2. Preliminaries on stacks and tangent spaces

Everything in the next two sections is well known; if it is not due to either Kas [Reference KasKas66] or Kodaira [Reference KodairaKod63], then it is folklore.

The stack $\overline {\mathcal {E}\ell \ell }$ is the Deligne–Mumford stack over ${\mathbb {C}}$ of stable generalized elliptic curves; that is, an $S$-point of $\overline {\mathcal {E}\ell \ell }$ is a flat projective morphism $Y\to S$ with a section $S_0$ contained in the relatively smooth locus of $Y\to S$ 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

\[ y^2=4x^3-g_4x-g_6, \]

where $g_4$ and $g_6$ are not both zero, so that $\overline {\mathcal {E}\ell \ell }$ is the quotient stack ${\mathbb {P}}(4,6)=({\mathbb {A}}^2-\{0\})/{\mathbb {G}}_m$, where ${\mathbb {G}}_m$ acts on ${\mathbb {A}}^2$ with weights $4,6$. Note that ${\mathbb {G}}_m$ acts on ${\mathbb {A}}^2$ via a homomorphism ${\mathbb {G}}_m\to {\mathbb {G}}_m^2$ (whose kernel is $\mu _2$) and the standard action of ${\mathbb {G}}_m^2$ on ${\mathbb {A}}^2$, so that there is a residual action of ${\mathbb {G}}_m$ on $\overline {\mathcal {E}\ell \ell }$. This exhibits ${\mathbb {G}}_m$ as the full automorphism group of $\overline {\mathcal {E}\ell \ell }$; cf. Theorem 8.1 of [Reference Behrend and NoohiBN06] and the calculation there on p. 139.

The geometric quotient $[\overline {\mathcal {E}\ell \ell }]$ of $\overline {\mathcal {E}\ell \ell }$ is the compactified $j$-line ${\mathbb {P}}^1_j$; if $\rho :\overline {\mathcal {E}\ell \ell }\to {\mathbb {P}}^1_j$ is the quotient morphism, then the automorphism group of each fibre of $\rho$ is ${\mathbb {Z}}/2$, except over $j=j_6=0$, where it is ${\mathbb {Z}}/6$, and over $j=j_4=1728$, where it is ${\mathbb {Z}}/4$. So $\deg \rho =1/2$. As is well known, it is possible to write down a generalized elliptic curve over the open locus $U$ of ${\mathbb {P}}^1_j$ defined by $j\ne 0,1728$, so that there is a section of $\rho$ over $U$. Moreover, $\rho ^{-1}(U)$ is isomorphic to $U\times B({\mathbb {Z}}/2)$, but there is no global section of $\rho$.

There are two obvious line bundles on $\overline {\mathcal {E}\ell \ell }$: 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 $T_{\overline {\mathcal {E}\ell \ell }}$.

The objects of the stack $\overline {\mathcal {M}}_1$ are stable curves of genus $1$; the geometric fibres are isomorphic to stable generalized 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 $\mathcal {C}\to \overline {\mathcal {M}}_1$ and $\mathcal {E}\to \overline {\mathcal {E}\ell \ell }$ denote the universal objects and let $\mathcal {G}\to \overline {\mathcal {E}\ell \ell }$ denote the Néron model of $\mathcal {E}\to \overline {\mathcal {E}\ell \ell }$, so that $\mathcal {G}$ is the open substack of $\mathcal {E}$ obtained by deleting the singular point of the fibre over $j=\infty$.

The next result is due to Altman and Kleiman [Reference Altman and KleimanAK80], 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.

Sending $\mathcal {C}\to \overline {\mathcal {M}}_1$ to $\widetilde {G}\to \overline {\mathcal {M}}_1$ defines a morphism $\pi : \overline {\mathcal {M}}_1\to \overline {\mathcal {E}\ell \ell }$.

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.

Note that $G\to \overline {\mathcal {M}}_1$ is isomorphic to the pull back under $\pi$ of the Néron model $\mathcal {G}\to \overline {\mathcal {E}\ell \ell }$.

As already remarked, a general simple surface $f:X\to C$ determines, and is determined by, a morphism $F=F_f:C\to \overline {\mathcal {M}}_1$ that is unramified over $j=\infty$. Let $\phi =\phi _f=\pi \circ F:C\to \overline {\mathcal {E}\ell \ell }$ denote the composite, so that the induced Jacobian elliptic surface is the compactified relative automorphism group scheme of $f:X\to C$.

For example, if $f:X\to {\mathbb {P}}^1$ is a primary Hopf surface, then $\phi _f$ is constant: the relative automorphism group scheme is a constant relative group scheme $E\times {\mathbb {P}}^1\to {\mathbb {P}}^1$.

We next consider various tangent spaces.

Regard $B\mathcal {G}$ as the quotient of $\overline {\mathcal {E}\ell \ell }$ by $\mathcal {G}$. Since $\overline {\mathcal {M}}_1$ is isomorphic to $B\mathcal {G}$, locally on $\overline {\mathcal {E}\ell \ell }$, the tangent complex $T^\bullet _{\overline {\mathcal {M}}_1}$ is a $2$-term complex, which is obtained by descending a $2$-term complex on $\overline {\mathcal {E}\ell \ell }$. In degree $0$, this complex is $T_{\overline {\mathcal {E}\ell \ell }}$, in degree $-1$ it is the adjoint bundle $\operatorname {Ad} \mathcal {G}$ and the differential is the derivative of the action of $\mathcal {G}$ on $\overline {\mathcal {E}\ell \ell }$.

