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Finite subschemes of abelian varieties and the Schottky problem

The Castelnuovo-Schottky theorem of Pareschi-Popa characterizes Jacobians, among indecomposable principally polarized abelian varieties of dimension g, by the existence of g+2 points in general position with respect to the principal polarization, but special with respect to twice the polarization, and furthermore states that such collections of points must be contained in an Abel-Jacobi curve. Building on the ideas in the original paper, we give here a self contained, scheme theoretic proof of the theorem, extending it to finite, possibly nonreduced subschemes.


Introduction
There is a classical result in projective geometry, due to Castelnuovo, saying that a large, but finite, collection of points in P r which is in linearly general position, but in sufficiently special position with respect to quadrics, is contained in a unique rational normal curve.
Pareschi-Popa [13] have discovered an analogy for g-dimensional indecomposable principally polarized abelian varieties (A, Θ), where divisors algebraically equivalent to Θ play the role of hyperplanes, and divisors algebraically equivalent to 2Θ play the role of quadrics. The Castelnuovo result of Pareschi-Popa says that if we have g + 2 points on A, in a suitable sense general with respect to Θ, but special with respect to 2Θ, then A is the Jacobian of a curve C, and the g + 2 points are contained in an Abel-Jacobi curve, i.e. a translate of C, embedded into its Jacobian. Thus Abel-Jacobi curves play the role of rational normal curves, and the analogue of Castelnuovo's result contains a Schottky statement (precise definitions of the terms occurring in the assumption are given in the next section): Theorem 1.1. Let Γ ⊂ A be a theta-general finite subscheme of degree g + 2, imposing less than g + 2 conditions on general 2Θ-translates. Then the following holds: The first author thanks the Royal Institute of Technology in Stockholm (KTH), where he was a postdoctoral fellow while much of this work was carried out. The second author thanks the Università di Roma "Tor Vergata" and Giuseppe Pareschi for his kind hospitality during the first steps of this work. He also thanks the KTH for the one week invitation that allowed to strengthen the collaboration with the first author. The second author was partially supported by the Proyecto de Investigación MTM2009-14163-C02-01.
(1) Schottky: The indecomposable principally polarized abelian variety (A, Θ) is isomorphic to a Jacobian J(C) of a curve C, with its canonical polarization. (2) Castelnuovo: There is an isomorphism A ∼ = J(C) as above such that the subscheme Γ is contained in an Abel-Jacobi curve.
The theorem was proved by Pareschi-Popa [13] for reduced subschemes Γ. The aim of this article is the extension to possibly nonreduced subschemes Γ, as Eisenbud-Harris did in the projective case for the classical Castelnuovo result [5,Thm. 2.2]. Moreover, in the proof of Pareschi-Popa for reduced subschemes, the scheme structure needed to prove the existence of trisecants is implicit (see Remark 4.3). Since we work scheme theoretically from the start, our work may serve to clarify the situation.
Grushevsky gave also a very similar result [7,Thm. 3] in the reduced case. His hypothesis are slightly different and he uses the analytic theory of theta functions to prove it. He uses this result to find equations for the locus of hyperelliptic Jacobians.
The Schottky and Castelnuovo statements are proved in Sections 4 and 5, respectively. Both results depend on an analysis of theta-duals and dependence loci of subschemes of Γ; these concepts are introduced in the preliminary Section 2 and further analysed in Section 3. The proof of the Schottky result is based on the criterion of Gunning and Welters, characterizing Jacobians among indecomposable principally polarized abelian varieties by trisecants to the associated Kummer variety.
The full trisecant conjecture, which characterizes Jacobians by the existence of a single trisecant, has now been proved by Krichever [9, Thm. 1.1] over the complex numbers. Pareschi-Popa have kindly pointed out that the trisecant conjecture, which was not proved when their work appeared, would simplify their argument. It seems to us that our version of the argument can not be shortened substantially, as the construction produces a family of trisecants in any case. An advantage of using only Gunning-Welters is that our proof of Theorem 1.1 is entirely algebraic and valid in arbitrary characteristic different from 2 (see [16,Remark 0.7]).
