n − E ( p + · · · + p n − + 2 p ) ⊗ ( N ( δ )) ∨ . Then ψ δ : Y δ → X δ is the base change of φ ( n − : C ( n − → E ( n − by the ´etalecovering of degree 4; Y δ ψ δ −→ X δ pr −→ E pr ↓ (cid:3) ↓ pr (cid:3) ↓ ( − J ( E ) ◦ ι p C ( n − −→ φ ( n − E ( n − −→ γ δ,p J ( E ) , where ( − J ( E ) ◦ ι p : E → J ( E ) is given by( − J ( E ) ◦ ι p : E ( k ) −→ Pic ( E ) = J ( E )( k ); p E (2 p − p ) . (cid:3) Lemma 5.10. β δ | Y ◦ δ : Y ◦ δ → ( B δ ∩ P ) \ D δ, sing is an isomorphism. In particular, ( B δ ∩ P ) \ D δ, sing is a nonsingular variety.Proof. By Lemma 5.5, the image β δ ( Y ◦ δ ) = ( B δ ∩ P ) \ D δ, sing is a closed subset in W δ \ W δ, sing . We show that β δ | Y ◦ δ : Y ◦ δ → W δ \ W δ, sing is a closed immersion. Let β : C ( n − × E → C ( n ) be the morphism given in the proof of Lemma 4.3. Since β δ = β δ ◦ β and β δ : C ( n ) → J ( C ) induces the isomorphism β δ : C ( n ) \ ( β δ ) − ( W δ, sing ) ∼ = −→ W δ \ W δ, sing , it is enough to show that the finite morphism β : ( C ( n − × E ) \ ( β δ ) − ( W δ, sing ) −→ C ( n ) \ ( β δ ) − ( W δ, sing )is a closed immersion. We remark that it is injective by Lemma 4.3. For y =( q + · · · + q n − , φ ( q )) ∈ ( C ( n − × E ) \ ( β δ ) − ( W δ, sing ), we prove that the homo-morphism T y ( C ( n − × E ) −→ T β ( y ) ( C ( n ) ) RYM-TORELLI THEOREM FOR DOUBLE COVERINGS OF ELLIPTIC CURVES 13 on the tangent spaces is injective. If φ ( q ) / ∈ { φ ( q ) , . . . , φ ( q n − ) } , then the point y ′ = ( q + · · · + q n − , q + σ ( q )) ∈ C ( n − ( k ) × C (2) ( k ) is not contained in the ram-ification divisor of the natural covering C ( n − × C (2) → C ( n ) . Since the morphism E ( k ) → C (2) ( k ); φ ( q ) q + σ ( q )is a closed immersion, the homomorphism T y ( C ( n − × E ) ֒ → T y ′ ( C ( n − × C (2) ) ∼ = T β ( y ) ( C ( n ) )is injective. We consider the case when y = ( q + · · · + q n − − i + iq , φ ( q )) and q / ∈ { q , . . . , q n − − i } for some i ≥
1. First we assume that σ ( q ) = q . Thenby Lemma 4.3, we have i = 1. The point ˜ y = ( q + · · · + q n − , q , φ ( q )) ∈ C ( n − × C × E is not contained in the ramification divisor of the natural cov-ering C ( n − × C × E → C ( n − × E , and the point ˜ y ′ = ( q + · · · + q n − , q ) ∈ C ( n − × C (3) is not contained in the ramification divisor of the natural covering C ( n − × C (3) → C ( n ) . Since the morphism C ( k ) × E ( k ) → C (3) ( k ); ( q ′ , φ ( q )) q ′ + q + σ ( q )is a closed immersion, the homomorphism T y ( C ( n − × E ) ∼ = T ˜ y ( C ( n − × C × E ) ֒ → T ˜ y ′ ( C ( n − × C (3) ) ∼ = T β ( y ) ( C ( n ) )is injective. We assume that σ ( q ) = q . Then by Lemma 4.3, we have σ ( q ) / ∈{ q , . . . , q