Since this action is trivial, the differential in $T^\bullet _{\overline {\mathcal {M}}_1}$ is zero. Moreover, since $\mathcal {G}$ has no characters (it is generically an elliptic curve) it follows that $T^\bullet _{\overline {\mathcal {M}}_1}$ is quasi-isomorphic to the pull back of the complex $\operatorname {Ad} \mathcal {G}[1]\oplus T_{\overline {\mathcal {E}\ell \ell }}[0]$ on $\overline {\mathcal {E}\ell \ell }$.

Note that $(\operatorname {Ad}\mathcal {G})^\vee$ is exactly the line bundle $M$ of modular forms of weight $1$.

Now fix a point $X$ of $\mathcal {SE}$. That is, we fix a general simple elliptic surface $f:X\to C$. This equals the datum of a morphism $F:C\to \overline {\mathcal {M}}_1$. Set $\phi =\pi \circ F:C\to \overline {\mathcal {E}\ell \ell }$. Then $F^*T^\bullet _{\overline {\mathcal {M}}_1}$ is quasi-isomorphic to $\phi ^*\operatorname {Ad} \mathcal {G}[1]\oplus \phi ^*T_{\overline {\mathcal {E}\ell \ell }}$. There is a distinguished triangle

\[ T_C\to F^*T^\bullet_{\overline{\mathcal{M}}_1}\to K^\bullet, \]

where $K^\bullet$ is a $2$-term complex of coherent sheaves on $C$, $K^{-1}=\phi ^*\operatorname {Ad} \mathcal {G}$, $K^0$ is the skyscraper sheaf $\mathrm {coker} (T_C\to \phi ^*T_{\overline {\mathcal {E}\ell \ell }})$ and the differential in $K^\bullet$ is zero.

Let $Z$ denote the ramification divisor $Z=\operatorname {Ram}_\phi =\operatorname {Ram}_F$ on $C$. Then $K^0$ is an invertible sheaf on $Z$, so that there is a non-canonical isomorphism $K^0\mathop {\to }\limits ^{\cong }\mathcal {O}_Z$.

The distinguished triangle just mentioned gives an exact sequence

\begin{align*} {0}&\to H^0(C,T_C) \to{\mathbb{H}}^0(C,F^*T^\bullet_{\overline{\mathcal{M}}_1})\to{\mathbb{H}}^0(C,K^\bullet)\\ &\to {H^1(C,T_C)} \to {\mathbb{H}}^1(C,F^*T^\bullet_{bsm_1})\to{\mathbb{H}}^1(C,K^\bullet)\to 0. \end{align*}

Then $H^2(C,\phi ^*\operatorname {Ad} G)=0$ and $H^1(C,\phi ^*T_{\overline {\mathcal {E}\ell \ell }})=0$, since

\[ \deg\phi^*T_{\overline{\mathcal{E}\ell\ell}}=10\chi(X,\mathcal{O}_X)>2q-2, \]

from the assumption (1.2), so that ${\mathbb {H}}^1(C,K^\bullet )={\mathbb {H}}^1(C,F^*T^\bullet _{\overline {\mathcal {M}}_1})=0$ and there is an exact commutative diagram

in which the two middle rows are canonically split.

Suppose that $f:X\to C$ is a point of $\mathcal {JE}^{\rm gen}$ and defines $\phi :C\to \overline {\mathcal {E}\ell \ell }$. Set $Z=\operatorname {Ram}_\phi$.

Since $Z$ is $0$-dimensional and $K^0$ is an invertible sheaf on $Z$, $H^0(Z,K^0)$ is non-canonically isomorphic to $H^0(Z,\mathcal {O}_Z)$.

We shall usually write $\phi ^*M=L=f_*\omega _{X/C}$. Then $\phi ^*T_{\overline {\mathcal {E}\ell \ell }}\cong L^{\otimes 10}$, so that

\[ \mathcal{O}_C(\operatorname{Ram}_\phi)\cong \phi^*T_{\overline{\mathcal{E}\ell\ell}}\otimes T_C^\vee \cong L^{\otimes 10}\otimes\mathcal{O}_C(K_C). \]

At this point we give a slight refinement of Saito's local Torelli theorem. The argument is essentially his.

Proof. From the main result of [Reference SaitoSai83] it is enough to consider the situation where the $j$-invariant is constant. Following [Reference SaitoSai83] it is enough to show that the natural homomorphisms

\[ \mu_1: H^0(C,f_*\Omega^2_X\!)\otimes H^1(C,f_*\Omega^1_X\!) \to H^1(C,f_*(\Omega^1_X\otimes\Omega^2_X\!)) \]

and

\[ \mu_2: H^0(C,f_*\Omega^2_X\!)\otimes H^0(C,R^1f_*\Omega^1_X\!) \to H^0(C,R^1f_*(\Omega^1_X\otimes\Omega^2_X)) \]

are surjective. Recall that $K_X\sim f^*(K_C+L)$, so that $f_*(\Omega ^1_X\otimes \Omega ^2_X\!)\cong \mathcal {O}_C(K_C+L)\otimes f_*\Omega ^1_X$, and that $\vert K_C+L\vert$ has no base points.

Lemma 2.11 If $\mathcal {F},\mathcal {G}$ are coherent sheaves on a $1$-dimensional projective scheme $C$ over a field $k$ and if $\mathcal {F}$ is generated by $H^0(C,\mathcal {F})$, then the natural multiplication

\[ H^0(C,\mathcal{F})\otimes_k H^1(C,\mathcal{G})\to H^1(C,\mathcal{F}\otimes_{\mathcal{O}_C} \mathcal{G}) \]

is surjective.

