Generic Torelli and local Schottky theorems for Jacobian elliptic surfaces

Suppose that $f:X\to C$ is a general Jacobian elliptic surface over ${\mathbb {C}}$ of irregularity $q$ and positive geometric genus $h$. Assume that $10 h>12(q-1)$, that $h>0$ and let $\overline {\mathcal {E}\ell \ell }$ denote the stack of generalized elliptic curves. (1) The moduli stack $\mathcal {JE}$ of such surfaces is smooth at the point $X$ and its tangent space $T$ there is naturally a direct sum of lines $(v_a)_{a\in Z}$, where $Z\subset C$ is the ramification locus of the classifying morphism $\phi :C\to \overline {\mathcal {E}\ell \ell }$ that corresponds to $X\to C$. (2) For each $a\in Z$ the map $\overline {\nabla }_{v_a}:H^{2,0}(X)\to H^{1,1}_{\rm prim}(X)$ defined by the derivative $per_*$ of the period map $per$ is of rank one. Its image is a line ${\mathbb {C}}[\eta _a]$ and its kernel is $H^0(X,\Omega ^2_X(-E_a))$, where $E_a=f^{-1}(a)$. (3) The classes $[\eta _a]$ form an orthogonal basis of $H^{1,1}_{\rm prim}(X)$ and $[\eta _a]$ is represented by a meromorphic $2$-form $\eta _a$ in $H^0(X,\Omega ^2_X(2E_a))$ of the second kind. (4) We prove a local Schottky theorem; that is, we give a description of $per_*$ in terms of a certain additional structure on the vector bundles that are involved. Assume further that $8h>10(q-1)$ and that $h\ge q+3$. (5) Given the period point $per(X)$ of $X$ that classifies the Hodge structure on the primitive cohomology $H^2_{\rm prim}(X)$ and the image of $T$ under $per_*$ we recover $Z$ as a subset of ${\mathbb {P}}^{h-1}$ and then, by quadratic interpolation, the curve $C$. (6) We prove a generic Torelli theorem for these surfaces. Everything relies on the construction, via certain kinds of Schiffer variations of curves, of certain variations of $X$ for which $per_*$ can be calculated. (In an earlier version of this paper we used variations constructed by Fay. However, Schiffer variations are slightly more powerful.)


Introduction
Suppose that M is a moduli stack of smooth projective varieties over C and and that per : M → P = D/Γ is a corresponding period map.The derivative of per is a homomorphism per * : T M → per * T P .
The local Torelli problem is that of describing the kernel of this homomorphism and the local Schottky problem is the problem of describing its image.We say that the local Torelli theorem holds at a point x of M if the derivative per * of per is injective at x and that the generic local Torelli theorem holds if it holds at every generic point of M. We also say that the generic Torelli theorem holds if per has degree 1 onto its image.The Schottky problem asks for a description of the image of the period map.As explained on p. 228 of [G], if the generic local Torelli theorem holds and if it can be proved that a variety X can be recovered from knowledge of the period point per(X) and the subspace per * (T M (X)) of the tangent space T P (X), then the generic Torelli theorem also holds.
In this paper we consider the problem for elliptic surfaces f : X → C with no multiple fibres (such surfaces we shall call simple) and show that the situation is closely parallel to that for curves, as follows.
Suppose that the geometric genus of X is h and its irregularity q, that b 1 (X) is even (so that, by a result of Miyaoka [Mi], X is Kähler), that 10h > 12(q − 1) and that h ≥ q + 3. Assume also that X is general, in a sense to be made precise later.Then we prove the following results, the first two of which are well known tautologies.
(1) There is a classifying morphism φ = φ f : C → M 1 , where M 1 is the stack of stable curves of genus 1. Set Z = Ram φ ⊂ C, the ramification t divisor.(Up to noise which is removed by the language of stacks, this is the locus where the derivative of the j-invariant vanishes.) (2) If also X is algebraic, then the tangent space at the point X to the stack of algebraic elliptic surfaces is naturally isomorphic to an invertible sheaf on Z.
(3) Every choice of a point a in Z and of a local co-ordinate on C at a defines a 1-parameter variation of X.This is based on the construction of the version of Schiffer variations that is described on p. 443 of [Ga].
(4) The derivative of the period map of this variation, which is a linear map ∇ a : H 2,0 (X) → H 1,1 prim (X), is of rank 1.
(5) There is a meromorphic 2-form η a ∈ H 0 (X, Ω 2 X (2E a )) of the second kind (that is, the residue of η a along E a vanishes) such that ∇ a = ω ∨ a ⊗ [η a ].(6) Assume also that h ≥ q + 3. Then the canonical model of X is a copy of C embedded as a curve of degree h+q −1 in a projective space P(H 2,0 (X) ∨ ) = P h−1 , and the set Z can be recovered, as a finite point set in P h−1 , from the finite subset {ω ∨ a ⊗ [η a ]} a∈Z of P((H 2,0 (X) ∨ ⊗ H 1,1 prim (X)) ∨ ).Indeed, we exploit this set of N points in projective space as an analogue of the theta divisor on the Jacobian of a curve.
(7) Assume that h ≥ q + 3 and that 4h > 5(q − 1).Then the curve C can be recovered from Z in P h−1 via quadratic interpolation.
(8) Given C and Z, we then prove a generic Torelli theorem for Jacobian elliptic surfaces.
Remark: It is clear that some of these constructions can be still be made when the phrase "elliptic curve" is replaced by "Calabi-Yau variety whose compactified moduli stack is a smooth 1-dimensional Deligne-Mumford stack whose first Chern class is positive."An essential difference between the case of curves and that of elliptic surfaces, however, is that for curves these variations arise for any point a on C while for surfaces they only arise for points of the ramification divisor Z. Indeed, for other points x of C there is no meromorphic 2-form of the second kind with double poles along E x .(I am grateful to Richard Thomas for explaining this to me.) We now give some more details.
Definition 1.1 An elliptic surface is Jacobian if it has a specified section.Jacobian implies simple but not conversely.
