A proof of the Trisecant Identity through the Fourier-Mukai transform
Daniel Hernández Serrano, Francisco José Plaza Martín, José M. Muñoz Porras
aa r X i v : . [ m a t h . AG ] N ov A PROOF OF THE TRISECANT IDENTITY THROUGHTHE FOURIER-MUKAI TRANSFORM.
D. HERN ´ANDEZ SERRANOJ. M. MU ˜NOZ PORRASF. J. PLAZA MART´IN
Abstract.
Using the technique of the Fourier-Mukai transform we givean explicit set of generators of the ideal defining an algebraic curve asa subscheme of its Jacobian. Essentially, these ideals are generated bythe Fay’s trisecant identities. Introduction.
Let C be a smooth algebraic curve of genus g defined over a field k and J its Jacobian. Our goal is to give explicit equations of the subschemes W i ⊂ J ( i ≤ g ) defined by images of the Abel morphisms S i C → J . Inparticular, for the case i = 1 we prove that the equations of C ∼→ W as asubvariety of J are generated by the trisecant identities proved by Fay [2].Some of the results presented in this paper were previously proved in [7]but we give a different proof and approach and we also aim to clarify someimprecisions in that paper.Finally, we give a reformulation of the trisecant conjecture characterizingJacobians recently proved by Krichever [8]. We hope that our methods willbe useful for understanding the proof of this conjecture in geometric terms.Section 2 offers a brief overview on abelian varieties and the Fourier-Mukaitransform for such varieties. Section 3 is devoted to giving the main resultof the paper; using the theory of the Fourier-Mukai transform we explicitlycompute global equations for certain subschemes of a principally polarizedabelian variety. The importance of these equations becomes apparent insection 4, where the case of Jacobians of smooth algebraic curves is studied.It turns out that the subschemes mentioned above are translations of certainsymmetric products of the curve and that the equations computed generalizeFay’s trisecant identity and Gunning’s relations. Finally, in section 5, wereformulate Krichever’s result on the classical Riemann-Schottky problemin terms of the theory developed in the previous sections. Date : September 25, 2018. : 14K05 (Primary) 14H40, 14H42, (Sec-ondary).
Key words : Abelian Varieties, Fourier-Mukai, Jacobians, Schottky problemThis work is partially supported by research contracts MTM2006-0768 of DGI andSA112A07 of JCyL. The first author is also supported by MTM2006-04779.
E-mail addresses : [email protected], [email protected],[email protected] . Abelian varieties and the Fourier-Mukai transform.
Everything in this section is extracted from [12, 1].2.1.
Abelian varieties.
Let X be an abelian variety of dimension g over a field k of characteristic p = 2. We shall use an additive notation and will denote: m : X × X → X (the group law)( x, y ) x + yι X : X → X (the inverse morphism) x
7→ − xτ x : X → X (the translation morphism) y τ x ( y ) := x + y The neutrum element of X will be denoted 0, and the projections onto thefactors of X × X will be written with π and π .Let ˆ X be the dual abelian variety of X , which represents the degree 0Picard functor of X . Therefore, there exists a universal line bundle P on X × ˆ X , called the Poincar´e bundle . Universality means that given a variety T and a line bundle L on X × T (such that the first Chern class of itsrestriction to the fibers of π T : X × T → T vanishes), there exists a uniquemorphism ϕ : T → ˆ X such that: L ∼→ (1 × ϕ ) ∗ P ⊗ π ∗ T N , where N is a line bundle on T . Thus, if a point ξ ∈ ˆ X corresponds to a linebundle L on X , one has: P ξ = P | X ×{ ξ } ∼→ L . Analogously, we shall denote P x = P |{ x }× ˆ X if x ∈ X .Let us normalize the Poincar´e bundle by the condition that P = P |{ }× ˆ X is the trivial line bundle on ˆ X .Let us consider the line bundle m ∗ L ⊗ π ∗ L − on X × X . By the universalproperty of the Poincar´e bundle and its normalization, there exists a uniquemorphism: ϕ L : X → ˆ X such that: (1 × ϕ L ) ∗ P ∼→ m ∗ L ⊗ π ∗ L − ⊗ π ∗ L − , and one checks that ϕ L ( x ) = τ ∗ x L ⊗ L − .2.2. The Fourier-Mukai transform for abelian varieties.