Proof. There is an exact sequence

\[ H^0(C,\mathcal{F})\otimes_k\mathcal{O}_C \to \mathcal{F}\to 0; \]

tensoring this with $\mathcal {G}$ gives an exact sequence

\[ H^0(C,\mathcal{F})\otimes_k\mathcal{G}\to \mathcal{F}\otimes \mathcal{G}\to 0. \]

Taking $H^1$ of this sequence gives the result.

In particular, taking $\mathcal {F}=\Omega ^1_C\otimes L=f_*\Omega ^2_X$ and $\mathcal {G}=f_*\Omega ^1_X$ shows that $\mu _1$ is surjective.

Now consider $\mu _2$. Set $A=\sum _1^r a_i$, the critical subset of $C$. The exact sequences (4.17) and (4.18) of [Reference SaitoSai83] are

\[ 0\to \Omega^1_C\to f_*\Omega^1_X \to\mathcal{O}_C(L-A)\to 0 \]

and

\[ 0\to\mathcal{O}_C(K_C+A-L)\to R^1f_*\Omega^1_X \to \mathcal{O}_C\oplus \mathcal{T}^1\to 0, \]

where $\mathcal {T}^1$ is a skyscraper sheaf. The second sequence then gives

\[ 0\to\mathcal{O}_C(K_C+A-L+\delta)\to R^1f_*\Omega^1_X \to \mathcal{O}_C\to 0, \]

where $\delta \ge 0$, via the process of saturating a subsheaf. Since $\deg A>\deg L$, by assumption, this last sequence splits and

\[ R^1f_*\Omega^1_X=\mathcal{O}_C\oplus \mathcal{O}_C(K_C+A-L+\delta). \]

From its definition, $\mu _2$ is then the direct sum

\[ \mu_2=\mu_2'\oplus\mu_2'' \]

of multiplication maps

\[ \mu_2':H^0(C,\mathcal{O}_C(K_C+L))\otimes{\mathbb{C}} \to H^0(C,\mathcal{O}_C(K_C+L)) \]

and

\[ \mu_2'':H^0(C,\mathcal{O}_C(K_C+L))\otimes H^0(C,\mathcal{O}_C(K_C-L+A+\delta)) \to H^0(C,\mathcal{O}_C(2K_C+A+\delta)). \]

The first of these is obviously surjective, while the surjectivity of the second follows from [Reference MumfordMum70, Theorem 6, p. 52] and the facts that $\deg (K_C+L)\ge 2q$ and $\deg (K_C-L+A+\delta )\ge 2q+1$.

3. The comparison between $\mathcal {SE}^{\rm gen}$ and $\mathcal {JE}^{\rm gen}$

The morphism $\pi :\overline {\mathcal {M}}_1\to \overline {\mathcal {E}\ell \ell }$ defines a morphism $\Pi :\mathcal {SE}^{\rm gen}\to \mathcal {JE}^{\rm gen}$. If we fix a point $f:X\to C$ of $\mathcal {JE}^{\rm gen}$, then $\Pi ^{-1}(X)$ is identified with $H^1(C,\mathcal {H})$, where $\mathcal {H}\to C$ is the Néron model of $X\to C$.

The sheaves $L^\vee$ and $\mathcal {H}$ are group schemes over $C$. Define $\mathcal {F}=R^1f_*{\mathbb {Z}}$; this is a constructible sheaf of ${\mathbb {Z}}$-modules on $C$ that is of generic rank $2$.

The following results are due, in essence, to Kodaira. In particular, Proposition 3.2 is a variant of [Reference KodairaKod63, Theorem 11.7, p. 1341]; there he proves only that $H^2(C,\mathcal {F})$ is finite, but he does not assume that $f:X\to C$ is general.

Lemma 3.1 There is a short exact sequence

\[ 0\to\mathcal{F}\to L^\vee\to\mathcal{H}\to 0 \]

of sheaves of commutative groups on $C$.

Proof. It is enough to show that $H^2(C,\mathcal {F})=0$.

For any ring $A$, there is a Leray spectral sequence

\[ E^{pq}_{2,A}=H^p(C,R^qf_*A)\Rightarrow H^{p+q}(X,A). \]

Since the fibres of $f$ are irreducible curves and $X\to C$ has a section, the hypotheses of Théorème 1.1 of [Reference Deligne and KatzDK, XVIII] are satisfied, so that this degenerates at $E_2$. Take $A={\mathbb {Z}}$; then there is a short exact sequence

\[ 0\to H^2(C,\mathcal{F})\to H^3(X,{\mathbb{Z}})\to H^1(C,{\mathbb{Z}})\to 0. \]

Thus, $H^2(C,\mathcal {F})$ is identified with the torsion subgroup $\operatorname {Tors} H^3(X,{\mathbb {Z}})$ of $H^3(X,{\mathbb {Z}})$.

Suppose that $\ell$ is a prime dividing the order of $\operatorname {Tors} H^3(X,{\mathbb {Z}})$. Since $H^4(X,A)$ is isomorphic to $A$, it follows from taking cohomology of the short exact sequence

\[ 0\to{\mathbb{Z}}\mathop{\to}^{{.\ell}}{\mathbb{Z}}\to{\mathbb{Z}}/\ell\to 0 \]

that $H^3(X,{\mathbb {Z}}/\ell )\cong H^3(X,{\mathbb {Z}})\otimes {\mathbb {Z}}/\ell$. In addition, Poincaré duality gives an isomorphism $H^3(X,{\mathbb {Z}}/\ell )\to H^1(X,{\mathbb {Z}}/\ell )^\vee,$ where ${}^\vee$ denotes the dual ${\mathbb {Z}}/\ell$-vector space.