In this paper the things of primary concern are the stacks SE and J E of simple and Jacobian elliptic surfaces f : X → C that are smooth and relatively minimal.We also consider the stack J E whose objects are the relative canonical models of surfaces in J E; given f : X → C in J E the relative canonical model is obtained by contracting all vertical (−2)-curves in X that are disjoint from the given section.There is an obvious morphism J E → J E that is a bijection on geometric points.At the level of miniversal deformation spaces, this morphism can be described by taking the geometric quotient by an action of the relevant Weyl group, as was shown by Artin and Brieskorn.
We shall say that a surface X in SE is general if its j-invariant C → P 1 j is non-constant and its singular fibres are all of type I 1 .We let SE gen and J E gen denote the stacks of general simple surfaces and general Jacobian surfaces; these are open substacks of SE and J E. Note that J E gen maps isomorphically to its image in J E so is also naturally an open substack of J E. Then to give a point in SE gen is equivalent to giving a classifying morphism F : C → M 1 that is nonconstant and unramified over j = ∞.The stack M 1 is not the same as the stack Eℓℓ of stable generalized elliptic curves; these stacks will be discussed in more detail in Section 2. Giving a point in J E gen is equivalent to giving a morphism φ : C → Eℓℓ that is non-constant over the j-line and unramified over j = ∞.
Assume that f : X → C is general in J E. Let Z denote the ramification divisor in C of φ.Say h = p g (X) and q = h 1 (O X ), so that q is also the genus of C. We shall assume throughout this paper that 10h > 12(q − 1) and h > 0. (1.2) These assumptions ensure that deg φ * T Eℓℓ > 2q − 2, which in turns ensures the vanishing of certain obstruction spaces.From Section 6 onwards we shall make the stronger assumptions that 8h > 10(q − 1) and h > q + 2. (1.3) These assumptions make it possible to apply theorems of Mumford and Saint-Donat about the defining equations of linearly normal projective curves.
Write N = 10h + 8(1 − q).Then, as is well known, SE is smooth at the point corresponding to X and Then we shall prove effective forms of both a generic local Torelli theorem and a generic Torelli theorem for the weight 2 Hodge structure on X, in the following sense.
If X is a surface in J E with specified section σ and fibre ξ then H 2 prim (X) and H 1,1 prim (X) will denote the orthogonal complement σ, ξ ⊥ .If X is in SE but is not necessarily algebraic then H 2 prim (X) and H 1,1 prim (X) will denote ξ ⊥ /Zξ; these two definitions are equivalent for X ∈ J E. Observe that h 1,1 prim (X) = N.In fact dim J E = N also, so that dim J E = h 1,1 prim (X).We shall use this coincidence in Section 5 to enhance the local structure of the derivative of the period map.
From the description of J E gen as the stack that parametrizes those nonconstant morphisms from curves to Eℓℓ that are unramified over j = ∞ we shall prove the following theorem, which appears as Theorem 4.10.It is the main result of the paper; everything else follows from it.
Theorem 1.4 Fix a surface f : X → C that is a point X of J E gen and corresponds to φ : C → Eℓℓ.For P ∈ C put E P = f −1 (P ) and let Z denote the ramification divisor Ram φ .
(1) Given a point a of Z there is a tangent line Cv a to J E at the point X.
(3) The corresponding map is of rank 1.Its kernel is the space H 0 (X, Ω 2 X (−E a )) of 2-forms that vanish along E a and its image is the line generated by the class [η a ] of η a , modulo ξ.
In order to prove this theorem we shall use Schiffer variations to construct, for each point a ∈ Ram φ where the ramification index of φ : C → Eℓℓ is m (so that a is of multiplicity m−1 in Z) an (m−1)-parameter deformation C → ∆ m−1 of C whose derivative can be calculated.So, when X is a point of J E gen , we have a detailed description of an N-dimensional subspace of the tangent space to moduli inside the tangent space to the period domain as the subspace spanned by certain explicit tensors of rank 1. (Masa-Hiko Saito [S] has proved the local Torelli theorem for simple elliptic surfaces with non-constant j-invariant and for many surfaces with constant j-invariant.We shall extend his result slightly; see Theorem 2.10 below.) It is a matter of linear algebra to recover Z as a subset of the projective space P h−1 in which C is embedded as the canonical model of X, under the assumption that Z is reduced.We then use a theorem of Mumford [Mu] and Saint-Donat [SD], to the effect that linearly normal curves of genus q and degree at least 2q + 2 are intersections of quadrics, to show that C is determined by quadratic interpolation through Z.We go on to prove that from the pair (C, Z) we can recover the classifying morphism φ : C → Eℓℓ, modulo the automorphism group G m of Eℓℓ provided that φ is generic.This recovery of C and φ from the period data we regard as an effective theorem.It shows that any failure of generic Torelli for Jacobian surfaces can be detected in a pencil that is the closure of the G morbit thorough a generic point of J E gen .(The fact that the automorphisms of Eℓℓ obstruct a direct deduction of generic Torelli from knowledge of C and Z was observed by Cox and Donagi [CD].) Once we know that the base curve C is determined by Hodge-theoretical data of weight 2 we go on to prove the generic Torelli theorem for Jacobian elliptic surfaces via ideas similar to those used by Chakiris [C1] [C2] to prove generic Torelli when C = P 1 , but reinforced by the Minimal Model Program.
There is also some further structure on the period map for J E gen : the relevant vector bundles and homomorphisms between them can be described in terms of line bundles on the universal ramification divisor Z gen over J E gen of the universal classifying morphism to Eℓℓ.This can be seen as a local solution to the Schottky problem.The details are stated in Theorem 4.11.
We also give a variational form of a partial solution to the global Schottky problem.
If X is a Deligne-Mumford stack then [X ] will denote its geometric quotient.
I am very grateful to Phillip Griffiths, Ian Grojnowski, Richard Thomas and Tony Scholl for some valuable discussion.
Everything in the next two sections is well known; if it is not due to either Kas [Ka] or Kodaira [Ko] then it is folklore.