Let us denote by D b ( X ) (resp. D b ( ˆ X )) the derived category of boundedcoherent complexes on X (resp. on ˆ X ). We shall denote the natural projec-tions by π X : X × ˆ X → X and π ˆ X : X × ˆ X → ˆ X . Theorem 2.1. [10, Theorem 2.2]
The integral functor defined by P : S : D b ( X ) → D b ( ˆ X ) E •
7→ S ( E • ) := R π ˆ X ∗ ( π ∗ X ( E • ) ⊗ P ) RISECANT IDENTITY THROUGH THE FOURIER-MUKAI TRANSFORM 3 is a Fourier-Mukai transform; that is, an equivalence of categories.
Definition 2.2. [1, Def. 1.6] A coherent sheaf F on X is WIT i if its Fourier-Mukai transform reduces to a single coherent sheaf ˆ F located in degree i ;that is S ( F ) ∼→ ˆ F [ − i ]. We shall say that F is IT i if in addition ˆ F is locallyfree.Let L be an ample line bundle on X . From [1, Prop. 3.19] it follows that L is IT and thus: S ( L ) = π ˆ X ∗ ( π ∗ X L ⊗ P )and this is a locally free sheaf on ˆ X . The following fact is also well-known,but since we shall make extensive use of this kind of computation along thispaper we shall not omit its proof. Lemma 2.3. ϕ ∗L S ( L ) ∼→ Γ( X, L ) ⊗ k L − .Proof. ϕ ∗L S ( L ) ∼→ ϕ ∗L π ˆ X ∗ ( π ∗ X L ⊗ P ) ∼→ π ∗ (cid:0) (1 × ϕ L ) ∗ ( π ∗ X L ⊗ P ) (cid:1) ∼→∼→ π ∗ ( π ∗ L ⊗ (1 × ϕ L ) ∗ P ) ∼→ π ∗ ( m ∗ L ⊗ π ∗ L − ) ∼→∼→ π ∗ (cid:0) ( m, π ) ∗ ( π ∗ L ⊗ π ∗ L − ) (cid:1) ∼→ π ∗ ( π ∗ L ⊗ π ∗ L − ) ∼→∼→ π ∗ π ∗ L ⊗ L − ∼→ Γ( X, L ) ⊗ k L − . (cid:3) Let us now recollect some definitions and facts that will be needed lateron.
Definition 2.4.
The
Pontrjagin product of two coherent sheaves E and F on X is the sheaf: E ⋆ F := m ∗ ( π E ⊗ π F ) . This product has a derived functor: R ⋆ : D b ( X ) × D b ( X ) → D b ( X )( E • , F • )
7→ E • R ⋆ F • := R m ∗ ( π ∗ E • L ⊗ π ∗ F • ) Proposition 2.5. [10][1, Prop.3.13]
The Fourier-Mukai transform inter-twines the Pontrjagin and the tensor product; that is: S ( E • R ⋆ F • ) ∼→ S ( E • ) L ⊗ S ( F • ) S ( E • L ⊗ F • ) ∼→ S ( E • ) R ⋆ S ( F • )[ g ] . Definition 2.6.
A sheaf E on X will be called Mukai regular , or simply
M-regular , if: codim (cid:0)
Supp R i S ( E ) (cid:1) > i ∀ i > . Sheaves satisfying the IT condition are trivially M-regular. Theorem 2.7. [13, Thm.2.4]
Let E be a coherent sheaf and L an invertiblesheaf supported on a subvariety Y of the abelian variety X (possibly X itself ).If both E and L are M-regular as sheaves on X , then E ⊗ L is generated byits global sections.
Corollary 2.8. [13]
Let ( X, Θ) be a polarized abelian variety and E a co-herent sheaf on ( X, Θ) . If E ( − Θ) is M-regular, then E is generated by itsglobal sections. D. HERN ´ANDEZ SERRANO, J. M. MU ˜NOZ PORRAS AND F. J. PLAZA MART´IN Global equations.
First results.
Let H be a closed and finite subscheme of X and let L be an ample linebundle on X . Consider the following exact sequence of sheaves on X :0 → I H ⊗ L → L → L | H → , where I H denotes the sheaf of ideals of H . Since L | H is concentrated overa 0-dimensional variety, using Grauert’s theorems and base change one canshow that S j ( L | H ) = 0 for all j > S ( L | H ) = π ˆ X ∗ (cid:0) ( π ∗ X L ⊗ P ) | H × ˆ X (cid:1) is a locally free sheaf on ˆ X ; that is, L | H is IT . Therefore, the restrictionmap L → L H induces a morphism of locally free sheaves on ˆ X between itstransforms:(3.1) α ( L , H ) : S ( L ) → S ( L | H ) Definition 3.1.