The spectral sequence $E^{pq}_{r,{\mathbb {Z}}/\ell }$ shows that $\beta :H^1(C,{\mathbb {Z}}/\ell )\to H^1(X,{\mathbb {Z}}/\ell )$ is injective, so if it is not surjective, then the map $i^*: H^1(X,{\mathbb {Z}}/\ell )\to H^1(\sigma,{\mathbb {Z}}/\ell )$ induced by the inclusion $i:\sigma \hookrightarrow X$ of the zero section is not injective. Then there is an étale ${\mathbb {Z}}/\ell$-cover $\alpha :\widetilde X\to X$ that is split over $\sigma$. Thus, $\widetilde X\to C$ is elliptic and has a section $\widetilde {\sigma }$ such that $N_{\widetilde {\sigma },\widetilde X}\cong N_{\sigma /X}$. However,

\[ \deg N_{\widetilde{\sigma},\widetilde X} =-\chi(\widetilde X,\mathcal{O}_{\widetilde X})= -\ell\chi(X,\mathcal{O}_X)=\ell\deg N_{\sigma/X}. \]

Thus, $\beta$ is an isomorphism, so that $H^3(X,{\mathbb {Z}}/\ell )\cong ({\mathbb {Z}}/\ell )^{2q}$ and, therefore, $H^3(X,{\mathbb {Z}})$ is torsion-free.

Let $\mathcal {SE}_{h,q}$ denote the substack of $\mathcal {SE}$ that consists of surfaces whose geometric genus is $h$ and whose irregularity is $q$. This is a union of connected components of $\mathcal {SE}$.

The next result is an immediate corollary of Proposition 3.2.

Now suppose that $Y\in \Pi ^{-1}(X)$. According to [Reference KodairaKod63, Theorem 11.5, p. 1338], $Y$ is algebraic if and only if it defines a torsion element of $H^1(C,\mathcal {H})$.

Proof. We must show that the image of $H^1(C,R^1f_*{\mathbb {Z}})\otimes {\mathbb {Q}}$ in $H^1(C,R^1f_*\mathcal {O}_X\!)$ is dense.

Let $\xi ^\perp$ denote the orthogonal complement of $\xi$ in $H^2(X,{\mathbb {Z}})$. Then, via the Leray spectral sequence, there is a commutative square

where the vertical arrows are isomorphisms. Then $\beta$ 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

Fix a curve $C$, a point $a\in C$, a local coordinate $z$ on $C$ at $a$, an integer $e\ge 2$ and a real number $\delta$ such that $0<\delta \ll 1$ and the region $\vert z\vert <\delta$ is an open disc $D=D_z$ in $C$. We begin by recalling, from [Reference GarabedianGar49, p. 443], the construction of certain variations that we shall call Schiffer variations: these are families $\pi :\mathcal {C}\to \Delta ^{e-1}=\Delta ^{e-1}_{\underline {t}}$ over $(e-1)$-dimensional polydiscs with coordinates $\underline {t}=(t_2,\ldots,t_{e})$ whose closed fibre $\pi ^{-1}(\underline {0})$ is $C$. They are constructed as follows.

In ${\mathbb {C}}^{e-1}_{\underline {t}}$ take the polydisc $\Delta ^{e-1}_{\underline {t}}$ defined by $\vert t_j\vert <\delta ^{4e}$ for all $j$. In $C\times \Delta ^{e-1}_{\underline {t}}$ take the complement $U$ of the closed subset $\bigcup _j(\vert z\vert ^{4e}\le \vert t_j\vert )$. Thus, $U$ is a thickening of the punctured curve $C-\{a\}$. Then put $G=U\cap (\vert z\vert <\delta )$.

In ${\mathbb {C}}^2_{z,w}\times \Delta ^{e-1}_{\underline {t}}$ take the subset $F$ defined by

\[ \vert t_j\vert < \vert z\vert^{4e} \ \forall j,\ \vert z\vert <\delta,\ \vert w-z\vert < \vert z\vert ^{2e}\ \textrm{and}\ w^e+e\sum_0^{e-2} t_{e-j}w^{j}=z^e. \]

Then there are projections $p:F\to U$ and $q:F\to {\mathbb {C}}_w\times \Delta ^{e-1}_{\underline {t}}$.

Proof. It is enough to show that they are unramified and separate points.

The ramification locus of $p$ is defined by

\[ w^{e-1}+\sum jt_{e-j}w^{j-1}=0. \]

Since $\vert t_{e-j}\vert <\vert z\vert ^{4e}$ this gives $\vert w\vert <\vert z\vert ^4$. Then $\vert z\vert -\vert z\vert ^{2e}<\vert z\vert ^4$, so that $z=0$ and $p$ is unramified.