The stack Eℓℓ is the Deligne-Mumford stack over C of stable generalized elliptic curves; that is, an S-point of Eℓℓ is a flat projective morphism Y → S with a section S 0 contained in the relatively smooth locus of Y → S and whose geometric fibres are reduced and irreducible nodal curves of arithmetic genus 1.Such a curve is then, locally on S, a plane cubic with affine equation where g 4 and g 6 are not both zero, so that Eℓℓ is the quotient stack P(4, 6) = (A 2 − {0})/G m , where G m acts on A 2 with weights 4, 6.Note that G m acts on A 2 via a homomorphism G m → G 2 m (whose kernel is µ 2 ) and the standard action of G 2 m on A 2 , so that there is a residual action of G m on Eℓℓ.This exhibits G m as the full automorphism group of Eℓℓ.Cf.Theorem 8.1 of [BN] and the calculation there on p. 139.
The geometric quotient [Eℓℓ] of Eℓℓ is the compactified j-line P 1 j ; if ρ : Eℓℓ → P 1 j is the quotient morphism then the automorphism group of each fibre of ρ is Z/2, except over j = j 6 = 0, where it is Z/6, and over j = j 4 = 1728, where it is Z/4.So deg ρ = 1/2.As is well known, it is possible to write down a generalised elliptic curve over the open locus U of P 1 j defined by j = 0, 1728, so that there is a section of ρ over U.Moreover, ρ −1 (U) is isomorphic to U × B(Z/2), but there is no global section of ρ.
There are two obvious line bundles on Eℓℓ: the bundle M of modular forms of weight 1, which is identified with the conormal bundle of the zero-section of the universal stable generalized elliptic curve, and the tangent bundle T Eℓℓ .The objects of the stack M 1 are stable curves of genus 1; the geometric fibres are isomorphic to stable generalised elliptic curves, but no section is given.This is an Artin stack, but not Deligne-Mumford.Indeed, the word "stable" in this context is an abuse of language, but I am optimistic that it will cause no confusion.
Let C → M 1 and E → Eℓℓ denote the universal objects and let G → Eℓℓ denote the Néron model of E → Eℓℓ, so that G is the open substack of E obtained by deleting the singular point of the fibre over j = ∞.
The next result is due to Altman and Kleiman [AK], although reformulated here in the language of stacks.We have chosen to include a slightly different proof that emphasizes automorphism groups rather than Picard varieties so that the relevant classifying stacks enter more easily.
Theorem 2.2 There is a morphism π : M 1 → Eℓℓ via which M 1 is isomorphic to the classifying stack BG over Eℓℓ.
PROOF: Let G → M 1 denote the connected component of the relative automorphism group scheme of C → M 1 .So G → M 1 is elliptic over the open substack M 1 of M 1 defined by j = ∞ and over j = ∞ the fibre of G is the multiplicative group G m .
Lemma 2.3 There is a unique open embedding G ֒→ G, over M 1 , where G → M 1 is a stable generalised elliptic curve and G is its relative smooth locus.
PROOF: To construct G we need to patch the puncture of G → M 1 that lies over j = ∞.In a suitable neighbourhood S of the locus j = ∞ in M 1 the process of patching the puncture is a matter of "reversing the process of deleting a closed point from a normal 2-dimensional analytic space (or scheme)", so the patch is unique if it exists.Therefore it is enough to exhibit the patch locally on M 1 , in the neighbourhood S.
Put C S = C × M 1 S. Since S is local and C S → S is generically smooth, there is a section over S of C S → S that is contained in the relative smooth locus.Use this section to put the structure of a stable generalised elliptic curve on C S → S.This structure on C S → S provides the patch for G S = G × M 1 S → S and yields G → S; the lemma is proved. Sending A gerbe is a morphism of stacks that is locally surjective on both objects and morphisms.A neutral gerbe is a gerbe with a section.Lemma 2.4 π is a neutral gerbe.
PROOF: It is enough to show that (1) π has a section (then it is certainly locally surjective on objects) and (2) π is locally surjective on morphisms.
For (1), use the forgetful morphism Eℓℓ → M 1 to get a section of π.
(2) is equally clear: a morphism of stable generalised elliptic curves is, in particular, a morphism of the underlying stable curves.
Finally, suppose that X → Y is a neutral gerbe with section s : Y → X .The stabiliser group scheme is a group scheme As already remarked, a general simple surface f : X → C determines, and is determined by, a morphism denote the composite, so that the induced Jacobian elliptic surface is the compactified relative automorphism group scheme of f : X → C.
For example, if f : X → P 1 is a primary Hopf surface then φ f is constant: the relative automorphism group scheme is a constant relative group scheme E × P 1 → P 1 .Lemma 2.5 If X → C has non-constant j-invariant then the irregularity q of X equals the geometric genus of C.
PROOF: This is well known, and easy.
Lemma 2.6 Suppose that f : X → C is a general simple surface. (1) PROOF: Since c 2 (X) equals the total number of nodes in the singular fibres of X → C, (1) follows from consideration of the inverse image of the locus j = ∞, the fact that deg ρ = 1/2 and Noether's formula.The rest follows immediately.
We next consider various tangent spaces.
Regard BG as the quotient of Eℓℓ by G. Since M 1 is isomorphic to BG, locally on Eℓℓ, the tangent complex T • M 1 is a 2-term complex, which is obtained by descending a 2-term complex on Eℓℓ.In degree 0 this complex is T Eℓℓ , in degree −1 it is the adjoint bundle Ad G and the differential is the derivative of the action of G on Eℓℓ.
Since this action is trivial, the differential in T • M 1 is zero.Moreover, since G has no characters (it is generically an elliptic curve) it follows that T • M 1 is quasi-isomorphic to the pull back of the complex Ad G[1] ⊕ T Eℓℓ [0] on Eℓℓ.
Note that (Ad G) ∨ is exactly the line bundle M of modular forms of weight 1.Now fix a point X of SE.That is, we fix a general simple elliptic surface f : X → C.This equals the datum of a morphism Proposition 2.7 The tangent space T SE (X) is naturally isomorphic to the hypercohomology group H 0 (C, K • ) and the obstructions to the smoothness of SE at X lie in H 1 (C, K • ).