The main line of argument is borrowed from Pareschi-Popa, although many (set theoretic) statements did not seem to translate well into schematic ones, and had to be substituted. Moreover, new phenomena occur when Γ is nonreduced, e.g. already the fact that Γ is contained in a nonsingular curve (i.e. Γ is curvilinear) is not obvious. Logically, our work does not depend on loc. cit., and can be read independently.
The converse to the theorem is easy, as we explain in Section 3.1: a finite degree g + 2 subscheme Γ of an Abel-Jacobi curve C imposes less than g + 2 conditions on general 2Θ-translates in the Jacobian.

Preliminaries
Throughout, (A, Θ) denotes a principally polarized abelian variety over an algebraically closed field of characteristic different from 2. We assume that the divisor Θ is symmetric and irreducible. For each point a ∈ A we denote by t a : A → A the translation map. For any subscheme The support Supp F of a coherent sheaf F is defined as a subscheme by the annihilator ideal. We will also make use of the scheme structure, on the same underlying set, defined by the Fitting ideal of F , which has the advantage of respecting base change. This will be called the Fitting support and denoted Fitt F .
Inclusions, intersections and unions of subschemes are always to be understood scheme theoretically. Unions are usually defined by intersection of ideals; we also make use of the scheme structure defined by the product of ideals, but point this out whenever it is of importance. When Y ⊆ X are two subschemes of some ambient scheme, we write 2.1. The Fourier-Mukai transform. Using the principal polarization Θ, we identify A with its dual Pic 0 (A). Thus, the Poincaré line bundle P is where m denotes the group law and p 1 and p 2 are the projections. The restriction of P to A × {a} is the homogeneous line bundle P a = O A (Θ a − Θ).
Following Mukai [11], we define a left exact endofunctor S on the category of O A -modules by S (E ) = p 2 * (p * 1 (E ) ⊗ P). The Fourier-Mukai functor is the total derived functor of S , and is an autofunctor on the derived category of A. We will not make use of this, and will just work with S and its right derived functors R i S , and will sloppily refer to these as Fourier-Mukai functors.
= 0 for all a ∈ A and all p = i.
By the base change theorem in cohomology, IT i implies WIT i , and the Fourier-Mukai transform of a sheaf satisfying IT is locally free.
We refer to Mukai's original paper [11] for the basic properties of the Fourier-Mukai functor.
2.2. Theta-duality. Given a morphism f : T → A, we define translation t f along f to be the composite Thus, as a set, T (Y ) consists of those points a ∈ A for which the thetatranslate Θ a contains Y as a scheme. We want to show that the theta-dual always exists as a scheme, and equals the object with the same name defined in Pareschi-Popa [14].  We call the scheme Z π (φ) ⊆ S in part (1) the relative zero scheme of φ. Before proving the Lemma, we conclude that theta-duals exist.
Proof of Lemma 2.4. The second part follows from the first: take S ′ ⊆ S to be the relative zero scheme of the inclusion φ : Then the condition f * T (φ) = 0 is equivalent to I X ′ T /X T = 0. For the first part, replace φ : F → G with a twist with a sufficiently relatively ample invertible sheaf. Then we may assume that π * π * F → F , π * π * G → G are surjective (i.e. F and G are globally generated over S) and, using flatness of G , that π * G is locally free. The zero locus of π * (φ) : π * F → π * G , in the usual sense, is the required scheme Z π (φ). It is closed by Nakayama.