To check the separation of points, suppose that

\[ w_\alpha^e+e\sum t_{e-j}w_\alpha^j-z^e=0 \]

for $\alpha =1,2$ and $w_1\ne w_2$. Then

\[ \prod_{r=1}^{e-1}(w_1-\zeta_e^r w_2)=e\sum_{j=1}^{e-2} t_{e-j}\prod_{s=1}^{j-1}(w_1-\zeta_j^sw_2), \]

where, for any integer $n$, $\zeta _n$ is a primitive $n$th root of unity. Since $\vert w_1-z\vert <\vert z\vert ^{2e}$ and $\vert \zeta _j^s w_2-\zeta _j^s z\vert <\vert z\vert ^{2e}$ it follows that

\[ \vert w_1-\zeta_j^s w_2\vert < 2\vert z\vert^{2e}+\vert z-\zeta_j^sz\vert =2\vert z\vert^{2e}+\vert 1-\zeta_j^s\vert\vert z\vert< 4\vert z\vert. \]

Thus,

\[ \biggl\vert e\sum_{j=1}^{e-2} t_{e-j}\prod_{s=1}^{j-1}(w_1-\zeta_j^sw_2)\biggr\vert \le 4e(e-2)\vert z\vert^{4e+1}. \]

On the other hand, $\vert w_1-z\vert <\vert z\vert ^{2e}$ and $\vert \zeta _e^r-\zeta _e^rz\vert <\vert z\vert ^{2e},$ so that

\[ \vert w_1-\zeta_e^r w_2\vert\ge \vert z-\zeta_e^rz\vert- 2\vert z\vert^{2e} \ge \lambda\vert z\vert, \]

where $\lambda =\vert \sin (2\pi /e)\vert /2$. Thus,

\[ \biggl\vert \prod_{r=1}^{e-1}(w_1-\zeta_e^rw_2)\biggr\vert \ge\lambda^{e-1}\vert z\vert^{e-1} \]

and, therefore,

\[ \lambda^{e-1}\le 4e(e-2)\vert z\vert^{3e+2}. \]

This contradiction proves the result for $p$.

The argument for $q$ is similar but easier, so we omit it.

For each $\underline {t}\in \Delta ^{e-1}_{\underline {t}}$ the intersection $q(F)\cap ({\mathbb {C}}_w\times \{\underline {t}\})$ is an annulus $A_{\underline {t}}$ in ${\mathbb {C}}_w\times \{\underline {t}\}$ that surrounds zero. Let $R_{\underline {t}}$ denote the open simply connected region in ${\mathbb {C}}_w\times \{\underline {t}\}$ that contains $0$ and has the same outer boundary as $A_{\underline {t}}$, and put $H=\bigsqcup _{\underline {t}}R_{\underline {t}}$. Thus, $H$ is a tubular neighbourhood of $\{0\}\times \Delta ^{e-1}_{\underline {t}}$ in ${\mathbb {C}}_w\times \Delta ^{e-1}_{\underline {t}}$ and $q(F)\subseteq H$.

Define $\mathcal {C}$ to be the result of glueing $U$ and $H$ together along $F$ via the maps $p$ and $q$. This is Hausdorff and, after shrinking $\Delta ^{e-1}_{\underline {t}}$ if necessary, the morphism $\pi :\mathcal {C}\to \Delta ^{e-1}_{\underline {t}}$ is proper and is the morphism that we sought. This is sometimes expressed by saying that $\mathcal {C}_{\underline {t}}$ is constructed from $C$ by deleting a small $z$-disc around $a$ and glueing in a $w$-disc, where $w$ is defined implicitly by $w^e+e\sum _0^{e-2} t_{e-j}w^{j}=z^e.$

Until after Theorem 4.11 we fix a point $f:X\to C$ of $\mathcal {JE}^{\rm gen}$ and a point $a$ in the ramification divisor $Z=\operatorname {Ram}_\phi$ of the classifying morphism $\phi :C\to \overline {\mathcal {E}\ell \ell }$. In particular, $\phi$ is unramified over $j=\infty$. Put $E_a=f^{-1}(a)$ and denote by $e=e(a)$ the ramification index at $a$ of $\phi$, 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 $X$.

Proof. Given a local coordinate $s$ on $\overline {\mathcal {E}\ell \ell }$ we have $\phi ^*s=z^e$ for some local coordinate $z$ on $C$. Then we define $\Phi$ on $H$ by

\[ \Phi^*s=w^e+e\sum_{j=0}^{e-2}t_{e-j}w^{j} \]

and on $U$ we define $\Phi$ by composing $\phi$ with the projection $U\to C-\{a\}$.

Any Jacobian deformation of the surface $X$ determines, for each point $a\in Z$, a deformation of the finite scheme $V_{e(a)}=\textbf {Spec}{\mathbb {C}}[t]/(t^{e(a)})$, so that there is a morphism of local analytic deformation spaces $\Psi :\operatorname {Def}_X\to \prod _{a\in Z}\operatorname {Def}_{V_{e(a)}}.$ Recall that $\operatorname {Def}_{V_{e(a)}}$ is smooth of dimension $e(a)-1$.

Thus, we have morphisms $\mathcal {X} \mathop {\to }\limits ^{F}\mathcal {C}\to \Delta ^{e-1}_{\underline {t}}$ and $F:\mathcal {X}\to \mathcal {C}$ is a family, parametrized by $\Delta ^{e-1}_{\underline {t}}$, of Jacobian elliptic surface whose fibre over $0\in \Delta _{\underline {t}}^{e-1}$ is $f:X\to C$.