PROOF: This follows from the identification of the points of SE with morphisms from curves to M 1 .In this latter context the result is well known.
Let Z denote the ramification divisor Z = Ram φ = Ram F on C. Then K 0 is an invertible sheaf on Z, so that there is a non-canonical isomorphism The distinguished triangle just mentioned gives an exact sequence in which the two middle rows are canonically split.
Proposition 2.8 (1) The stack SE is smooth at the point X.
(3) The degree of the ramification divisor PROOF: Except for (4), which is well known, this follows from the preceding discussion.Suppose that f : X → C is a point of J E gen and defines φ :

Proposition 2.9
(1) There is a short exact sequence (2) T J E (X) is naturally isomorphic to H 0 (Z, K 0 ).
(3) J E is smooth at the point X and its dimension there is N.
PROOF: This is proved in the same way as Propositions 2.7 and 2.8.
We shall usually write At this point we give a slight refinement of M.-H.Saito's local Torelli theorem.
The argument is essentially his.
Theorem 2.10 Suppose that f : Then the local Torelli theorem holds for X.
PROOF: From the main result of [S] it is enough to consider the situation where the j-invariant is constant.Following [S] it is enough to show that the natural homomorphisms and X , and that |K C + L| has no base points.Lemma 2.11 If F , G are coherent sheaves on a 1-dimensional projective scheme C over a field k and if F is generated by H 0 (C, F ), then the natural multiplication

is surjective.
PROOF: There is an exact sequence tensoring this with G gives an exact sequence Taking H 1 of this sequence gives the result.
In particular, taking The exact sequences (4.17) and (4.18) of [S] are where T 1 is a skyscraper sheaf.The second sequence then gives where δ ≥ 0, via the process of saturating a subsheaf.Since deg A > deg L, by assumption, this last sequence splits and From its definition, µ 2 is then the direct sum The first of these is obviously surjective, while the surjectivity of the second follows from [Mu], Theorem 6, p. 52 and the facts that

The comparison between SE gen and J E gen
The morphism π : M 1 → Eℓℓ defines a morphism Π : The sheaves L ∨ and H are group schemes over C.
The following results are due, in essence, to Kodaira.In particular, Proposition 3.2 is a variant of [Ko], p. 1341, Theorem 11.7; there he proves only that of sheaves of commutative groups on C.
PROOF: This follows from the exponential exact sequence on X and the identification For any ring A, there is a Leray spectral sequence Since the fibres of f are irreducible curves and X → C has a section, the hypotheses of Théorème 1.1 of [SGA7II] XVIII are satisfied, so that this degenerates at E 2 .Take A = Z; then there is a short exact sequence Suppose that ℓ is a prime dividing the order of Tors H 3 (X, Z).Since H 4 (X, A) is isomorphic to A, it follows from taking cohomology of the short exact sequence , where ∨ denotes the dual Z/ℓ-vector space.
The spectral sequence E pq r,Z/ℓ shows that β : induced by the inclusion i : σ ֒→ X of the zero section is not injective.Then there is an étale Z/ℓ-cover α : X → X that is split over σ.So X → C is elliptic and has a section σ such that N σ, X ∼ = N σ/X .However, So β is an isomorphism, so that H 3 (X, Z/ℓ) ∼ = (Z/ℓ) 2q and therefore H 3 (X, Z) is torsion-free.Let SE h,q denote the substack of SE that consists of surfaces whose geometric genus is h and whose irregularity is q.This is a union of connected components of SE.
The next result is an immediate corollary of Proposition 3.2.
According to [Ko], p. 1338, Theorem 11.5, Y is algebraic if and only if it defines a torsion element of H 1 (C, H).
Let ξ ⊥ denote the orthogonal complement of ξ in H 2 (X, Z).Then, via the Leray spectral sequence, there is a commutative square where the vertical arrows are isomorphisms.Then β has dense image, from the Kähler property of X, and the proposition is proved.
4 Schiffer variations for elliptic surfaces and the derivative of the period map This contradiction proves the result for p.
The argument for q is similar but easier, so we omit it.
Lemma 4.2 The image of p is G.
PROOF: This is a matter of showing that, given (z, t) ∈ G, we can solve the equation w e + e t e−j w j − z e = 0 with |w − z| < |z| 2e .Without the inequality there are e solutions; if |w − z| ≥ |z| 2e for all of them then we get a contradiction to z e − w e = e t e−j w j and the inequalities satisfied by the t e−j .
For each t ∈ ∆ e−1 t the intersection q(F ) ∩ (C w × {t}) is an annulus A t in C w × {t} that surrounds zero.Let R t denote the open simply connected region in C w × {t} that contains 0 and has the same outer boundary as A t , and put Define C to be the result of glueing U and H together along F via the maps p and q.This is Hausdorff and, after shrinking ∆ e−1 t if necessary, the morphism π : C → ∆ e−1 t is proper and is the morphism that we sought.This is sometimes expressed by saying that C t is constructed from C by deleting a small z-disc around a and glueing in a w-disc, where w is defined implicitly by w e + e e−2 0 t e−j w j = z e .
Until after Theorem 4.11 we fix a point f : X → C of J E gen and a point a in the ramification divisor Z = Ram φ of the classifying morphism φ : C → Eℓℓ.In particular, φ is unramified over j = ∞.Put E a = f −1 (a) and denote by e = e(a) the ramification index at a of φ, so that a has multiplicity e−1 in Z. Assume that the disc D is sufficiently small, so that it contains no other points of Z.We use the Schiffer variations of C that we have just described to construct variations of X.Any Jacobian deformation of the surface X determines, for each point a ∈ Z, a deformation of the finite scheme V e(a) = Spec C[t]/(t e(a) ), so that there is a morphism of local analytic deformation spaces Ψ : Def X → a∈Z Def V e(a) .Recall that Def V e(a) is smooth of dimension e(a) − 1.
(2) The universal ramification divisor Z is smooth over C.

