Schottky via the punctual Hilbert scheme

We show that a smooth projective curve of genus $g$ can be reconstructed from its polarized Jacobian $(X, \Theta)$ as a certain locus in the Hilbert scheme $\mathrm{Hilb}^d(X)$, for $d=3$ and for $d=g+2$, defined by geometric conditions in terms of the polarization $\Theta$. The result is an application of the Gunning--Welters trisecant criterion and the Castelnuovo--Schottky theorem by Pareschi--Popa and Grushevsky, and its scheme theoretic extension by the authors.


Introduction
Let (X, Θ) be an indecomposable principally polarized abelian variety (ppav) of dimension g over an algebraically closed field k of characteristic different from 2. The polarization Θ is considered as a divisor class under algebraic equivalence, but for notational convenience, we shall fix a representative Θ ⊂ X. (X, Θ) being indecomposable means that Θ is irreducible.
The geometric Schottky problem asks for geometric conditions on (X, Θ) which determine whether it is isomorphic, as a ppav, to the Jacobian of a nonsingular genus g curve C. The Torelli theorem then guarantees the uniqueness of the curve C up to isomorphism. One may ask for a constructive version: can you "write down" the curve C, starting from (X, Θ)? Even though one may embed C in its Jacobian X, there is no canonical choice of such an embedding, so one cannot reconstruct C as a curve in X without making some choices along the way. We refer to Mumford's classic [10] for various approaches and answers to the Schottky and Torelli problems, and also to Beauville [1] and Debarre [2] for more recent results.
In this note, we show that any curve C sits naturally inside the punctual Hilbert scheme of its Jacobian X. We give two versions: firstly, using the Gunning-Welters criterion [6,13], characterizing Jacobians by having many trisecants, we reconstruct C as a locus in Hilb 3 (X). Secondly, using the Castelnuovo-Schottky theorem, quoted below, we reconstruct C as a locus in Hilb g+2 (X). In fact, for any indecomposable ppav (X, Θ), we define a certain locus in the Hilbert scheme Hilb d (X) for d ≥ 3, and show that this locus is either empty, or one or two copies of a curve C, according to whether (X, Θ) is not a Jacobian, or the Jacobian of the hyperelliptic or nonhyperelliptic curve C. Then we characterize the locus in question for d = 3 in terms of trisecants, and for d = g + 2 in terms of being in special position with respect to 2Θ-translates.
To state the results precisely, we introduce some notation. For any subscheme V ⊂ X, we shall write V x ⊂ X for the translate V − x by x ∈ X. Let ψ : X → P 2 g−1 be the (Kummer) map given by the linear system |2Θ|.
Theorem A. Let Y ⊂ Hilb 3 (X) be the subset consisting of all subschemes Γ ⊂ X with support {0}, with the property that to a disjoint union of two copies of C.
The proof is by reduction to the Gunning-Welters criterion; more precisely to the characterization of Jacobians by inflectional trisecants. Note that the criterion defining Y only depends on the algebraic equivalence class of Θ, and not the chosen divisor.
For the second version, we need some further terminology from [11] and [5].
Again note that these conditions depend only on the algebraic equivalence class of Θ. The term "special position" makes most sense for Γ of small degree, at least not exceeding dim H 0 (O X (2Θ x )) = 2 g .
Our second version reads: Theorem B. Let Y ⊂ Hilb g+2 (X) be the subset consisting of all subschemes Γ ⊂ X with support {0}, which are theta-general and in special position with respect to 2Θ-translates. Then Y is locally closed, and (1) if X is not a Jacobian, then Y = ∅.
(2) if X ∼ = Jac(C) for a hyperelliptic curve C, then Y is isomorphic to the curve C minus its Weierstraß points. (3) if X ∼ = Jac(C) for a non-hyperelliptic curve C, then Y is isomorphic to a disjoint union of two copies of C minus its Weierstraß points.
The proof of Theorem B is by reduction to the Castelnuovo-Schottky theorem, which is the following: Theorem 1.3. Let Γ ⊂ X be a finite subscheme of degree g + 2, in special position with respect to 2Θ-translates, but theta-general. Then there exist a nonsingular curve C and an isomorphism Jac(C) ∼ = X of ppavs, such that Γ is contained in the image of C under an Abel-Jacobi embedding.
Here, an Abel-Jacobi embedding means a map C → Jac(C) of the form p → p − p 0 for some chosen base point p 0 ∈ C. This theorem, for reduced Γ, is due to Pareschi-Popa [11] and, under a different genericity hypothesis, Grushevsky [3,4]. The scheme theoretic extension stated above is by the authors [5]. The scheme theoretic generality is clearly essential for the application in Theorem B.
We point out that the Gunning-Welters criterion is again the fundamental result that underpins Theorem 1.3, and thus Theorem B. More recently, Krichever [8] showed that Jacobians are in fact characterized by the presence of a single trisecant (as opposed to a positive dimensional family of translations), but we are not making use of this result.