Fix a suitable basis of $H_2(X,{\mathbb {Z}})$, which we identify with $H_2(\mathcal {X}_{\underline {t}},{\mathbb {Z}})$. Then there are holomorphic $(e+1)$-forms $\Omega ^{(1)},\ldots,\Omega ^{(h)}$ on $\mathcal {X}$ such that the residues

\[ \omega^{(j)}(\underline{t})= \operatorname{Res}_{\mathcal{X}_{\underline{t}}} \biggl( {\begin{array}{c}{\Omega^{(j)}}\\ {\underline{t}}\\ \end{array}}\biggr) \]

form a normalized basis of $H^0(\mathcal {X}_{\underline {t}},\Omega ^2_{\mathcal {X}_{\underline {t}}})$ for all $\underline {t}$. In particular, there are $2$-cycles $A_1,\ldots,A_h$ on $\mathcal {X}_{\underline {t}}$ such that $\int _{A_i}\omega ^{(j)}(\underline {t})=\delta _i^j$, the Kronecker delta.

Since the line bundle $\Omega ^{e+1}_{\mathcal {X}}$ pulls back from a line bundle on $\mathcal {C}$, we can expand $\Omega ^{(j)}$ as

\[ \Omega^{(j)}=(-1)^{e-1}\sum_{p\ge0,\underline{q}\ge\underline{0}} b_{p,\underline{q}}^{(j)} w^p\underline{t}^{\underline{q}}\, dw\wedge \,d{\underline{t}}\wedge \,dv, \]

where $v$ is a fibre coordinate.

We substitute this into the expansion of $\Omega ^{(j)}$. Since

\[ \biggl(1-\sum t_iz^{-i}\biggr)^p\biggl(1+\sum(i-1)t_iz^{-i}\biggr)= 1+\sum(i-p-1)t_iz^{-i} \]

modulo $(\underline {t})^2$, we get

(4.6)\begin{equation} \omega^{(j)}(\underline{t})=\sum b_{p,\underline{q}}^{(j)} z^p \biggl(1+\sum(i-p-1)t_iz^{-i}\biggr)\underline{t}^{\underline{q}}\,dz\wedge \,dv. \end{equation}

modulo $(\underline {t})^2$.

Next, write $\omega ^{(j)}(t)=\omega ^{(j)}+ \sum _{i=0}^{e-2} t_{e-i}\eta _{e-i}^{(j)}$ modulo $(\underline {t})^2$, where

\[ \omega^{(j)}=\omega^{(j)}\vert_{\underline{t}=\underline{0}}\quad \text{and}\quad \eta_i^{(j)}=\frac{\partial}{\partial t_i}\omega^{(j)}\vert_{\underline{t}=\underline{0}}. \]

Moreover, every $2$-cycle $\gamma$ on $X$ that is disjoint from $E$ is identified, via a $C^\infty$ collapsing map, with a $2$-cycle on $\mathcal {X}_{\underline {t}}$ and

(4.7)\begin{equation} \int_\gamma\omega^{(j)}(t)=\int_\gamma\omega^{(j)}+\sum_i t_i\int_\gamma\eta_i^{(j)}\quad \text{for all such } \gamma. \end{equation}

Consider the class $[\omega ^{(j)}(t)]\in H^2(\mathcal {X}_{\underline {t}},{\mathbb {C}})$. Let $\mathcal {E}\subset \mathcal {X}_{\underline {t}}$ be a fibre. Then $[\omega ^{(j)}(t)]\in \mathcal {E}^\perp$. Moreover, from the exact sequence

\[ {\mathbb{Z}}\xi={\mathbb{Z}}[E_a]\to H^2(X,{\mathbb{Z}})\to H^2(X-E_a,{\mathbb{Z}}) \]

and the formula (4.7) it follows that $\eta _i^{(j)}$ defines a class

\[ [\eta_i^{(j)}]\in \xi^\perp/{\mathbb{Z}}\xi=H^2_{\rm prim}(X) \]

such that

\[ \frac{\partial}{\partial t_i} \big([\omega^{(j)}(t)]\big)\vert_{t=0}=[\eta_i^{(j)}]\pmod{{\mathbb{Z}}\xi}. \]

Griffiths transversality shows that, in fact, $[\eta _i^{(j)}]$ lies in ${\it Fil}^1={\it Fil}^1H^{2}_{\rm prim}(X)$. (We let ${\it Fil}^i$ refer to the $i$’th piece of the Hodge filtration of $H^{2}_{\rm prim}(X)$.)

We have constructed, for each $a\in Z$ of ramification index $e(a)=e$, an $(e-1)$-parameter variation $\mathcal {X} \to \Delta ^{e-1}$ of $X$ such that the tangent space $T_{\Delta ^{e-1}}(\underline {0})$ 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 $v\in L_a$ consider the derivative

\[ \nabla_v: H^{2,0}(X)={\it Fil}^2\to {\it Fil}^1. \]

Let

\[ \overline{\nabla}_v:H^{2,0}\to H^{1,1}_{\rm prim}(X) \]

denote the composite of $\nabla _v$ with the projection ${\it Fil}^1\to H^{1,1}_{\rm prim}(X)$. If $L_a$ is of length $1$, then we write $\nabla _a$ rather than $\nabla _v$.

Assume that the geometric genus $h$ of $X$ is not zero. In addition, given $\omega \in H^0(X,\Omega ^2_X)$ and $P\in C$, put $\omega (P)=({\omega }/{dz\wedge dv})(Q)$ for an arbitrary point $Q\in E_P$ and identify $\omega$ with the pullback of a tensor on $C$. Denote by $\sigma \subset X$ the given zero-section of $f:X\to C$.