Next we compare with the theta-dual as defined by Pareschi-Popa [14], which we recall: let D(−) denote the dualization functor H om(−, O). Then, working in the derived category, the theta-dual is defined in loc. cit. as the support of the g'th cohomology sheaf of (−1) * RS RD(I Y (Θ)). We have (In the third line we used the identity (1 × (−1)) * P ∼ = P −1 , in the fourth line we used P = O(m * Θ − p * 1 Θ − p * 2 Θ) and in the fifth line we used the projection formula.) The twist by Θ clearly does not affect the supports of the cohomologies of this complex, and the i'th derived functor of p 2 * • D is the relative Ext-sheaf E xt i p 2 (−, O A×A ). Thus we find that the thetadual according to loc. cit. is the support of E xt g p 2 (I Y ×A (m * Θ), O A×A ). The following thus shows that our theta-dual agrees with the theta-dual of Pareschi-Popa: Proof. We view T (Y ) as the zero locus Z p 2 (ϑ) of the canonical section ϑ of O Y ×A (m * Θ), relative to second projection p 2 : A × A → A. This agrees with Proposition 2.5, where T (Y ) was constructed as the relative zero locus for Let us temporarily denote the support of E xt g . Apply relative Ext with respect to the second projection p 2 : A×A → A to the short exact sequence to obtain the right exact By relative duality, the sheaf in the middle is dual to , and the leftmost homomorphism has T ′ (Y ) as its vanishing locus. Now let D ⊂ A be an effective divisor not containing Y . Then T ′ (Y ) is also the vanishing locus of the composite . If D is sufficiently ample, we may apply relative duality to see that this composite map is a homomorphism between locally free sheaves, dual to Remark 2.7. It is obvious from the universal property of the theta-dual that, as subschemes of Example 2.8. Let S ⊂ A be a nonreduced degree two subscheme supported in a closed point a. Then "translation along S" defines an infinitesimal deformation of Θ a , namely the scheme m −1 Θ ∩ (S × A) as a family over S. The theta-dual T (S) ⊂ Θ a is the vanishing locus of the corresponding section (defined up to scale, corresponding to the choice of an isomorphism Example 2.9. Let J(C) be the Jacobian of a nonsingular projective curve C of genus g, and choose a base point of C. Let W i be the image of the canonical map C (i) → J(C) (defined using the base point). We take Θ = W g−1 as the polarization. Then it is clear that . This inclusion is in fact an equality, as is shown in [14, Section 8.1] (in the reference there is a sign change, which can be traced to the choice of identification between J(C) and its dual).
Example 2.10. Let J(C) be the Jacobian of a nonsingular projective curve C of genus g. For any two distinct points p, q in C ֒→ Pic 1 (C), we have the following equality [12, Lecture IV] of sets in Pic g−1 (C) where K is a canonical divisor of C. The translations are to be understood inside Pic(C), so that, e.g. (W g−2 ) −p = W g−2 + p is a subset of Pic g−1 (C). For fixed q and generic p, both sides of the equality are reduced in any case, and form a flat family over an open subset of C. Taking flat limits we extend the family to all of C.
We conclude that, when S is a subscheme of degree 2 of an Abel-Jacobi curve C ⊂ A, supported in two possibly coinciding points a and b, there is a schematic equality with α and β depending linearly on a and b, and if the two ±W g−2 -translates on the right coincide, their union is to be understood by perturbing b and taking the flat limit. Thus T (S) is either the union of two distinct ±W g−2translates, or a scheme structure of multiplicity two on a single W g−2translate. Moreover, the latter happens only in the hyperelliptic case: it follows from example 2.9 that W g−2 and −W g−2 coincide up to translation if and only if C and −C do, which is equivalent to C being hyperelliptic.
whereas the other two sheaves are IT 0 . Thus the Fourier-Mukai functor gives a short exact sequence Choosing D = 0 in the proof of Proposition 2.5, which is indeed sufficiently ample on the finite scheme Γ, we find that the maps named F in that proof and in (1) coincide up to twist by Θ. Thus the theta-dual T (Γ) is precisely the zero locus of F in (1). Note that the fibre of F over a point a ∈ A is which vanishes precisely when Γ ⊂ Θ a . Thus the zero locus of F is, from the outset, a natural scheme structure on the set of such points a. The definition of Pareschi-Popa can be seen as a generalization of this observation, where the lack of base change for O Y (Θ) is handled by working with the dual of F instead.
Next we define theta-genericity: recall that a finite subscheme Γ in P r is in linearly general position if every subscheme Γ ′ ⊆ Γ of degree d ≤ r + 1 spans a linear subspace of dimension d − 1. Equivalently, for any pair of Γ ′′ ⊂ Γ ′ of subschemes of Γ, such that there exists a hyperplane containing Γ ′′ but not Γ ′ . Phrased in this way, the condition of linear independence can be carried over to (A, Θ), with Θ-translates replacing hyperplanes.
Remark 2.14 (correction to Remark 3.5 in [13]). The condition in the definition demands that the inclusion T (Γ ′ ) ⊂ T (Γ ′′ ) is set theoretically strict, and not just scheme theoretically. As an example, consider the surface of schemes is strict, but it is an equality of sets. This might suggest that it is more natural to work with the weaker notion of theta-generality given by demanding that T (Γ ′ ) ⊂ T (Γ ′′ ) is a strict inclusion of schemes. We will however continue to use the stronger, set theoretic, notion here. We will in fact study the number of independent conditions imposed by Γ on all the linear systems associated to H 0 (A, O A (D) ⊗ P a ) for a ∈ A. Since D is ample, the collection of these linear systems coincides with the collection of the translated systems |D a |.
Definition 2.15. The superabundance of a finite subscheme Γ ⊂ A with respect to D is the value for general a ∈ A. Equivalently, it is the minimal value of the right hand side, over a ∈ A. The subscheme Γ is superabundant if its superabundance is nonzero.
Remark 2.16. We deviate slightly from the literature (e.g. Griffiths-Harris [6]), where superabundance ω(Γ, D) is defined without the twist by a generic P a , but otherwise as above.
It is also useful to study the locus of points a ∈ A such that Γ does not impose independent conditions on H 0 (A, O A (D) ⊗ P a ).
Definition 2.17. The dependence locus ∆(Γ, D) is the Fitting support of Remark 2.18. Note that R i S (I Γ (D)) vanish for all i > 1. Hence, by base change, the fibre of R 1 S (I Γ (D)) at a is which is nonzero precisely when Γ fails to impose independent conditions on the linear system associated to O A (D) ⊗ P a .
Let Γ ′ ⊂ Γ be a pair of finite subschemes. There is an exact sequence Applying the Fourier-Mukai functor, we obtain a right exact sequence where the union is defined by the product of the corresponding ideals.

Residual subschemes. Following Eisenbud-Green-Harris [3]
, we define a scheme theoretic version of "complement" as follows: Definition 2.22. Let Γ be a finite scheme and Γ ′ ⊂ Γ a subscheme. The residual subscheme of Γ ′ in Γ is the support of the ideal of Γ ′ in Γ. If I Γ ′ /Γ is a principal ideal, then we say that Γ ′′ is well formed.
Remark 2.23. When the residual subscheme is well formed, we may identify I Γ ′ /Γ with the structure sheaf O Γ ′′ , so there is a short exact sequence In particular, there is an equality between the underlying zero-cycles.
Remark 2.24. Let the union Γ ′ ∪ Γ ′′ denote the subscheme, inside some ambient scheme, defined by the product of the corresponding ideals. Then it is immediate from the definition of the residual subscheme (not necessarily well formed) that Γ ⊂ Γ ′ ∪ Γ ′′ .
In particular, if D is an effective divisor containing Γ ′ , then there is an inclusion of ideals Example 2.25. If Γ ′ has degree deg Γ−1, then the ideal I Γ ′ /Γ is isomorphic to the residue field k(x) of the unique closed point x where Γ ′ and Γ differ. Thus x is the residual point of Γ ′ in Γ, and it is well formed.
Definition 2.26 (Le Barz [10]). A finite subscheme Z of a variety X is curvilinear if there is a reduced curve C ⊂ X containing Z, and such that C is smooth along the support of Z.
Example 2.27. If Γ is curvilinear, and Γ ′ ⊂ Γ is arbitrary, then the residual subscheme Γ ′′ is well formed. In fact, it is uniquely determined by (3).