Subschemes of Abel-Jacobi curves
be the closed subset consisting of all degree d subschemes Γ ⊂ X such that (i) the support of Γ is the origin 0 ∈ X, (ii) there exists a smooth curve C ⊂ X containing Γ, such that the induced map Jac(C) → X is an isomorphism of ppav's. We give Y d the induced reduced scheme structure.
We shall now prove analogues of (1), (2) and (3) in Theorems A and B for Y d with d ≥ 3: to a disjoint union of two copies of C.
As preparation for the proof, consider a Jacobian X = Jac(C) for some smooth curve C of genus g. It is convenient to fix an Abel-Jacobi embedding C ֒→ X; any other curve C ′ ⊂ X for which Jac(C ′ ) → X is an isomorphism is of the form ±C x for some x ∈ X. Such a curve ±C x contains the origin 0 ∈ X if and only if x ∈ C. Hence Y d is the image of the map that sends x ∈ C to the unique degree d subscheme Γ ⊂ ±C x supported at 0, with the positive sign on the first copy of C and the negative sign on the second copy.
More precisely, φ is defined as a morphism of schemes as follows. Let m : X × X → X denote the group law, and consider as a family over C via first projection. The fibre over p ∈ C is C p . Let This family defines φ + : C → Hilb d (X), and we let φ − = −φ + (where the minus sign denotes the automorphism of Hilb d (X) induced by the group inverse in X).
In the proof of the Lemma, we shall make use of the difference map δ : C × C → X, sending a pair (p, q) to the degree zero divisor p − q. We let C −C ⊂ X denote its image. If C is hyperelliptic, we may and will choose the Abel-Jacobi embedding C ⊂ X such that the involution −1 on X restricts to the hyperelliptic involution ι on C. Thus, when C is hyperelliptic, C − C coincides with the distinguished surface W 2 , and the difference map δ can be factored via the symmetric product C (2) : We note that the double cover C × C → C (2) , that sends an ordered pair to the corresponding unordered pair, is branched along the diagonal, so that via 1 × ι, the branching divisor becomes the "antidiagonal" (1, ι) : C ֒→ C × C.
As is well known, the surface C − C is singular at 0, and nonsingular everywhere else. The blowup of C − C at 0 coincides with δ : C × C → C − C when C is nonhyperelliptic, and with the addition map C (2) → W 2 when C is hyperelliptic.