Proof. Up to now we have identified the tangent space $T_{\mathcal {JE}}(X)$ with a line bundle on $Z$; we can also identify it with $H^1(X,\Theta _X(-\log \sigma ))$. Since $\Theta _X(-\log \sigma )\otimes \Omega ^2_X$ is isomorphic to $\Omega ^1_X(\log \sigma )(-\sigma )$ we get a short exact sequence

\[ 0\to \Theta_X(-\log\sigma)\to \Omega^1_X(\log\sigma)(-\sigma) \to \bigoplus_{P\in(\omega)_0}\mathcal{F}_P\to 0, \]

where $\mathcal {F}_P$ is a rank 2 vector bundle on $E_P$ that fits into a short exact sequence

\[ 0\to\mathcal{O}_{E_P}(-\sigma_P)\to\mathcal{F}_P\to\mathcal{O}_{E_P}\to 0. \]

The coboundary map $H^0(E_P,\mathcal {O}_{E_P})\to H^1(E_P,\mathcal {O}_{E_P}(-\sigma _P))$ is identified with the Kodaira–Spencer map, so is an isomorphism from our assumption about $(\omega )_0$. Thus, $H^0(E_P,\mathcal {F}_P)=0$ and then the homomorphism

\[ H^1(X, \Theta_X(-\log\sigma))\to H^1(X, \Omega^1_X(\log\sigma)(-\sigma)) \]

is injective. This homomorphism factors through the cup product with which we are concerned, and the proposition is proved.

Write $b_{p,\underline {0}}^{(j)}=b_p$. Then $\omega ^{(j)}=\sum b^{(j)}_{p\ge 0}z^pdz\wedge dv$ and $b^{(j)}_{0}=\omega ^{(j)}(a)$.

Proof. This follows from further consideration of the expansion (4.6). To begin, write $b^{(j)}_{p,\underline {0}}=b^{(j)}_0$. Then we get

\[ \omega^{(j)}(\underline{t})= \sum_{p,\underline{q}}b^{(j)}_{p,\underline{q}}z^p \biggl(1+\sum_i(i-p-1)t_iz^{-i}\biggr)\,dz\wedge dv, \]

modulo $(\underline {t}).H^0(X,\Omega ^2_X)$, so that

\[ \omega^{(j)}=\sum_{p}b^{(j)}_{p}z^p\,dz\wedge dv\quad \textrm{and}\quad \eta^{(j)}_i=\sum_{p} (i-p-1)b^{(j)}_{p}z^{p-i}\,dz\wedge dv, \]

where the second equality holds modulo $H^0(X,\Omega ^2_X)$. Observe that

\[ \dim H^0(X,\Omega^2_X(iE_a))\le h+i\quad \textrm{and}\quad H^0(X,\Omega^2_X(E_a))_{{\rm 2nd\ kind}}=H^0(X,\Omega^2_X\!), \]

so that $\dim H^0(X,\Omega ^2_X(iE_a))_{\rm 2nd\ kind}\le h+i-1.$ 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 $\underline {\omega }$ denote the vector $[\omega ^{(1)},\ldots,\omega ^{(h)}]$ and $\underline {\omega }_{(i)}$ its $i$th derivative with respect to $z$.

Theorem 4.10 For each $a\in Z$ and each $k=2,\ldots,e(a)$ there is an explicit meromorphic $2$-form

\[ \eta_{a,k}\in H^0(X,\Omega^2_X(kE_a))_{{\rm 2nd\ kind}} \]

such that $[\eta _{a,k}]$ lies in ${\it Fil}^1$ and the image of $\overline {\nabla }_{\partial /\partial t_k}:{\it Fil}^2\to {\it Fil}^1$ is spanned by the classes $[\eta _{a,2}],\ldots, [\eta _{a,k}]$. The kernel of $\overline {\nabla }_{\partial /\partial t_k}$ contains $H^0(X,\Omega ^2_X(-(k-1)E))$.

Moreover, given the identification $H^0(X,\Omega ^2_X)\cong H^0(X,\Omega ^2_X\!)^\vee$ provided by the basis $\underline {\omega }$, $\overline {\nabla }_{\partial /\partial t_k}$ is, as an element of $H^0(X,\Omega ^2_X\!)\otimes {\it Fil}^1$, a linear combination of the tensors $\underline {\omega }(a)\otimes [\eta _{a,k}],\ldots,\underline {\omega }_{(k-2)}(a)\otimes [\eta _{a,2}]$.

Finally, if $h+q-1\ge e(a)$, then $\overline {\nabla }_{\partial /\partial t_k}$ is of rank $k-1$.

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 $\underline {\omega }(a),\ldots, \underline {\omega }_{(e(a)-2)}$ are linearly independent. This follows from the cohomology of the exact sequence

\[ 0\to\mathcal{O}_C(K_C+L-(e(a)-1)a)\to \mathcal{O}_C(K_C+L)\to \mathcal{O}_{(e(a)-1)a}(K_C+L)\to 0 \]

and the assumption that $h+q-1\ge e(a)$.

Proof. (1) We use the same family $\mathcal {X}\to \mathcal {C}\to \Delta ^{e(a)-1}$ corresponding to the point $a$ as before. This variation is constant outside a small neighbourhood of $a$ and the morphism to $\overline {\mathcal {E}\ell \ell }$ is also constant outside this neighbourhood. Thus, the ramification locus $Z$ is also constant there.