PROOF:
(1) is a an immediate consequence of the formula used to define Φ * s, and (2) follows at once.

So we have morphisms
and F : X → C is a family, parametrized by ∆ e−1 t , of Jacobian elliptic surface whose fibre over 0 Fix a suitable basis of H 2 (X, Z), which we identify with H 2 (X t , Z).Then there are holomorphic (e + 1)-forms Ω (1) , ..., Ω (h) on X such that the residues Xt ) for all t.In particular, there are 2-cycles A 1 , ..., A h on X t such that A i ω (j) (t) = δ j i , the Kronecker delta.Since the line bundle Ω e+1 X pulls back from a line bundle on C, we can expand Ω (j) as where v is a fibre co-ordinate.
. PROOF: Immediate from w e + e t e−i w i = z e .
Remark: Lemma 4.5 1 shows that, in terms of H 1 (C, T C ), the first order deformation obtained from C → ∆ e−1 t is the one that arises from integrating the space of Čech 1-cocycles with values in T C , with respect to the cover {D, C − {a}} of C, that is generated by the vector fields z −1 d/dz, . . ., z −(e−1) d/dz on D − {a}.However, there are different kinds of Schiffer variation that lead to the same space of cocycles; this feature is part of their strength.
We substitute this into the expansion of Ω (j) .Since e−i modulo (t) 2 , where ω (j) = ω (j) | t=0 and η Moreover, every 2-cycle γ on X that is disjoint from E is identified, via a C ∞ collapsing map, with a 2-cycle on X t and and the formula (4.7) it follows that η Griffiths transversality shows that in fact [η . (We let F il i refer to the i'th piece of the Hodge filtration of H 2 prim (X).)We have constructed, for each a ∈ Z of ramification index e(a) = e, an (e − 1)-parameter variation X → ∆ e−1 of X such that the tangent space T ∆ e−1 (0) is identified with the cyclic skyscraper sheaf L a of length e − 1 on C that is supported at a and determined by Z.For any v ∈ L a consider the derivative Assume that the geometric genus h of X is not zero.Also, given ω ∈ H 0 (X, Ω 2 X ) and P ∈ C, put ω(P ) = ω dz∧dv (Q) for an arbitrary point Q ∈ E P and identify ω with the pullback of a tensor on C. Denote by σ ⊂ X the given zero-section of f : prim (X) is injective.PROOF: Until now we have identified the tangent space T J E (X) with a line bundle on Z; we can also identify it with where F P is a rank 2 vector bundle on E P that fits into a short exact sequence The coboundary map H 0 (E P , O E P ) → H 1 (E P , O E P (−σ P )) is identified with the Kodaira-Spencer map, so is an isomorphism from our assumption about (ω) 0 .So H 0 (E P , F P ) = 0 and then the homomorphism ) is injective.This homomorphism factors through the cup product with which we are concerned, and the proposition is proved.
Write b (4) More generally, for each ℓ with 2 ≤ ℓ ≤ e(a), there exists a form η a,ℓ ∈ H 0 (X, Ω 2 X (ℓE a )) 2 nd kind such that the classes [η prim (X).PROOF: This follows from further consideration of the expansion (4.6).To begin, write b where the second equality holds modulo Then inspection of these coefficients and the linear independence provided by Proposition 4.8 are enough to prove the theorem.
We can restate all this in more intrinsic terms, as follows.Let ω denote the vector [ω (1) , . . ., ω (h) ] and ω (i) its i'th derivative with respect to z.
Theorem 4.10 For each a ∈ Z and each k = 2, . . ., e(a) there is an explicit meromorphic 2-form PROOF: As remarked, this is, except for the final statement, nothing more than a restatement in intrinsic terms of what we have just proved.For the final part we need to know that the vectors ω(a), . . ., ω (e(a)−2) are linearly independent.This follows from the cohomology of the exact sequence and the assumption that h + q − 1 ≥ e(a).(3) The classes [η a,i ] form a basis of H 1,1 prim (X) as a runs over the points of Z and i runs from 2 to e(a).
(4) If Z is reduced then the classes [η a,2 ] form an orthogonal basis of H 1,1 prim (X) as a runs over the points of Z.
PROOF: (1): We use the same family X → C → ∆ e(a)−1 corresponding to the point a as before.This variation is constant outside a small neighbourhood of a and the morphism to is also constant outside this neighbourhood .So the ramification locus Z is also constant there.
The point b moves in a family of points b(t Similarly to what we did before, we write for some meromorphic (e(a) + 1)-form Then the same kind of calculation in terms of a power series expansion as before shows that ∇ ∂/∂t i η b(t),k is an element of H 0 (X, Ω 2 X (iE a + kE b )) whose residues along both E a and E b are zero.Therefore ∇ a (η b,k ) is, modulo H 2,0 (X), a linear combination as described.
(2): Choose an element ω of H 2,0 (X) that does not vanish along E a .So ∇ ∂/∂t i ω is a non-zero multiple of η a,i .We can assume that (3) and ( 4) follow from the linear independence of the [η a,i ] and the fact that there are N of them, where N = dim H 1,1 prim (X).Remark: The fact that the η a,i form a basis of H 1,1 prim (X) follows from Theorem 3.24 and Remark 3.29 of [CZ], or from Proposition 4.8.However, the orthogonality seems to be new.
Until now f : X → C has been a point in J E gen .Now, however, allow X to have A 1 -singularities, so that f : X → C is defined by a classifying morphism φ : C → Eℓℓ that is simply ramified over j = ∞.Let X → X be the minimal model, so that X has singular fibres of types I 1 and I 2 and X → C is the relative canonical model of X → C. Suppose that D 1 , . . ., D r are the exceptional (−2)curves on X. Suppose that a 1 , . . ., a r are the points in Z lying over j = ∞ and that a r+1 , . . ., a N are the other points of Z.For a = a i with i ≤ r define [η a ] = [D i ].The surface X is a point in the stack J E whose points are minimal models of points of J E. We can extend Theorem 4.11 to this context, as follows.