Proof of Lemma 2.2.
To prove that φ + is a closed embedding, we need to show that its restriction to any finite subscheme T ⊂ C of degree 2 is nonconstant, i.e. that the family Z| T is not a product T × Γ. For this it suffices to prove that if Γ is a finite scheme such that then the degree of Γ is at most 2.
Consider the following commutative diagram: (2) The claim is then simply that C p ∩ C q , or equivalently its translate C ∩ C q−p , is at most a finite scheme of degree 2. Diagram (2) identifies the fibre δ −1 (q − p) on the right with precisely C ∩ C q−p on the left. But δ −1 (q − p) is a point when C is nonhyperelliptic, and two points if C is hyperelliptic. Next suppose T ⊂ C is a nonreduced degree 2 subscheme supported in p. Assuming Γ satisfies (1), we have Γ ⊂ C p , so We have T p ⊂ C − C, and Diagram (2) identifies δ −1 (T p ) on the right with m −1 (C) ∩ (T p × C) on the left. Suppose C is nonhyperelliptic. Then δ is the blowup of 0 ∈ C − C, and δ −1 (T p ) is the diagonal ∆ C ⊂ C × C together with an embedded point of multiplicity 1 (corresponding to the tangent direction of T p ⊂ C − C). Diagram (2) identifies the diagonal in C × C on the right with {0} × C on the left. Thus is {p} × C p with an embedded point, say at (p, q). This contains no constant family T × Γ except for Γ = {q}, so Γ has at most degree 1.
Next suppose C is hyperelliptic. We claim that δ −1 (T p ) is the diagonal ∆ C ⊂ C × C with either two embedded points of multiplicity 1, or one embedded point of multiplicity 2. As in the previous case, this implies that m −1 (C) ∩ (T × C p ) is {p} × C p with two embedded points of multiplicity 1 or one embedded point of multiplicity 2, and the maximal constant family T × Γ it contains has Γ of degree 2. It remains to prove that δ −1 (T p ) is as claimed.
We have W 2 = C − C, and the blowup at 0 is C (2) → W 2 = C − C. The preimage of T p is the curve (1 + ι) : C → C (2) , together with an embedded point of multiplicity 1, say supported at q + ι(q). Now the two to one cover C ×C → C (2) is branched along the diagonal 2C ⊂ C (2) , If q = ι(q), then the preimage in C × C is just (1, ι) : C → C × C, together with two embedded points of multiplicity 1, supported at (q, ι(q)) and (ι(q), q). If q = ι(q), i.e. q is Weierstraß, then we claim the preimage in C × C is (1, ι) : C → C × C together with an embedded point of multiplicity 2. This follows once we know that the curves 2C and (1 + ι)(C) in C (2) intersect transversally. And they do, as the tangent spaces of the two curves (1, 1)(C) (the diagonal) and (1, ι)(C) in C × C are invariant under the involution exchanging the two factors, with eigenvalues 1 and −1, respectively. (1) is obvious, so we may assume X = Jac C. By Lemma 2.2, φ + is a closed embedding and hence so is φ − = −φ + . If C is hyperelliptic, we have chosen the embedding C ⊂ X such that the involution −1 on X extends the hyperelliptic involusion ι on C. It follows that C p = −C ι(p) , and thus φ − = φ + • ι. Thus the two maps φ + and φ − have coinciding image, and (2) follows.

Proof of Proposition 2.1. Point
For (3), it remains to prove that if C is nonhyperelliptic, then the images of φ − and φ + are disjoint, i.e. we never have C p ∩ N d = (−C q ) ∩ N d for distinct points p, q ∈ C. In fact, C p ∩ (−C q ) is at most a finite scheme of degree 2: the addition map C × C → X is a degree two branched cover of C (2) ∼ = W 2 (using that C is nonhyperelliptic), and its fibre over p + q ∈ W 2 is isomorphic to C p ∩ (−C q ).

Proof of Theorem A
In view of Proposition 2.1, it suffices to prove that Y in Theorem A agrees with Y 3 in Proposition 2.1. This is a reformulation of the Gunning-Welters criterion: given Γ ∈ Hilb 3 (X), consider the set Then Gunning-Welters says that V Γ has positive dimension if and only if (X, Θ) is a Jacobian. Moreover, when V Γ has positive dimension, it is a smooth curve, the canonical map Jac(V Γ ) → X is an isomorphism, and Γ is contained in V Γ (see [14,Theorem (0.4)]). Thus Y in Theorem A agrees with Y 3 in Proposition 2.1.