Example 2.28. If Γ = Spec k[x, y]/(x 2 , xy, y 2 ) and Γ ′ is the origin, then it is clear that the data given does not distinguish any degree 2 subscheme of Γ. Indeed, the residual subscheme is just the origin again, so it is not well formed.
Proof. If Γ ′ and Γ differ at two distinct points x and y, then the ideal of Γ ′ in Γ is just k(x) ⊕ k(y) and thus Γ ′′ = {x, y}.
On the other hand, if Γ ′′ and Γ differ at a single point x, then locally at x, we have Γ = Spec(R) for a local Artin Gorenstein ring R. The ideal of Γ ′′ in R is two dimensional as a vector space. Hence it is either a principal ideal, or it is generated by two linearly independent elements of the socle of R. The Gorenstein assumption rules out the latter possibility, so the ideal of Γ ′′ in R is principal.

Superabundance and dependence loci
From here on, we fix D = 2Θ and abbreviate ∆(Γ, 2Θ), ∆(Γ ′ , Γ, 2Θ) and ω(Γ, 2Θ) to ∆(Γ), ∆(Γ ′ , Γ) and ω(Γ). Proof. Since Θ y contains Γ ′ , but not Γ, we have a commutative diagram with exact rows. Twist with 2Θ, use that 2Θ − Θ y is linearly equivalent to Θ −y , and apply the Fourier-Mukai transform to arrive at the commutative diagram with exact rows, and where F is the Fourier-Mukai transform of the WIT 1 sheaf I {a} (Θ −y ). A small calculation shows that F ∼ = P a | Θ a−y . By definition, ∆(Γ ′ , Γ) is the support of Ker(φ). Since the kernel of φ is a quotient of F , it follows that its support is contained in the support of F , which gives the claim. 3.1. Superabundant subschemes. It is to be expected that the superabundance ω(Γ) (always with respect to 2Θ) vanishes as long as Γ has small degree. We begin by establishing that the minimal degree of a superabundant theta-general subscheme is g + 2.
Proposition 3.3. Let Γ ⊂ A be a theta-general finite subscheme of degree at most g + 1.
The above bound is sharp: on a Jacobian A there exist superabundant subschemes of degree g + 2. In fact, by Riemann-Roch, an Abel-Jacobi curve C ⊂ A imposes g + 1 independent conditions on H 0 (A, O A (2Θ) ⊗ P x ) for any x. Hence a finite subscheme Γ of C, no matter how big, cannot impose more than g + 1 conditions. See Pareschi-Popa [13,Example 3.7] for a more precise statement. Our main Theorem 1.1 says that subschemes of Abel-Jacobi curves are the only (theta-general) examples of superabundant subschemes of degree g + 2.
Proof. By the same argument as in the proof of the proposition, the existence of a point in T (Γ ′ , Γ) would imply that ∆(Γ) = A, hence Γ could not be superabundant. Proof. Let Γ 0 ⊂ Γ be a component, so Γ 0 = Spec A for a local Artin ring A. We need to show that the socle Soc(A), i.e. the elements in A annihilated by its maximal ideal, is one dimensional as a vector space. For contradiction, assume f, g ∈ Soc(A) are linearly independent elements. Since the ideal generated by any collection of socle elements coincides with the vector space they span, the ideals (f, g) and (f ) in A determine subschemes Γ ′′ ⊂ Γ ′ in Γ, of degree g and g + 1, respectively (precisely, Γ ′ is the union of Γ \ Γ 0 with the subscheme of Γ 0 defined by (f ), and similarly for Γ ′′ ). By thetagenericity, there exists a theta-translate Θ a containing Γ ′′ but not Γ ′ . Let ϑ ∈ A be a local equation for Θ a . Then ϑ is a socle element, since ϑ ∈ (f, g), and hence defines a subscheme Z ⊂ Γ of degree g+1. But then Z is contained in Θ a , and Γ is not, contradicting the previous corollary.
(i) There exists a unique theta-translate Θ x containing Γ g but not Γ g+2 . (ii) The divisorial part of ∆(Γ g+1 ) is reduced and equals Θ b−x , where x is as above and b is the residual point of Γ g ⊂ Γ g+1 .