For every non-negative integer i , we shall write Z i ( L , H )to denote the closed subscheme of ˆ X defined by Λ i α ( L , H ) = 0. Remark . Notice that since S ( L ) is not generetad by its global sections,the equations Λ i α ( L , H ) = 0 are not global. We will give a procedure toexplicitly compute the global equations of Z i ( L , H ).Let ∆ ⊂ X × X denote the diagonal and let ( H,
0) + ∆ ⊂ X × X be theimage of H × X under the map ( m, π ) : X × X → X × X . Let β ( L , H ) bethe following morphism of sheaves on X :(3.2) β ( L , H ) : Γ( X, L ) ⊗ k O X → π ∗ ( π ∗ L | ( H, ) , defined by restriction. Definition 3.3.
For every non-negative integer i , we shall write U i ( L , H )to denote the closed subscheme of X defined by Λ i β ( L , H ) = 0. Lemma 3.4.
One has the following diagram: ϕ ∗L S ( L ) ⊗ O X L ϕ ∗L (cid:0) α ( L ,H ) (cid:1) ⊗ / / ≀ (cid:15) (cid:15) ϕ ∗L S ( L | H ) ⊗ O X L ≀ (cid:15) (cid:15) Γ( X, L ) ⊗ k O X β ( L ,H ) / / π ∗ ( π ∗ L | ( H, ) Proof.
This follows from Lemma 2.3. (cid:3)
Corollary 3.5.
The preimage of Z i ( L , H ) by the morphism ϕ L is precisely U i ( L , H ) . Let us note that the above approach allows us to write down local equa-tions for the subschemes Z i ( L , H ), whose geometric interpretation will begiven in Section 4 for the case of Jacobian varieties. On the other hand, global equations for U i ( L , H ) are already at our disposal, and these sub-schemes are related to the relative position of the points of H . Our nexttask, motivated by Lemma 3.4, consists of comparing these subschemes aswell as obtaining global equations for Z i ( L , H ). RISECANT IDENTITY THROUGH THE FOURIER-MUKAI TRANSFORM 5
Global equations.
Henceforth X will denote a principally polarized abelian variety ( X, Θ),where Θ is a symmetric polarization and L will be a line bundle algebraicallyequivalent to O X (2Θ). Recall that the principal polarization O X (Θ) definesan isomorphism ϕ O X (Θ) : X ∼→ ˆ X . From now on D b ( X ) and D b ( ˆ X ) will beidentified. Under this identification, the Poincar´e bundle on X × X , whichwe shall still denote by P , takes the shape: P not = (1 × ϕ O X (Θ) ) ∗ P ∼→ m ∗ O X (Θ) ⊗ π ∗ O X ( − Θ) ⊗ π ∗ O X ( − Θ) . We shall use Z i ( L , H ) to denote the subscheme of X defined by ϕ − O X (Θ) ( Z i ( L , H )).Under the isomorphism ϕ O X (Θ) : X ∼→ ˆ X , one has the following commuta-tive diagram: X ϕ L / / X (cid:15) (cid:15) ˆ XX ϕ O X (Θ) ? ? ~~~~~~~ where 2 X denotes the multiplication by 2. We shall use the same notationboth for ϕ L and 2 X . Theorem 3.6.