n − − i } . The point ˜ y = ( q + · · · + q n − − i + iq , q ) ∈ C ( n − × C is notcontained in the ramification divisor of the covering id C ( n − × φ : C ( n − × C → C ( n − × E , and the point ˜ y ′ = ( q + · · · + q n − − i + ( i + 1) q , σ ( q )) ∈ C ( n − × C isnot contained in the ramification divisor of the natural covering C ( n − × C → C ( n ) .Since the morphism C ( n − ( k ) × C ( k ) −→ C ( n − ( k ) × C ( k );( q ′ + · · · + q ′ n − , q ) ( q ′ + · · · + q ′ n − + q, σ ( q ))is a closed immersion, the homomorphism T y ( C ( n − × E ) ∼ = T ˜ y ( C ( n − × C ) ֒ → T ˜ y ′ ( C ( n − × C ) ∼ = T β ( y ) ( C ( n ) )is injective. (cid:3) Let X ′ δ = Ψ BJ ( C ) ,δ (( B δ ∩ P ) \ D δ, sing ) be the Zariski closure of the image of therestricted Gauss map Ψ BJ ( C ) ,δ | ( B δ ∩ P ) \ D δ, sing in P ( H ( E, η ) ∨ ). Lemma 5.11. ν δ ( X δ ) = X ′ δ .Proof. By Lemma 5.8, we haveΨ BJ ( C ) ,δ (( B δ ∩ P ) \ D δ, sing ) = ν δ ( ψ δ ( Y ◦ δ )) ⊂ ν δ ( X δ ) , hence X ′ δ = ν δ ( ψ δ ( Y ◦ δ )) ⊂ ν δ ( X δ ) and Y ◦ δ ⊂ ( ν δ ◦ ψ δ ) − ( X ′ δ ). Since Y ◦ δ is dense in Y δ , we have Y δ ⊂ ( ν δ ◦ ψ δ ) − ( X ′ δ ) and ν δ ( X δ ) = ( ν δ ◦ ψ δ )( Y δ ) ⊂ X ′ δ . (cid:3) Lemma 5.12. If N ( δ ) − η / ∈ J ( E ) \ { } , then ν δ : X δ → X ′ δ is the normalizationof X ′ δ . Proof.
We set morphisms α ± δ : E ( n − × E → E ( n ) , µ δ : E ( n − × E → E ( n ) and ν δ : E ( n − × E (2) → E ( n ) by α + δ : E ( n − ( k ) × E ( k ) −→ E ( n ) ( k );( p + · · · + p n − , p ) p + · · · + p n − + 2 p + t η − N ( δ ) ( p ) ,α − δ : E ( n − ( k ) × E ( k ) −→ E ( n ) ( k );( p + · · · + p n − , p ) p + · · · + p n − + 2 p + t N ( δ ) − η ( p ) ,µ δ : E ( n − ( k ) × E ( k ) −→ E ( n ) ( k );( p + · · · + p n − , p ) p + · · · + p n − + p + t η − N ( δ ) ( p ) + t N ( δ ) − η ( p )and ν δ : E ( n − ( k ) × E (2) ( k ) −→ E ( n ) ( k );( p + · · · + p n − , p + p ′ ) p + · · · + p n − + p + t η − N ( δ ) ( p ) + p ′ + t η − N ( δ ) ( p ′ ) . By the natural inclusion X ′ δ ⊂ | η | ⊂ E ( n ) , the subset U = X ′ δ \ (Image ( α + δ ) ∪ Image ( α − δ ) ∪ Image ( µ δ ) ∪ Image ( ν δ )) , is open dense in X ′ δ , where we consider as Image ( ν δ ) = ∅ if n = 3. We show thatthe morphism ν δ : ν − δ ( U ) −→ E ( n ) \ (Image ( α + δ ) ∪ Image ( α − δ ) ∪ Image ( µ δ ) ∪ Image ( ν δ ))is a closed immersion. For u = p + · · · + p n ∈ U ( k ), we assume that p + t η − N ( δ ) ( p ) ≤ p + · · · + p n and p ′ + t η − N ( δ ) ( p ′ ) ≤ p + · · · + p n for some p = p ′ ∈ E ( k ). Since u / ∈ Image ( ν δ ), we have t η − N ( δ ) ( p ) = p ′ or p = t η − N ( δ ) ( p ′ ) , and furthermore u / ∈ Image ( µ δ ) implies that t η − N ( δ ) ( p ) = p ′ and p = t η − N ( δ ) ( p ′ ) , hence N ( δ ) − η ∈ J ( E ) \ { } . This means that ν δ : ν − δ ( U ) → U is bijective if N ( δ ) − η / ∈ J ( E ) \ { } . In the following, we prove that the homomorphism T x ( E ( n − × E ) −→ T ν δ ( x ) ( E ( n ) )on the tangent spaces is injective for x ∈ ν − δ ( U ). Let ˜ ν δ : E ( n − × E → E ( n − × E (2) be the morphism defined by˜ ν δ : E ( n − ( k ) × E ( k ) −→ E ( n − ( k ) × E (2) ( k );( p + · · · + p n − , p ) ( p + · · · + p n − , p + t η − N ( δ ) ( p )) . If N ( δ ) − η / ∈ J ( E ) \ { } , then the morphism ˜ ν δ is a closed immersion. For x ∈ ν − δ ( U ), the image ˜ ν δ ( x ) is not contained in the ramification divisor of thenatural covering E ( n − × E (2) → E ( n ) , because ν δ ( x ) / ∈ Image ( α + δ ) ∪ Image ( α − δ ).Hence the homomorphism T x ( E ( n − × E ) ֒ → T ˜ ν δ ( x ) ( E ( n ) × E (2) ) ∼ = T ν δ ( x ) ( E ( n ) ) RYM-TORELLI THEOREM FOR DOUBLE COVERINGS OF ELLIPTIC CURVES 15 is invective. By Lemma 5.9, the finite birational morphism ν δ : X δ → X ′ δ gives thenormalization of X ′ δ . (cid:3) Remark . If N ( δ ) − η ∈ J ( E ) \ { } , then ν δ : X δ → X ′ δ is a covering of degree2.5.3. The branch locus of the restricted Gauss maps.
Let R δ ⊂ Y δ be thedivisor defined by R δ ( k ) = { ( q + · · · + q n − , p ) ∈ Y δ ( k ) | p i = σ ( p j ) for some i = j } . Lemma 5.14. β δ ( R δ ) ⊂ W δ, sing .Proof. It is a consequence of Lemma 4.3, because β δ ( R δ ) ⊂ B δ . (cid:3) Let S δ,r ⊂ Y δ be the divisor defined by S δ,r ( k ) = { ( q + · · · + q n − , p ) ∈ Y δ ( k ) | q + · · · + q n − ≥ r } for r ∈ Ram ( φ ). Then the ramification divisor of ψ δ : Y δ → X δ isRam ( ψ δ ) = R δ ∪ [ r ∈ Ram ( φ ) S δ,r . Lemma 5.15. β δ ( S δ,r ) * W δ, sing , and moreover β δ ( S δ,r ) ∩ W δ, sing = ∅ if n = 3 .Proof. Let W δ,r ⊂ J ( C ) be the subvariety defined by W δ,r ( k ) = { L ∈ Pic ( C ) | h ( C, L ⊗ O C ( − r ) ⊗ δ ) > } , and let T δ,r ⊂ J ( C ) be the image of the morphism C ( n − ( k ) × E ( k ) −→ Pic ( C ) = J ( C )( k );( q + · · · + q n − , φ ( q )) C ( q + · · · + q n − + r + q + σ ( q )) ⊗ δ ∨ . Since C is not a hyperelliptic curve, by Martens’ theorem [12, Theorem 1], we havedim W δ,r ≤ n − T δ,r * W δ,r , hence dim T δ,r = n −
2. We remark that W δ,r = ∅ in the case when n = 3. Since β δ ( S δ,r ) = T δ,r ∩ P is the fiber of the composition T δ,r ⊂ J ( C ) N → J ( E )at 0 ∈ J ( E ), we have dim β δ ( S δ,r ) = n −