Remark 3.7. By Corollary 3.4, the theta-translate Θ x in part (i) cannot contain Γ g+1 . Thus Θ x is also the unique theta-translate containing Γ g but not Γ g+1 .
Before proving the Lemma, we explain a consequence that will be important for proving the Castelnuovo part of Theorem 1.1. Proof. Since Γ is Gorenstein, it suffices to show that every subscheme Γ g+1 ⊂ Γ of degree g + 1 is also Gorenstein. In other words, Gorenstein means the choice of a point a ∈ Γ uniquely determines a subscheme Γ g+1 ⊂ Γ with residual point a. If also the choice of a residual point b ∈ Γ g+1 uniquely determines Γ g ⊂ Γ g+1 , then Γ is curvilinear [4, Lemma 1.4].
Thus we suppose that Γ 1 g and Γ 2 g are two subschemes of Γ g+1 of degree g with residual point b. By the first part of the Lemma, there are unique points x 1 and x 2 such that Γ i g is contained in Θ x i , but Γ g+1 is not. By the second part of the lemma, the divisorial part of ∆(Γ g+1 ) equals Θ b−x i , for either i, and so x 1 = x 2 . Call this point x. Then Θ x contains both Γ 1 g and Γ 2 g , but not Γ g+1 , which is impossible unless Γ 1 g = Γ 2 g .
Proof of Lemma 3.6. By theta-genericity, there exists a theta-translate Θ x that contains Γ g , but not Γ g+1 (or equivalently, not Γ g+2 , by Remark 3.7). We claim there are inclusions where the union on the right hand side is scheme theoretically defined by taking the product of the corresponding ideals. Since ∆(Γ g ) has codimension at least two, by Lemma 3.2, this immediately gives part (ii) of the Lemma. Then also part (i) follows, since Θ b−x and thus x is uniquely determined by the pair Γ g ⊂ Γ g+1 . Now we prove (4). The finite scheme Γ g+2 is Gorenstein by Corollary 3.5. Thus, by Lemma 2.29, the residual scheme S of Γ g in Γ g+2 is well formed, so it has degree two. Let a be the residual point of Γ g+1 in Γ g+2 . By Corollary 3.4, the intersection Γ g+2 ∩ Θ x cannot have degree g + 1, so it must equal Γ g . In other words, locally, the ideal of Γ g is generated by the ideal of Γ g+2 together with a local equation for Θ x . From this one deduces that there is an exact commutative diagram: Now twist this diagram by 2Θ, note that 2Θ − Θ x is linearly equivalent to Θ −x , and apply the Fourier-Mukai transform. Using the short exact sequence (1) with Z = {b}, we find With Z = S, the vanishing locus T (S) of F in (1) has codimension two, hence the cokernel I S (Θ −x ) is torsion free of rank one, i.e. it is a twist of the ideal of T (S). It follows that and so we arrive at the exact commutative diagram: We claim that ν is injective. First note that ρ is not an isomorphism, since its domain has ∆(Γ g+2 ) = A as support (Γ g+2 is superabundant), whereas its codomain is torsion, supported in ∆(Γ g+1 ). It follows that µ is nonzero, and having torsion free domain of rank one, it is injective. By a diagram chase we find that also ν is injective. The vertical short exact sequence on the right thus shows Thus we are done once we have shown that R 1 S (I Γg (2Θ)) ∼ = R 1 S (I Γg/Θx (2Θ)), and in fact, such an isomorphism is obtained from the short exact sequence by twisting with 2Θ and then applying the Fourier-Mukai transform.