For simplicity we state it with the assumption that Z is reduced.
Theorem 4.12 Suppose that Z is reduced.Put η a,2 = η a for each a ∈ Z.
(1) Each point a in Z defines a line v a in the tangent space H 1 ( X, T X ) = T J E X such that the covariant derivative is proportional to the rank one tensor ω ∨ a ⊗ η a .( 2) The classes [η a ] a∈Z form an orthogonal basis of H 1,1 prim ( X). PROOF: We only need to prove (1) when a = a i for i ≤ r.Regard the surface X as the specialization of a surface in J E gen .It follows by continuity that ∇ a = ω ∨ a ⊗ θ a for some class θ a .To see that θ a is proportional to [D a ] we argue as follows.
Suppose that Γ = Γ j is a configuration of (−2)-curves on a smooth surface X that contracts to a single du Val singularity P ∈ X.Then we identify H 2 ( X, Ω 2 X ) = H 0 (X, ω X ) and put There is a short exact sequence which gives a commutative diagram Taking Γ = D a leads to the fact that θ a is proportional to [D a ].

A local Schottky theorem
In this section we use the coincidence that dim H 1,1 prim = dim J E to put further structure on the derivative of the period map.
The vector spaces H 1,1 prim (X) fit together into a vector bundle H = H 1,1 prim on J E gen .We shall restrict attention to the open substack J E ss of J E gen over which the universal ramification divisor Z is étale, so that, under the projection ρ : Z ss → J E ss , the sheaf ρ * O Z ss is a sheaf of semi-simple rings on J E ss .Proposition 5.1 On J E ss the vector bundle H is naturally a line bundle B on Z ss .
PROOF: Essentially, we do this componentwise, using the orthogonal basis of H 1,1 prim (X) that is provided by the classes ([η a ]) a∈Z described in the previous section.
Let x ∈ J E ss , and choose an analytic neighbourhood where f a is a function on U. Now define hs by the formula Proposition 5.2 There is a perfect O Z ss -bilinear pairing β : B × B → O Z ss such that T r(ρ * β) is the intersection product on H.

PROOF:
We define the pairing β in terms of the notation used in the proof of Proposition 5.1, as follows: It is clear that β is perfect.
Recall that, tautologically, the tangent bundle T J E gen is naturally a line bundle S on Z.That is, T J E gen = ρ * S. The derivative per * of the period map is an O J E gen -linear map per * : ρ * S → Hom J E gen (H 2,0 , ρ * B).
Theorem 6.1 Assume that both surfaces have the same IVHS.Then C 1 = C 2 and Z 1 = Z 2 .
PROOF: Our assumptions mean that the two surfaces give the same point and the same tangent space under the period map after each cohomology group H 2 (X i , Z) has been appropriately normalized.In particular, such a normalization gives a normalized basis i , ..., ω i ] of H 0 (X i , Ω 2 X i ) for each i.We can therefore regard the curves C i as embedded in the same projective space P h−1 .That is, any point P of C i is identified with the point ω i ( P ) in P h−1 , where P ∈ X is any point of X that lies over P .The basis ω i also gives an identification of H 0 (X i , Ω 2 X i ) with its dual.We show first that Z 1 = Z 2 .We then deduce, by quadratic interpolation, that By the results of Section 4, especially Theorem 4.10, the tangent space ) is a direct sum a∈Z 1 L a where L a is a line described by Theorem 4.10.We make the identifications (1) U and V are vector spaces such that dim U = h and dim V = N, (2) u 1 , ..., u N ∈ U, (3) no two of the u i are linearly dependent, (4) v 1 , ..., v N ∈ V and form a basis of V , (5) ξ ∈ U ⊗ V and there is a linear relation Then there is a unique index i such that ξ is proportional to so x is a multiple of u j .Since no two of the u i are linearly dependent, this index j is unique, so y = µ j v j and then prim (X 1 ) and write From Theorem 4.10 and the assumption that the IVHS of the two surfaces X 1 and X 2 are isomorphic, the tensors x ′ l ⊗ y ′ l span the same N-dimensional subspace of U ⊗ V as do the tensors x k ⊗ y k .In particular, each x ′ l ⊗ y ′ l is a linear combination of the tensors x k ⊗ y k .Then, by Lemma 6.2, for each index l there is a unique index m such that x l is proportional to x ′ m .That is, a l = a ′ m , and therefore Z 1 = Z 2 = Z, say.
By the assumptions (1.3) each C i is non-degenerately embedded in P h−1 by a complete linear system, and deg C i ≥ 2q + 2, so that, by the results of [Mu] and [SD], C i is an intersection of quadrics.Since again by (1.3), each C i equals the intersection of the quadrics through Z, so that We can reformulate this as follows.
Theorem 6.4 If the ramification divisor Ram φ of the classifying morphism φ : C → Eℓℓ is reduced then the IVHS of the surface X determines the base curve C, the divisor Ram φ and the line bundle L = φ * M.
PROOF: What remains to be done is to show that we can recover the bundle L.
For this, observe that our argument has shown that the IVHS of X determines the embedding i : C ֒→ P h−1 as the canonical model of X and that i * O(1) = O C (K C + L).
Remark: (1) Theorems 6.1 and 6.4 lead to the problem of trying to recover a morphism φ : C → Eℓℓ from knowledge of C, the divisor Ram φ and the line bundle L. However, it is impossible to do more than recover φ modulo the action of the automorphism group G m of Eℓℓ.In this direction we shall prove Proposition 8.7.
(2) If instead of Ram φ being reduced we assume only that h + q − 1 ≥ e(a) + e(b) − 1 for all a, b ∈ Ram φ then a refinement of the argument given here shows that Theorem 6.4 still holds.
7 The structure of the tangent bundle to SE gen Recall from Section 3 that we have a morphism π : M 1 → Eℓℓ that gives, locally on Eℓℓ, an isomorphism M 1 → BG, so that the tangent complex T • M 1 is locally isomorphic to the 2-term complex whose differential is zero.The morphism π determines a morphism Π : SE gen → J E gen .