3. Let T δ, r ⊂ T δ,r be the image of themorphism C ( n − ( k ) × E ( k ) −→ Pic ( C ) = J ( C )( k );( q + · · · + q n − , φ ( q )) C ( q + · · · + q n − + 2 r + q + σ ( q )) ⊗ δ ∨ . Since 2 r = r + σ ( r ), we have T δ, r ⊂ B δ ⊂ W δ, sing by Lemma 4.3. For L ∈ ( T δ,r ( k ) ∩ W δ, sing ( k )) \ T δ, r ( k ), there is ( q + · · · + q n − , φ ( q )) ∈ C ( n − ( k ) × E ( k )such that L = O C ( q + · · · + q n − + r + q + σ ( q )) ⊗ δ ∨ and r / ∈ { q , . . . , q n − } . If q ′ + · · · + q ′ n − i + ir ∈ | Ω C ( − q − · · · − q n − − r − q − σ ( q )) | = | Ω C ⊗ ( L ⊗ δ ) ∨ | and r / ∈ { q ′ , . . . , q ′ n − i } , then by Lemma 5.2, the number i is odd. By the same way,any member in the linear system | Ω C ( − q ′ − · · · − q ′ n − i − ir ) | = | L ⊗ δ | has an odd multiplicity at r . It implies that h ( C, L ⊗ O C ( − r ) ⊗ δ ) = h ( C, L ⊗ δ ) >
1. Hencewe have T δ,r ∩ W δ, sing = T δ, r ∪ ( T δ,r ∩ W δ,r ) . When n = 3, it implies that T δ,r ∩ W δ, sing = ∅ . When n ≥ β δ ( S δ,r ) ∩ W δ, sing = ( T δ, r ∩ P ) ∪ ( β δ ( S δ,r ) ∩ W δ,r )is a proper closed subset of β δ ( S δ,r ) = T δ,r ∩ P , because dim ( T δ, r ∩ P ) ≤ n − W δ,r ≤ n − (cid:3) Let Z δ,r = ψ δ ( S δ,r ) be the image of S δ,r by ψ δ : Y δ → X δ . Then Z δ,r ( k ) = { ( p + · · · + p n − , p ) ∈ X δ | φ ( r ) ≤ p + · · · + p n − } . Lemma 5.16. If n ≥ , then Z δ,r is irreducible. If n = 3 , then X δ ∼ = E , and Z δ,r ⊂ X δ is a J ( X δ ) -orbit by the natural action of J ( X δ ) on the curve X δ ofgenus .Proof. Let γ δ,p : E ( n − → J ( E ) be the morphism given in the proof of Lemma 5.9for fixed p ∈ E ( k ), and let i r : E ( n − → E ( n − be the morphism defined by i r : E ( n − ( k ) −→ E ( n − ( k ); p + · · · + p n − p + · · · + p n − + φ ( r ) . If n ≥
4, then Z δ,r is a P n − -bundle over E by the base change Z δ,r −→ X δ pr −→ E ↓ (cid:3) ↓ pr (cid:3) ↓ ( − J ( E ) ◦ ι p E ( n − −→ i r E ( n − −→ γ δ,p J ( E )of the P n − -bundle γ δ,p ◦ i r : E ( n − → J ( E ), hence Z δ,r is irreducible. If n = 3,then pr : X δ → E is an isomorphism, and Z δ,r ∼ = { p ∈ E ( k ) | O E ( φ ( r ) + 2 p ) ∼ = N ( δ ) } is an orbit of J ( E ) -action. (cid:3) We denote by Ram ( ψ ◦ δ ) ⊂ Y ◦ δ the ramification divisor of ψ ◦ δ = ψ δ | Y ◦ δ : Y ◦ δ → X δ . Lemma 5.17. ψ ◦ δ (Ram ( ψ ◦ δ )) = [ r ∈ Ram ( φ ) Z δ,r . Proof.
Since Ram ( ψ δ ) = R δ ∪ S r ∈ Ram ( φ ) S δ,r , by Lemma 5.14, we have Ram ( ψ ◦ δ ) = S r ∈ Ram ( φ ) S δ,r ∩ Y ◦ δ . By Lemma 5.15, S δ,r ∩ Y ◦ δ = ∅ for n ≥
3, and S δ,r ∩ Y ◦ δ = S δ,r for n = 3. Since ψ δ is a finite morphism, ψ ◦ δ ( S δ,r ∩ Y ◦ δ ) is of dimension n −
3. ByLemma 5.16, we have ψ ◦ δ ( S δ,r ∩ Y ◦ δ ) = ψ δ ( S δ,r ) = Z δ,r . (cid:3) Let H r ⊂ P ( H ( E, η ) ∨ ) be the hyperplane corresponding to the subspace H ( E, η ⊗ O E ( − φ ( r ))) ⊂ H ( E, η ) . Lemma 5.18. H r is the unique hyperplane with the property ν δ ( Z δ,r ) ⊂ H r . RYM-TORELLI THEOREM FOR DOUBLE COVERINGS OF ELLIPTIC CURVES 17
Proof.