The sheaf S ( L ) ⊗ L is generated by its global sections.Proof. For the sake of simplicity, one can reduce to the case in which L = O X (2Θ). By Corollary 2.8, it suffices to prove that: S ( L ) ⊗ L ⊗ O X ( − Θ) ∼→ S ( L ) ⊗ O X (Θ)is M-regular (see Definition 2.6). Indeed, let us set: µ : X × X → X ( x, y ) x − y We have: S ( L ) ⊗ O X (Θ) = π ∗ (cid:0) π ∗ L ⊗ P (cid:1) ⊗ O X (Θ) ∼→∼→ π ∗ (cid:0) π ∗ O X (Θ) ⊗ m ∗ O X (Θ) (cid:1) ∼→∼→ π ∗ (cid:0) ( m, π ) ∗ ( π ∗ O X (Θ) ⊗ π ∗ O X (Θ) (cid:1) ∼→∼→ µ ∗ (cid:0) π ∗ O X (Θ) ⊗ π ∗ O X (Θ) (cid:1) ∼→∼→ m ∗ (1 × ι X ) ∗ (cid:0) π ∗ O X (Θ) ⊗ π ∗ O X (Θ) (cid:1) ∼→∼→ m ∗ (1 × ι X ) ∗ (cid:0) π ∗ O X (Θ) ⊗ (1 × ι X ) ∗ π ∗ O X (Θ) (cid:1) ∼→∼→ m ∗ (cid:0) (1 × ι X ) ∗ π ∗ O X (Θ) ⊗ π ∗ O X (Θ) (cid:1) ∼→∼→ m ∗ (cid:0) π ∗ ι ∗ X O X (Θ) ⊗ π ∗ O X (Θ) (cid:1) ∼→∼→ O X (Θ) ⋆ O X (Θ) , where the last step uses the fact that O X (Θ) is symmetric; that is ι ∗ X O X (Θ) ∼→ O X (Θ).Now, using Proposition 2.5 and bearing in mind that S (cid:0) O X (Θ) (cid:1) ∼→ O X ( − Θ)one has: S (cid:0) O X (Θ) ⋆ O X (Θ) (cid:1) ∼→ S (cid:0) O X (Θ) (cid:1) ⊗ S (cid:0) O X (Θ) (cid:1) ∼→ O X ( − . This implies that S ( L ) ⊗ L ⊗ O X ( − Θ) is IT , and is therefore M-regular. (cid:3) D. HERN ´ANDEZ SERRANO, J. M. MU ˜NOZ PORRAS AND F. J. PLAZA MART´IN
Corollary 3.7.
Let H = { c , . . . , c n +2 } be a set of n + 2 pairwise differentpoints of X . The subscheme Z i ( L , H ) of Definition 3.1 coincides with thesubscheme defined by the global equations Λ i δ ( L , H ) = 0 , where: δ ( L , H ) : Γ( X × X, π ∗ L ⊗ P ⊗ π ∗ L ) → Γ( X × X, ( π ∗ L ⊗ P ⊗ π ∗ L ) | H × X ) . Proof.
By definition, the subscheme Z i ( L , H ) is defined by the local equa-tions Λ i α ( L , H ) = 0, where α ( L , H ) is the morphism of equation (3.1) (recallthat now we are identifying X with ˆ X ). This map induces a morphism:(3.3) α ( L , H ) ⊗ S ( L ) ⊗ L → S ( L | H ) ⊗ L . Let us denote by δ ( L , H ) the morphism induced by α ( L , H ) ⊗ δ ( L , H ) : Γ( X, S ( L ) ⊗ L ) → Γ( X, S ( L | H ) ⊗ L ) , or, what is the same: δ ( L , H ) : Γ( X × X, π ∗ L ⊗ P ⊗ π ∗ L ) → Γ( X × X, ( π ∗ L ⊗ P ⊗ π ∗ L ) | H × X )by the properties of the pushforward. Since S ( L ) ⊗ L is generated by itsglobal sections, one concludes. (cid:3) Let { θ σ ( z ) , σ ∈ ( Z / Z ) g } be a basis for the vector space Γ (cid:0) X, O X (2Θ) (cid:1) .When k = C , the basis { θ σ } could be the classical basis of second-ordertheta functions of ( X, Θ). If the characteristic of k is = 2, we fix a thetastructure and the basis will be the Mumford theta functions (see [11]). Theorem 3.8.