Suppose that f : X → C corresponds to ψ : C → M 1 and maps under Π to Y → C. Say φ = π • ψ.Observe that Ram φ = Ram ψ = Z, say.The description just given of T • M 1 leads to a short exact sequence where L = φ * M. The subobject H 1 (C, −L) in this sequence is identified with the tangent space T Π −1 (Y ) X to the fibre of Π though X.So which is in turn naturally isomorphic to both H 2,0 (X) ∨ and to H 2,0 (Y ) ∨ .Let ξ denote the class of a fibre of X → C; then the period map gives a commutative diagram with exact rows Theorem 7.1 If Z is reduced and disjoint from the locus j = ∞ then the IVHS of the surface X determines the ramification locus Z, the base curve C and the line bundle φ * M.
which is the image of T J E (Y ) under per Y, * , is spanned by rank one tensors where the vectors y 1 , . . ., y N form a basis of V /ξ = H 1,1 prim (Y ).Let π : V → V /ξ denote the projection.
Suppose that the same IVHS arises also from another surface X ′ .Then there is a vector ξ ′ ∈ V that arises from X such that (1 U ⊗ π)(U ⊗ ξ ′ ) is a subspace of U ⊗ V /ξ and lies in the subspace of U ⊗ V /ξ that is spanned by the x i ⊗ y i .However, by Lemma 6.2, the x i ⊗ y i are the only rank one tensors in U ⊗ V /ξ, so that (1 U ⊗ π)(U ⊗ ξ ′ ) = 0 and therefore ξ ′ is proportional to ξ.Consider the vectors x ′ i , y ′ i that arise from X ′ ; then the tensors x i ⊗ y i and x ′ i ⊗ y ′ i lie in the same vector space U ⊗ V /ξ, and then we can use Lemma 6.2 again to conclude the proof.
Compare the case of M q : over the non-hyperelliptic locus Schiffer variations give a cone structure in the tangent bundle, where at each point C of M q the corresponding cone is the cone over the bicanonical model of C, the generators of the cone map, under the period map, to tensors (quadratic forms) of rank 1 and, again, these account for all the rank 1 tensors in the image.
8 Recovering information from C and Z We shall show (Proposition 8.7) that φ 1 : C → Eℓℓ is generic and if φ 2 : C → Eℓℓ is another morphism such that Ram φ 1 = Ram φ 2 and φ * 1 M ∼ = φ * 2 M, then φ 1 and φ 2 are equivalent modulo the action of Aut Eℓℓ = G m provided that also there are sufficiently many points a i ∈ Z such that φ 1 (a i ) is isomorphic to φ 2 (a i ).Therefore an effective form of generic Torelli holds for Jacobian elliptic surfaces modulo this action of G m .
To begin, we rewrite some results of Tannenbaum [T] in the context of Deligne-Mumford stacks.Assume that S is a smooth 2-dimensional Deligne-Mumford stack (the relevant example will be S = Eℓℓ × Eℓℓ), that C is a smooth projective curve and that π : C → S is a morphism that factors as where D is a projective curve with only cusps, π ′ is birational and i induces surjections of henselian local rings at all points.(We shall say that "π is birational onto its image".)Then there is a conormal sheaf N ∨ D , a line bundle on D, which is generated by the pull back under i of the kernel So the adjunction formula (or duality for the morphism i : D → S) gives an isomorphism is the tangent space to the functor that classifies deformations of the morphism i : D → S that are locally trivial in the étale (or analytic) topology.
These conditions imply that there is a reduced effective divisor R on C such that T this is a line bundle on C. Define J ⊂ O S to be the Jacobian ideal of the ideal I D .The next result is a slight variant of Lemma 1.5 of [T].
, by the nature of a cusp, and Corollary 8.2 H 0 (C, N ′ π ) is isomorphic to the tangent space of the deformation functor that classifies those deformations of the morphism π : C → S where the length of the O C -module coker(π * Ω 1 S → Ω 1 C ) is preserved.PROOF: By the lemma, H 0 (C, N ′ π ) is the tangent space to locally trivial deformations of the morphism i : D → S. Since D has only cusps, a deformation of i is locally trivial if and only if it preserves the length of coker(π * Ω 1 S → Ω 1 C ).
(3) If q = 0 then these morphisms φ 2 form a family of dimension at most 3.
Then no such φ 2 exist provided that one of the following is true: q ≥ 2; q = 1 and r ≥ 1; q = 0 and r ≥ 4.

PROOF:
The further constraints on φ 2 imply that, in the notation of the proof of Lemma 8.3, the curve D has r cusps that lie on the diagonal of S. Therefore there are r further constraints on φ 2 and the corollary follows.
Lemma 8.5 If φ : C → Eℓℓ is generic then there is no non-trivial factorization through a curve of the composite morphism γ : C → P 1 j .
PROOF: Suppose that By the assumption of genericity, the divisor Z = Ram φ is reduced and its image B = γ * Z in P 1 j consists of distinct points, none of which equals j 4 or j 6 .Suppose α is branched over y ∈ P 1 j and y = j 4 , j 6 .Then γ −1 (y) ⊂ Z, so that Z → B is not 1-to-1.So α is branched only over j 4 , j 6 , and so is the cyclic cover of P 1 j that is branched at these two points and is of degree a.Since φ is étale over j 4 , and over j 6 it follows that a|2 and a|3, contradiction.
Suppose that ∆ is any irreducible curve of bidegree (1, 1) in P 1 j × P 1 j , let ∆ ′ denote the fibre product We now take V to be a miniversal deformation space of a surface X with r = r(q) fibres of type I 2 and Y a germ of the domain D at the period point of X.
Suppose that H ⊂ O(H 2 prim ( X, Z)) is the automorphism group of the polarized Hodge structure of the X and that G ⊂ H is the subgroup consisting of those elements of H that preserve the image of V in Y .Since H is finite, we can identify the germs Y and V , which are smooth, with their tangent spaces.In turn, V = H 2,0 ( X) ∨ ⊗ H 1,1 prim ( X) and H acts on both components of this tensor product.Let W = W (A r 1 ) denote the Weyl group generated by reflexions σ [ηa] in the classes [η a ] where a lies over j = ∞; then σ [ηa] lies in H, so that W ⊂ H. Also (±1) is a subgroup of H and acts trivially on D.