The inclusion ν δ ( Z δ,r ) ⊂ H r is obvious. We prove the uniqueness of thehyperplane H r . Let z = ( p + · · · + p n − + φ ( r ) , p ) ∈ Z δ,r ( k ) be satisfying p i = t η − N ( δ ) ( p j ) for i = j . We take a point p ′ ∈ E ( k ) \ { p } such that O E (2 p ′ ) ∼ = O E (2 p )and O E ( p ′ − p ) ≇ η ⊗ N ( δ ) ∨ . Then z ′ = ( p + · · · + p n − + φ ( r ) , p ′ ) is containedin Z δ,r ( k ), and ν δ ( z ) = ν δ ( z ′ ). It implies the uniqueness in the case when n = 3.When n ≥
4, we show that ν δ ( Z δ,r ) ⊂ H r is a non-linear hypersurface in H r . Let l ⊂ H r be the line containing the two points ν δ ( z ) , ν δ ( z ′ ) ∈ ν δ ( Z δ,r ). Then the line l ⊂ P ( H ( E, η ) ∨ ) corresponds to the linear pencil | η ( − p − · · · − p n − − φ ( r )) | ⊂ | η | ∼ = P ( H ( E, η ) ∨ ) . For a point p ∈ E ( k ), there is a unique point p ′ ∈ E ( k ) such that p + p ′ ∈| η ( − p − · · · − p n − − φ ( r )) | . If O E (2 p ) ≇ O E (2 p ), O E (2 p ′ ) ≇ O E (2 p ) and p , p ′ / ∈ { t η − N ( δ ) ( p ) , . . . , t η − N ( δ ) ( p n − ) , t N ( δ ) − η ( p ) , . . . , t N ( δ ) − η ( p n − ) } , then the point p + · · · + p n − + φ ( r ) + p + p ′ ∈ | η | on the line l is not containedin ν δ ( Z δ,r ). (cid:3) Lemma 5.19.
The pull-back of the divisor H r by ν δ : X δ → P ( H ( E, η ) ∨ ) is ν ∗ δ H r = Z δ,r + M δ,r + M ′ δ,r , where M δ,y and M ′ δ,y are irreducible divisors on X δ defined by M δ,r ( k ) = { ( p + · · · + p n − , p ) ∈ X δ ( k ) | p = φ ( r ) } ,M ′ δ,r ( k ) = { ( p + · · · + p n − , p ) ∈ X δ ( k ) | p = t N ( δ ) − η ( φ ( r )) } . Proof.
Let I r be an irreducible divisor on E ( n ) defined by I r ( k ) = { p + · · · + p n ∈ E ( n ) ( k ) | p + · · · + p n ≥ φ ( r ) } , and let Z r , M r , M ′ r be irreducible divisors on E ( n − × E defined by Z r ( k ) = { ( p + · · · + p n − , p ) ∈ E ( n − ( k ) × E ( k ) | p + · · · + p n − ≥ φ ( r ) } ,M r ( k ) = { ( p + · · · + p n − , p ) ∈ E ( n − ( k ) × E ( k ) | p = φ ( r ) } ,M ′ r ( k ) = { ( p + · · · + p n − , p ) ∈ E ( n − ( k ) × E ( k ) | p = t N ( δ ) − η ( φ ( r )) } . Then the pull-back of the divisor I r by the morphism E ( n − ( k ) × E ( k ) −→ E ( n ) ( k );( p + · · · + p n − , p ) p + · · · + p n − + p + t η − N ( δ ) ( p )is the divisor Z r + M r + M ′ r on E ( n − × E . Since the restriction of I r to | η | ⊂ E ( n ) is the divisor H r on P ( H ( E, η ) ∨ ) ∼ = | η | , the pull-back ν ∗ δ H r is the restriction of Z r + M r + M ′ r to X δ . (cid:3) Corollary 5.20. ν ∗ δ H r − Z δ,r is an irreducible divisor on X δ if and only if N ( δ ) = η . We consider the dual variety (Φ | η | ( E )) ∨ ⊂ P ( H ( E, η ) ∨ ) of the image of theclosed immersion Φ | η | : E → P ( H ( E, η )).