Let L be the line bundle τ ∗ ξ O X (2Θ) and ξ ∈ X a point suchthat ξ = c + · · · + c n +2 . Then: U n +2 ( L , H ) = ϕ − L (cid:0) Z n +2 ( L , H ) (cid:1) is scheme-theoretically defined by the following system of global equations: det (cid:0) θ σ λi ( z − ξ + c j ) (cid:1) = 0 for every ( σ λ , . . . , σ λ n +2 ) ∈ (cid:0) ( Z / Z ) g (cid:1) n +2 . Furthermore, these equationsare the pullback under the isogeny ϕ L of the global equations of Z i ( L , H ) .Proof. By Definition 3.3, the equations of U n +2 ( L , H ) are Λ n +2 β ( L , H )=0.In this context, β ( L , H ) is the map: β ( L , H ) : Γ( X, L ) ⊗ k O X → π ∗ ( π ∗ L | ( H, ) ∼→ ⊕ n +2 i =1 Γ( X, τ ∗ c i L ) ⊗ k O X . Since L = τ ∗ ξ O X (2Θ) for some ξ ∈ X , then { θ σ ( z − ξ ) , σ ∈ ( Z / Z ) g } is a basisfor the vector space Γ( X, L ) (recall that the translation map τ ξ : X → X isdefined by τ ξ ( x ) = x + ξ ). Therefore, taking global sections in the morphismabove one has: β ( L , H ) : Γ( X, L ) → ⊕ n +2 i =1 Γ( X, τ ∗ c i L ) θ σ ( z − ξ ) (cid:0) θ σ ( z − ξ + c ) , . . . , θ σ ( z − ξ + c n +2 ) (cid:1) . and thus the global equations of U n +2 ( L , H ) are:det (cid:0) θ σ λi ( z − ξ + c j ) (cid:1) = 0for every ( σ λ , . . . , σ λ n +2 ) ∈ (cid:0) ( Z / Z ) g (cid:1) n +2 . RISECANT IDENTITY THROUGH THE FOURIER-MUKAI TRANSFORM 7
Let us see that these equations are the pullback under the isogeny ϕ L ofthe global equations of Z i ( L , H ). From Corollary 3.7, the global equationsof Z n +2 ( L , H ) are Λ i δ ( L , H ) = 0 and now we have that: Γ( X × X, π ∗ L⊗P⊗ π ∗ L ) δ ( L ,H ) −−−−→ Γ (cid:0) X × X, ( π ∗ L⊗P⊗ π ∗ L ) H × X (cid:1) ∼→ ⊕ n +2 i =1 Γ( X, τ ∗ ˜ c i L ) , where ˜ c i ∈ X is such that 2˜ c i = c i (the last isomorphism makes use of thesquare lemma). Recall that δ ( L , H ) is the map induced between the globalsections of the map α ( L , H ) ⊗ α ( L , H ) ⊗ ϕ L we obtain a mor-phism: Γ( X, L ) ⊗ ( L − ⊗ ∗ X L ) → ⊕ n +2 i =1 (cid:0) Γ( X, τ ∗ c i L ) (cid:1) ⊗ ( L − ⊗ ∗ X L ) . Since L − ⊗ ∗ X L is generated by its global sections (it is algebraicallyequivalent to O X ( m Θ) for m > (cid:3) Jacobians.
The aim of this section is to give a geometric meaning to the above equa-tions in the case in which the principally polarized abelian variety (p.p.a.v.)is the Jacobian of a smooth projective curve C of genus g ≥
1. We shall see,using Kempf’s results, that Z i ( L , H ) is a translation of the i -th symmetricproduct of the curve C. Moreover, when H consists of 3 pairwise differentclosed points in the Jacobian, then the equations computed in Theorem 3.8are Fay’s trisecant identity ([2]). If the degree of H is n + 2 the equationsare Gunning’s relations ([3]).4.1. The Jacobian case and the relation with Kempf ’s results.
Let J be the Jacobian of a smooth projective curve C of genus g ≥ J is a p.p.a.v. and the principal polarization is the so calledtheta divisor Θ, which is determined up to translation.We shall work with line bundles L that are algebraically equivalent to O J ( m Θ) for m > J :0 → I C ⊗ L → L → L | C → I C denotes the sheaf of ideals of the curve C . Proposition 4.1.
For m > , the twisted ideal sheaf I C ⊗ L satisfies the IT condition.Proof. The case m = 2 is exactly [13, Thm.4.1]. Since O J (Θ) is also IT (its Fourier-Mukai transform is O J ( − Θ)), using [13, Prop.2.9] one has that I C (3Θ) is IT . By applying this method recursively the result follows. (cid:3) Corollary 4.2.
The sheaf L | C satisfies the IT condition. Corollary 4.3.
The map ρ : S ( L ) → S ( L | C ) is surjective. Let H be a closed finite subscheme of C . Thus, we have a restriction map: α ′ ( L , H ) : S ( L | C ) → S ( L | H ) D. HERN ´ANDEZ SERRANO, J. M. MU ˜NOZ PORRAS AND F. J. PLAZA MART´IN and the map: α ( L , H ) : S ( L ) → S ( L | H )factorizes by ρ : S ( L ) α ( L ,H ) / / ρ HHHHHHHHH S ( L | H ) S ( L | C ) α ′ ( L ,H ) : : ttttttttt Therefore, the subscheme Z i ( L , H ) (see Definition 3.1) coincides withthe closed subscheme defined by Λ i α ′ ( L , H ) = 0. This scheme has beenintensively studied in [6] in dual form and its relevance lies in the followingfact. If: 2 g − ≥ mg − deg H ≥ g − , we define i by: i = 2 g − − mg + deg H .