3.3. Sums of theta-duals and dependence loci. The following Lemma will be an essential ingredient in our proof of the Castelnuovo statement. Lemma 3.9. Let Σ i ⊂ Γ g+1 ⊂ Γ g+2 be finite subschemes of A, of degree indicated by their subscripts, and assume that Γ g+2 is theta-general and superabundant. Define x to be the residual point of Γ g+1 in Γ g+2 , let j = g + 1 − i and let Σ i+1 and Λ j ⊂ Λ j+1 be subschemes of Γ g+2 such that the underlying zero cycles satisfy Then there is an inclusion of schemes where the left hand side denotes the scheme theoretic image of Proof. Note that the equalities of zero cycles define the various finite subschemes uniquely, as Γ g+2 is curvilinear by Corollary 3.8. We rephrase the statement a little: since formation of Fitting ideals commute with base change, it suffices to show that (6) T To understand the right hand side of (6), we begin with the short exact sequence Instead of first applying Fourier-Mukai, and then pulling back by ν, we encode both operations in the functor T sending a sheaf F on A to the sheaf Here p k and p kl denote the various projections from A×T (Σ i )×T (Λ j ). In standard terminology, T (or its total derived functor) is the Fourier-Mukai transform with kernel (8) (1 × ν) * P ∼ = p * 12 ( P| A×T (Σ i ) ) ⊗ p * 13 ( P| A×T (Λ j ) ).
Applying T to (7), we get a long exact sequence If F is a sheaf on A and p is maximal such that R p S (F ) = 0, then base change shows that ν * R p S (F ) ∼ = R p T (F ). Using this, we can rewrite the last few terms in the long exact sequence, and obtain As Γ g+2 is superabundant, the support of R 1 S (I Γ g+2 (2Θ)) is all of A, so the Fitting support of its pullback by ν is all of T (Σ i ) × T (Λ j ). So it is enough to prove the following claim.
Claim 3.10. The homomorphism φ in the long exact sequence (9) is sur- To prove the claim, we use the commutative diagram where the vertical arrows are the evaluation maps at x as in (7), and the horizontal arrows are multiplication maps. The top map is well defined by Remark 2.24. If we apply the functor T to the previous diagram, we get Using (8), we compute . Consider now the natural map (12) p 23 * (G 1 ) ⊗ p 23 * (G 2 ) ̺ → p 23 * (G 1 ⊗ G 2 ).
By (11) we recognize the codomain of ̺. To understand the domain, we define the functor T 1 , sending a sheaf F on A to the sheaf , and the vanishing locus ). This proves Claim 3.11, which concludes the proof of the Lemma.

Schottky
In this section we prove part (1) of Theorem 1.1 by constructing many trisecants to the Kummer variety of A.
with exact rows, where F is S (I Γ g−1 (2Θ)). Replace F with a locally free sheaf admitting a surjection to S ( Then we have the above diagram, with F locally free, and where the inclusions O(−Θ −y ) ⊂ F on the left are split. Now ∆(Γ g−1 , Γ g+1 ) is locally defined by the maximal minors of f 1 , and ∆(Γ g−1 , Γ g ), T (S) −y and Θ a−y are the vanishing loci of f 2 , f 3 and f 4 , respectively. The inclusion (15) can now be verified explicitly by locally representing f 1 by a matrix as indicated in Figure 1.
Proof. Since y ′ = x ∈ T (Γ g ) and b = a + (x − y ′ ), we have that S does not contain a + (x − y ′ ), so Y is well defined as a finite subscheme of degree three. We deal separately with the two possible cases.
Case i). Y ∼ = which implies that the three points [1,Prop. 11.9.3]), where the factor 1 2 means any counter image under the multiplication-by-two endomorphism of A.
Case ii). Y ∼ = Spec(k[ε]/ε 2 ) + Spec k, that is S is a nonreduced scheme supported in a = b. The inclusion (14) can be written Let s ∈ H 0 (O Θ (Θ)) be the section corresponding to T (S) −a (see Example 2.8). We will also choose a nonzero section θ t of H 0 (O(Θ t )) for all points t. Then s · θ y−y ′ vanish on Θ ∩ Θ y−x . By the exact sequence, Thus, using [16, proof of Thm 0.5, case ii)] we get that, with γ = 2a − y ′ , the line l passing through the two points ψ a + 1 2 (−y − γ) , ψ a + (x − y ′ ) + 1 2 (−y − γ) is tangent at the former. More precisely S + 1 2 (−y − γ) ⊂ ψ −1 (l). Remark 4.3. An inclusion of the type Θ ∩ Θ a ⊆ Θ b ∪ Θ c must hold scheme theoretically to give a trisecant to the Kummer variety. The argument [13,Theorem 5.2] is written set theoretically, so we hope that our schematic treatment is clarifying.