Lemma 5.21.
The projective curve Φ | η | ( E ) ⊂ P ( H ( E, η )) is reflexive. In partic-ular, Φ | η | ( E ) = ((Φ | η | ( E )) ∨ ) ∨ ⊂ P ( H ( E, η )) .Proof. If 1 ≤ i < n , then h ( E, η ⊗ O E ( − ip )) = n − i for any p ∈ E ( k ). If n = 3, then h ( E, η ⊗ O E ( − p )) = 0 for general p ∈ E ( k ). Hence h ( E, η ⊗O E ( − p )) > h ( E, η ⊗ O E ( − p )) for general p ∈ E ( k ). Then there is a hyperplane H ⊂ P ( H ( E, η )) which intersects Φ | η | ( E ) at Φ | η | ( p ) with the multiplicity 2. By[8, (3.5)], Φ | η | ( E ) ⊂ P ( H ( E, η )) is reflexive, because the characteristic of the basefield k is not equal to 2. (cid:3) Lemma 5.22. If N ( δ ) = η , then the dual variety of X ′ δ ⊂ P ( H ( E, η ) ∨ ) is Φ | η | ( E ) ⊂ P ( H ( E, η )) .Proof. By Lemma 5.21, we show that the dual variety of Φ | η | ( E ) is X ′ δ . For L ∈ ( B δ ∩ P ) \ D δ, sing ⊂ Pic ( C ), there is a unique effective divisor q + · · · + q n − + q + σ ( q ) ∈ C ( n ) ( k ) such that L ⊗ δ ∼ = O C ( q + · · · + q n − + q + σ ( q )) . Since L ∈ P ( k ), we have η = N ( δ ) = [ O E ( φ ( q ) + · · · + φ ( q n − ) + 2 φ ( q ))] , henceΩ C ⊗ L ∨ ⊗ δ ∨ ∼ = φ ∗ η ⊗ L ∨ ⊗ δ ∨ ∼ = O C ( σ ( q ) + · · · + σ ( q n − ) + σ ( q ) + q )and Ψ BJ ( C ) ,δ ( L ) ∈ P ( H ( E, η ) ∨ ) is defined by the effective divisor φ ( q ) + · · · + φ ( q n − ) + 2 φ ( q ) ∈ | η | ∼ = P ( H ( E, η ) ∨ ) . It means that the hyperplane in P ( H ( E, η )) corresponding Ψ BJ ( C ) ,δ ( L ) is tangentto the image Φ | η | ( E ). Hence we haveΨ BJ ( C ) ,δ (( B δ ∩ P ) \ D δ, sing ) ⊂ (Φ | η | ( E )) ∨ . Since (Φ | η | ( E )) ∨ and Ψ BJ ( C ) ,δ (( B δ ∩ P ) \ D δ, sing ) are irreducible hypersurfaces in P ( H ( E, η ) ∨ ), we have X ′ δ = (Φ | η | ( E )) ∨ . (cid:3) Key Propositions
Let L be an ample invertible sheaf on P which represents the the polarization λ P . Lemma 6.1. U D = Bs |L| \ D sing is nonsingular for any D ∈ |L| .Proof. Since D = D δ for some δ ∈ Pic n ( C ), it is a consequence of Lemma 4.6 andLemma 5.10. (cid:3) Let Ψ D : D \ D sing −→ P n − = Grass ( n − , H ( P, Ω P ) ∨ )be the Gauss map for D ∈ |L| , and let ν D : X D → X ′ D be the normalizationof X ′ D = Ψ D ( U D ) ⊂ P n − . Then by Lemma 6.1, there is a unique morphism RYM-TORELLI THEOREM FOR DOUBLE COVERINGS OF ELLIPTIC CURVES 19 ψ D : U D → X D such that Ψ D | U D = ν D ◦ ψ D . Let Z D = ψ D (Ram ( ψ D )) ⊂ X D bethe Zariski closure of the image of the ramification divisor of ψ D . Proposition 6.2.