Therefore, Z deg H ( L , H ) is a translation of − W i , where W i is the imageunder the Abel-Jacobi map of the i -th symmetric product of the curve C .Since ϕ L consists of multiplying by m , one has that m − Z deg H ( L , H ) = U deg H ( L , H ). Thus, in this case Theorem 3.8 generalizes Fay’s trisecantidentity for m = 2 and deg H = 3.4.2. Recovering Fay’s Trisecant Identity and Gunning’s relations.
The global equations obtained in Theorem 3.8 generalize Fay’s trisecantidentity ([2]) and Gunning’s relations ([3]). To see this, let p ∈ C be aclosed point and i : C ֒ → J the embedding defined by p . We fix a thetacharacteristic η (that is, η ⊗ ∼→ ω C , where ω C denotes the canonical linebundle on C ). These data allow us to determine a unique polarization Θon J with the condition Θ | C = η + p (where we also use η to refer to itsassociated divisor).Let { p , . . . , p n +2 } be pairwise different points of C and { c , . . . , c n +2 } their images in J , and let us choose a point ξ ∈ J such that 2 ξ = c + · · · + c n +2 . Let us define L := τ ∗ ξ O J (2Θ) and H = { c , . . . , c n +2 } ⊂ J .By applying Theorem 3.8 to ( J, L , H ) we recover Fay’s trisecant identityfor n = 1 and Gunning’s relations for arbitrary n .The geometric interpretation of the above results must be given in termsof the geometry of the Kummer variety associated with the Jacobian. Let { θ σ ( z ) , σ ∈ ( Z / Z ) g } be a basis for the vector space Γ (cid:0) X, O X (2Θ) (cid:1) ; thelinear series | | is identified with the projective space P N (where N =2 g − φ J : J → | | ∼→ P N x Θ x + Θ − x = ( . . . , θ σ ( x ) , . . . )whose schematic image is the Kummer variety K ( J ) (where Θ x is the imageof Θ under the translation morphism τ x ).Let p , p , p be three pairwise different points of C and c , c , c its imagesin J and let ξ be a point J such that 2 ξ = c + c + c . Recall that we areidentifying J with ˆ J , so the isogeny ϕ O X (2Θ) consists of multiplying by 2. RISECANT IDENTITY THROUGH THE FOURIER-MUKAI TRANSFORM 9
Theorem 4.4.
A point x ∈ ϕ ∗O X (2Θ) ( W − ξ ) , that is x + c + c + c ∈ W ,if and only if the points φ J ( x + c ) , φ J ( x + c ) , φ J ( x + c ) are collinear in P N . This theorem and its schematic equivalent Theorem 3.8 means that, up tothe isogeny ϕ O X (2Θ) , the ideal of the curve C as a subvariety of the Jacobianis generated by the trisecant identities.4.3. Relation with the Riemann-Schottky problem.
A geometrical characterization of Jacobians was proposed by Gunning[4, 5] based on Fay’s trisecant identity [2]. Gunning’s result claims thatthe existence of a family of trisecants is a necessary and sufficient conditionfor a p.p.a.v. to be the Jacobian of an algebraic curve. The first step informulating Gunning’s criterion in terms of equations was taken by Welters[15, 14] and its remarkable conjecture, which states that to characterizeJacobians the existence of only one trisecant, is sufficient. This conjecturehas been solved by Krichever [9, 8] in all three different configurations of theintersections points of the trisecant in the Kummer variety. Here we offerhere a reformulation of Krichever’s result ([8]).Let ( X, Θ) be a p.p.a.v. and let H = { c , c , c } be a set of 3 pairwisedifferent closed points of X . Let L be the line bundle O X (2Θ). Definition 4.5.
Let us define W ( X, H ) := − Z ( L , H ).Recall that Z ( L , H ) is defined by the vanishing of the map betweenexterior powers: Λ α ( L , H ) : Λ S ( L ) → Λ S ( L | H ) . From the theory developed in the previous sections, one can reformulateKrichever’s result ([8]) in the following fashion.
Theorem 4.6.
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