Proof of part (1) of Theorem 1.1. The Gunning-Welters criterion ( [8] or [16,Thm 0.5]) states that, if the algebraic subset V ⊂ A in Proposition 4.2 is at least one dimensional, then it is a smooth irreducible curve and A is its Jacobian. (This holds, according to Welters [16,Remark 0.7], in arbitrary characteristic different from 2, when Y is supported in at least two points, as is the case here.) Since T (Γ g−1 , Γ g ) is at least one dimensional, Proposition 4.2 thus implies that A is a Jacobian, and T (Γ g−1 , Γ g ) is an Abel-Jacobi curve.
Remark 4.4. As a first step in the direction of the Castelnuovo statement, Theorem 1.1 (2), we note: in Proposition 4.2, the residual subscheme S of Γ g−1 in Γ g+1 is contained in a translate of the algebraic set V ⊂ A. In fact, by [16, Rem. 0.6], the degree 3 subscheme Y is contained in a translate of V , and by definition Y contains S.

Castelnuovo
Proof of part (2) of Theorem 1.1. By part (1) of Theorem 1.1, we know that A = J(C) is the Jacobian of some curve C. Let i be maximal such that there exists a degree i subscheme Σ i ⊂ Γ g+2 , which is contained in a translate of ±C. Replace C with this translate of ±C, and fix such Σ i ⊂ C. Then i ≥ 2 by Remark 4.4, and the claim is that i = g + 2. For contradiction, assume i ≤ g + 1. Then Σ i is contained in a degree g + 1 subscheme Γ g+1 ⊂ Γ g+2 , which we fix.
With notation as in Lemma 3.9, there is a schematic inclusion where j = g + 1 − i ≤ g − 1. Now we use that C and W g−2 are theta-duals of each other (Example 2.9 with C = W 1 ). Both C and W g−2 are a priori defined up to translation; as we have fixed C, we may just as well fix W g−2 as T (C). We infer that the inclusion Σ i ⊂ C is equivalent to W g−2 ⊆ T (Σ i ), so T (Σ i , Σ i+1 ) contains W g−2 .
If i > 2, or equivalently j < g −1, then T (Λ j , Λ j+1 ) has dimension at least 2. Then W g−2 + T (Λ j , Λ j+1 ) necessarily equals all of A: clearly T (Λ j , Λ j+1 ) has dimension at least g − j ≥ 2, and it is well known that all the loci W g−k are geometrically nondegenerate, which implies that W g−k + Y = A for any Y ⊂ A of dimension k (see Ran [15,§II] and Debarre [2,Prop. 1.4]). Thus the left hand side in (16) is A, which is a contradiction, since ∆(Γ g+1 ) has codimension one.
The case i = 2, j = g − 1 remains. By Example 2.10, the theta-dual T (Σ 2 ) is the union W ∪ W ′ of two translates of ±W g−2 (the two copies may coincide, when C is hyperelliptic, in which case W ∪ W ′ is to be understood as a multiplicity two scheme structure on W ). By minimality of i, neither W nor W ′ are contained in T (Σ 3 ), so T (Σ 2 , Σ 3 ) also equals W ∪ W ′ . By Proposition 4.2 (and the proof of part (1) of Theorem 1.1), the locus T (Λ g−1 , Λ g ) is a translate of ±C. Thus, the left hand side of (16) contains a translate of the divisor W ∪ W ′ ± C. By the Key Lemma 3.6, the divisorial part of the right hand side of (16) is a theta-translate. So we have an inclusion (W ∪ W ′ ) c ± C ⊆ Θ for some point c, which says that (W ∪ W ′ ) c is contained in T (±C). But the latter is just ±W g−2 , which is integral, so it cannot contain a translate of W ∪ W ′ , and we have a contradiction.