Let D ⊂ P be a member in |L| \ Π L , where Π L ⊂ |L| is thesubset in Lemma 4.8. (1) If n = 3 , then X D is a nonsingular projective curve of genus , and Z D isa disjoint union of orbits Z D, , . . . , Z D, by the J ( X D ) -action. (2) If n ≥ , then Z D has n irreducible components Z D, , . . . , Z D, n . (3) For any subset Z D,i ⊂ Z D in (1) and (2) , there is a unique hyperplane H D,i ⊂ P n − such that ν D ( Z D,i ) ⊂ H D,i .Proof.
By Lemma 4.7, there is δ ∈ Pic n ( C ) such that N ( δ ) = η and L ∼ = L δ .By Lemma 4.6, there is s ∈ Pic ( E ) such that D = D δ + φ ∗ s . By the proof ofLemma 4.8, D / ∈ Π L implies s / ∈ J ( E ) \ J ( E ) . By Lemma 5.7, the Gaussmap Ψ D | U D : U D → P n − is identified with Ψ BJ ( C ) ,δ + φ ∗ s | U D : U D → P ( H ( E, η ) ∨ ).Since N ( δ + φ ∗ s ) − η = 2 s / ∈ J ( E ) \ { } , by Lemma 5.11 and Lemma 5.12,the normalization of X ′ δ + φ ∗ s = X ′ D is given by ν δ + φ ∗ s : X δ + φ ∗ s → X ′ δ + φ ∗ s , and byLemma 5.8 and Lemma 5.10, ψ D : U D → X D is identified with ψ ◦ δ + φ ∗ s : Y ◦ δ + φ ∗ s → X δ + φ ∗ s . Hence the statements (1), (2) and (3) are consequence of Lemma 5.16,Lemma 5.17 and Lemma 5.18. (cid:3) We define the subset Π ′L in the linear pencil |L| byΠ ′L = { D ∈ |L| \ Π L | ν ∗ D H D,i − Z D,i is irreducible for 1 ≤ i ≤ n } . Lemma 6.3. ♯ Π ′L = 4 .Proof. We use the same identification for Gauss maps as in the proof of Proposi-tion 6.2. Then by Corollary 5.20, D = D δ + φ ∗ s ∈ Π ′L ⇐⇒ N ( δ + φ ∗ s ) = η ⇐⇒ s ∈ J ( E ) , and by Lemma 4.4, we have ♯ Π ′L = ♯J ( C ) = 4. (cid:3) Let e + · · · + e n be the branch divisor of the original covering φ : C → E , andlet η ∈ Pic ( E ) be the invertible sheaf with φ ∗ η ∼ = Ω C . Proposition 6.4.
For any member D ∈ Π ′L , there is an isomorphism ( E, e + · · · + e n , η ) ∼ = (( X ′ D ) ∨ , H ∨ D, + · · · + H ∨ D, n , O ( P n − ) ∨ (1) | ( X ′ D ) ∨ ) , where H ∨ D,i ∈ ( P n − ) ∨ is the point corresponding to the hyperplane H D,i , and ( X ′ D ) ∨ ⊂ ( P n − ) ∨ is the dual variety of X ′ D ⊂ P n − .Proof. We use the same identification for Gauss maps as in the proof of Proposi-tion 6.2. When D ∈ Π ′L , we may assume that D = D δ and N ( δ ) = η by Corol-lary 5.20. Then the point H ∨ D,i is identified with the point H ∨ r = Φ | η | ( φ ( r )) for r ∈ Ram ( φ ), and ( X ′ D ) ∨ is identified with ( X ′ δ ) ∨ , which coincides with Φ | η | ( E ) ⊂ P ( H ( E, η )) by Lemma 5.22. (cid:3)
Remark . For a member D ∈ Π ′L the Gauss map Ψ D : D \ D sing → P n − is ofdegree 2 n , and X ′ D + P ni =1 H D,i is the branch divisor of Ψ D . But for D / ∈ Π ′L theGauss map Ψ D is not easy to compute. Acknowledgments.
The author would like to thank Juan Carlos Naranjo for theinformation on the recent paper [17].
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Department of Mathematics, School of Engineering, Tokyo Denki University,Adachi-ku, Tokyo 120-8551, Japan
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