PROOF: Let g ∈ G.We know that the subspace V of Y is based by the rank 1 tensors ω ∨ a ⊗ [η a ] and that these are the only rank 1 tensors in V .Therefore g permutes the lines Cω ∨ a and so acts on the pair (C, Z), where C is embedded in P h−1 via |K C + L|.This action is trivial, by the previous lemma, and so g acts as a scalar on H 2,0 ( X).We can then take this scalar to be 1.Then g fixes each line C[η a ].The classes [η a ] are orthogonal and can be normalized by imposing the condition that [η a ] 2 = −2 for all a ∈ Z.This normalization is unique up to a choice of sign for each a.Since Z r is irreducible, one choice of sign, for a point in Z r , determines all the other signs, except when a lies over j = ∞.This last ambiguity is exactly taken care of by the Weyl group and so G ⊂ (±1) × W .
PROOF: The theorem follows from Corollary 9.4 and the fact that, at the level of miniversal deformation spaces, the morphism J E → J E is a quotient map by the relevant Weyl group.We conclude via Proposition 9.7 and the fact that (±1) acts trivially.
10 The variational Schottky problem for Jacobian surfaces The Schottky problem is that of determining the image of a moduli space under a period map.As explained by Donagi [D], there is a variational approach to this.For curves of genus q that are neither hyperelliptic, trigonal nor plane quintics, his approach leads (see p. 257 of [D]) to the statement that the image of the variational period map lies in the Grassmannian that parametrizes 3q − 3dimensional quotient spaces W of Sym 2 V , where V is a fixed q-dimensional vector space, and the kernel of Sym 2 V → W defines a smooth linearly normal curve C of genus q in P(V ).(It follows from this that the embedding C ֒→ P(V ) can be identified with the canonical embedding of C.) For Jacobian elliptic surfaces of geometric genus h and irregularity q we get something equally concrete.
Let Z → J E gen be the universal ramification locus, of degree 10 + 8(1 − q) over J E gen .The image of Z lies in a tensor product U ⊗ V where dim U = h and dim V = 10h + 8(1 − q).Projecting to P(U) = P h−1 leads to the following variational partial solution to the Schottky problem.The solution is only partial because this projection factors through the quotient stack J E gen /G m .
Recall that J E gen can be described as follows.
Suppose that P = P h+1−q → M q is the universal Picard variety of degree h + 1 − q line bundles L on a curve C of genus q.Then writing the equation of X in affine Weierstrass form, namely as y 2 = 4x 3 − g 4 x − g 6 , shows that g n ∈ H 0 (C, nL) and J E gen is birationally equivalent to a B(Z/2)gerbe over a bundle over P whose fibre is the quotient stack (H 0 (C, 4L) ⊕ H 0 (C, 6L) − {(0, 0)})/G m .
The action of G m on Eℓℓ leads to an action of G m on J E gen and the quotient J E gen /G m is birationally equivalent to a B(Z/2)-gerbe over the universal |4L| × |6L|-bundle over P.
Theorem 10.1 The image of the variational period map for Jacobian elliptic surfaces lies in the locus V = V h,q of zero-cycles Z in P h−1 such that (1) deg Z = 10h + 8(1 − q); (2) the intersection of the quadrics through Z is a curve C of genus q and degree h + q − 1; (3) the divisor Z on C is linearly equivalent to 10L + K C and the hyperplane class PROOF: This follows at once from the results of the previous section.We can make this more precise.First, recall the idea of transvectants: if N is a line bundle on a variety V over a field k and m, n ∈ Z, then there is a homomorphism

Multiple surfaces
An elliptic surface f : X → C is multiple if it is not simple; that is, if it has multiple fibres.As far as the period map is concerned there is not much to be said about these.Kodaira proved that, given f : X → C, there is a simple elliptic surface g : Y → C such that X is obtained from Y by logarithmic transformations; in particular, there is a finite subset S of C such that, if C 0 = C − S, X 0 = f −1 (C 0 ) and Y 0 = g −1 (C 0 ), then X 0 and Y 0 are isomorphic relative to C 0 , but are not usually bimeromorphic.In this situation the sheaves f * ω X/C and g * ω Y /C are isomorphic ( [Sch], p. 234).It follows that f * Ω 2 X and g * Ω 2 Y are isomorphic and then that the weight 2 Hodge structures on X and Y are isomorphic.
We deduce that the period map can detect neither the presence nor the location of multiple fibres on an elliptic surface.
Lemma 2.1 T Eℓℓ ∼ = M ⊗10 , deg M = 1/24 and deg T Eℓℓ = 5/12.PROOF: Quite generally the Picard group of P(a, b) is generated by O(1), which has degree 1/ab, and T P(a,b) is isomorphic to O(a + b), which then has degree (a + b)/ab.

Proposition 4. 3
For some choice of local co-ordinate z on C at a the morphism φ : C → Eℓℓ lifts to a morphism Φ : C → Eℓℓ in such a way that the restriction of Φ to U factors through the projection U → C − {a}.PROOF: Given a local co-ordinate s on Eℓℓ we have φ * s = z e for some local co-ordinate z on C. Then we define Φ on H by Φ * s = w e + e e−2 j=0 t e−j w j and on U we define Φ by composing φ with the projection U → C − {a}.
Theorem 4.11 (1) If a, b ∈ Z are distinct then, for all v ∈ L a and for all η b,k with k ∈ [2, e(b)], ∇ v η b,k is a linear combination of the classes {[η a,i ]} i∈[2,e(a)] and {[η b,j ]} j∈[2,e(b)] .In particular, ∇ v η b,k lies in F il 1 .(2) If a, b ∈ Z are distinct then the classes [η a,i ] and [η b,k ] are orthogonal.
by direct calculation, and, from the definition of N ′ D , we have N