Proof of Theorem B
Let X be the Jacobian of C. For convencience, we fix an Abel-Jacobi embedding C ֒→ X. First, we shall analyse theta-genericity for finite subschemes of C.
Recall the notion of theta-duality: whenever V ⊂ X is a closed subscheme, we let It has a natural structure as a closed subscheme of X (see [12,Section 4] and [5, Section 2.2]); the definition as a (closed) subset is sufficient for our present purpose.
With this notation, theta-genericity means that for all chains of subschemes where Γ i has degree i, the corresponding chain of theta-duals, consists of strict inclusions of sets. We write F for the Fourier-Mukai transform [9,7] of a WIT-sheaf F on X [9, Def. 2.3]: F is a sheaf on the dual abelian variety, which we will identify with X using the principal polarization.
Proposition 4.1. Let Γ ⊂ C be a finite subscheme of degree at least g + 1.
Then Γ is theta-general, as a subscheme of Jac(C), if and only if dim H 0 (O C (Γ g )) = 1 for every degree g subscheme Γ g ⊂ Γ. In particular, if Γ is supported at a single point p ∈ C, then Γ is theta-general if and only if p is not a Weierstraß point.
Proof. For the last claim, note that the condition dim H 0 (O C (gp)) > 1 says precisely that p is a Weierstraß point.
For any effective divisor Γ g ⊂ C degree g, it is well known that dim H 0 (O C (Γ g )) = 1 if and only if Γ g can be written as the intersection of C ⊂ Jac(C) and a Θ-translate (this is one formulation of Jacobi inversion). If this is the case, then the point x ∈ X satisfying Γ g = C ∩ Θ x is unique.
Consider a chain (3). If there is a degree g subscheme Γ g ⊂ Γ not of the form C ∩ Θ x , then every Θ-translate containing Γ g also contains C, and in particular T (Γ g ) = T (Γ g+1 ). Hence Γ is not theta-general.
Suppose, on the other hand, that Γ g is of the form C ∩ Θ x . Then T (Γ g ) \ T (Γ g+1 ) consists (as a set) of exactly the point x. Thus there is a Zariski open neighbourhood U ⊂ X of x such that T (Γ g ) ∩ U = {x}. We claim that, for a possibly smaller neighbourhood U , there are regular functions . . . , f i ) for all i: in fact, apply the Fourier-Mukai functor to the short exact sequence Then F i is a section of a locally free sheaf of rank i, and its vanishing locus is exactly T (Γ i ). Choose trivializations of O Γ i over U for all i compatibly, in the sense that the surjections O Γ i+1 → O Γ i correspond to projection to the first i factors. Then F i = (f 1 , . . . , f i ) in these trivializations.
As T (Γ g ) ∩ U is zero dimensional, it follows that each T (Γ i ) ∩ U has codimension i in U . Hence all the inclusions T (Γ i ) ⊃ T (Γ i+1 ) are strict, and so Γ is theta-general. Now we can compare the locus Y in Theorem B with Y g+2 in Proposition 2.1 by means of the Castelnuovo-Schottky theorem: Proof. Theorem 1.3 immediately shows that if Γ is in special position with respect to 2Θ-translates, then Γ ∈ Y g+2 .
The converse is straight forward, and does not require the theta-genericity assumption. Indeed, we use that any curve C ′ ⊂ X for which Jac(C ′ ) → X is an isomorphism is of the form ±C p for some p ∈ X and we claim that if Γ ⊂ ±C p , then Γ is in special position with respect to 2Θ-translates. For ease of notation, we rename ±C p as C, so that Γ ⊂ C. Then H 0 (I C (2Θ x )) ⊂ H 0 (I Γ (2Θ x )), and the exact sequence shows that already the codimension of H 0 (I C (2Θ x )) in H 0 (O X (2Θ x )) is at most dim H 0 (O C (2Θ x )) = g + 1.
Theorem B now follows: The set Y defined there agrees with the thetageneral elements in Y g+2 , by the Corollary. By Proposition 4.1, Γ = φ ± (p) is theta-general if and only if the supporting point 0 of Γ is not Weierstraß in ±C p , i.e. p ∈ C is not Weierstraß.

Historical remark
Assume C is not hyperelliptic. Then C p ∩ (−C p ) is a finite subscheme of degree 2 supported at 0. Thus, for d = 2, we have φ + = φ − , and the argument in Lemma 2.2 shows that φ + is an isomorphism from C onto Y 2 . If C is hyperelliptic with hyperelliptic involution ι, however, we find that φ + factors through C/ι ∼ = P 1 , and Y 2 ∼ = P 1 , and we cannot reconstruct C from Y 2 alone.
In the nonhyperelliptic situation, it is well known that the curve C can be reconstructed as the projectivized tangent cone to the surface C − C ⊂ X at 0. This projectivized tangent cone is exactly Y 2 (when we identify the projectivized tangent space to X at 0 with the closed subset of Hilb 2 (X) consisting of nonreduced degree 2 subschemes supported at 0). To quote Mumford [10]: "If C is hyperelliptic, other arguments are needed." In the present note, these other arguments are to increase d!