The point $b$ moves in a family of points $b(\underline {t})\in \mathcal {C}_{\underline {t}}$, each $b(\underline {t})$ being of constant multiplicity $e(b)-1$ in the ramification divisor $Z_{\underline {t}}\subset \mathcal {C}_{\underline {t}}$. The fibre $E_b$ in $X$ moves in a family $\mathcal {E}_{b(\underline {t})}$. Then, for each $k\in [2, e(b)]$,

\[ \eta_{b(\underline{t}),k} =\eta_{b,k} + \sum t_i\nabla_{\partial/\partial t_i}\eta_{b(\underline{t}),k} \]

modulo $t^2$. Similarly to what we did before, we write

\[ \eta_{b(\underline{t}),k} = \operatorname{Res}_{\mathcal{X}_{\underline{t}}} \biggl( \begin{array}{c}H_{b(\underline{t}),k}\\ {\underline{t}}\\ \end{array} \biggr) \]

for some meromorphic $(e(a)+1)$-form

\[ H_{b(\underline{t}),k}\in H^0(\mathcal{X}, \Omega^{e(a)+1}_{\mathcal{X}}(k\mathcal{E}_{b(t)})). \]

Then the same kind of calculation in terms of a power series expansion as before shows that $\nabla _{\partial /\partial t_i}\eta _{b(\underline {t}),k}$ is an element of $H^0(X,\Omega ^2_X(iE_a + kE_b))$ whose residues along both $E_a$ and $E_b$ are zero. Therefore, $\nabla _a(\eta _{b,k})$ is, modulo $H^{2,0}(X)$, a linear combination as described.

(2) Choose an element $\omega$ of $H^{2,0}(X)$ that does not vanish along $E_a$. Thus, $\nabla _{\partial /\partial t_i}\omega$ is a non-zero multiple of $\eta _{a,i}$. We can assume that $\nabla _{\partial /\partial t_i}\omega =\eta _{a,i}$. Now $\langle \omega, \eta _{b,k}\rangle =0$, since $H^{2,0}(X)$ is orthogonal to $H^{1,1}(X)$, so that

\[ 0= \langle \nabla_{\partial/\partial t_i}\omega,\eta_{b,k}\rangle + \langle \omega, \nabla_{\partial/\partial t_i}\eta_{b,k}\rangle =\langle \eta_{a,i},\eta_{b,k}\rangle, \]

since, as we have just proved, $\nabla _{\partial /\partial t_i}\eta _{b,k}\in {\it Fil}^1$.

Parts (3) and (4) follow from the linear independence of the $[\eta _{a,i}]$ and the fact that there are $N$ of them, where $N=\dim H^{1,1}_{\rm prim}(X).$

Until now $f:X\to C$ has been a point in $\mathcal {JE}^{\rm gen}$. Now, however, allow $X$ to have $A_1$-singularities, so that $f:X\to C$ is defined by a classifying morphism $\phi :C\to \overline {\mathcal {E}\ell \ell }$ that is simply ramified over $j=\infty$. Let $\widetilde X\to X$ be the minimal model, so that $\widetilde X$ has singular fibres of types $I_1$ and $I_2$ and $X\to C$ is the relative canonical model of $\widetilde X\to C$. Suppose that $D_1,\ldots, D_r$ are the exceptional $(-2)$-curves on $\widetilde X$. Suppose that $a_1,\ldots, a_r$ are the points in $Z$ lying over $j=\infty$ and that $a_{r+1},\ldots, a_N$ are the other points of $Z$. For $a=a_i$ with $i\le r$ define $[\eta _a]=[D_i]$. The surface $\widetilde X$ is a point in the stack $\widetilde {\mathcal {JE}}$ whose points are minimal models of points of $\mathcal {JE}$. We can extend Theorem 4.11 to this context, as follows. For simplicity we state it with the assumption that $Z$ is reduced.

Proof. We only need to prove part (1) when $a=a_i$ for $i\le r$. Regard the surface $\widetilde X$ as the specialization of a surface in $\mathcal {JE}^{\rm gen}$. It follows by continuity that $\overline {\nabla }_a=\omega _a^\vee \otimes \theta _a$ for some class $\theta _a$. To see that $\theta _a$ is proportional to $[D_a]$ we argue as follows.

Suppose that $\Gamma =\sum \Gamma _j$ is a configuration of $(-2)$-curves on a smooth surface $\widetilde X$ that contracts to a single du Val singularity $P\in X$. Then we identify $H^2(\widetilde X,\Omega ^2_{\widetilde X}) =H^0(X,\omega _X)$ and put

\[ V(P)=H^0(X,\omega_X)/\mathfrak m_P H^0(X,\omega_X) . \]

There is a short exact sequence

\[ 0\to H^1(\widetilde X,T_{\widetilde X}(-\log \Gamma))\to H^1(X,T_X)\to \oplus H^1(\Gamma_j,\mathcal{N}_{\Gamma_j/\widetilde X}), \]

which gives the following commutative diagram.

Taking $\Gamma =D_a$ leads to the fact that $\theta _a$ is proportional to $[D_a]$.

Part (2) is proved as in Theorem 4.11.

5. A local Schottky theorem

In this section we use the coincidence that $\dim H^{1,1}_{\rm prim}=\dim \mathcal {JE}$ to put further structure on the derivative of the period map.

The vector spaces $H^{1,1}_{\rm prim}(X)$ fit together into a vector bundle $\mathcal {H}=\mathcal {H}^{1,1}_{\rm prim}$ on $\mathcal {JE}^{\rm gen}$. We shall restrict attention to the open substack $\mathcal {JE}^{ss}$ of $\mathcal {JE}^{\rm gen}$ over which the universal ramification divisor $\mathcal {Z}$ is étale, so that, under the projection $\rho :\mathcal {Z}^{ss}\to \mathcal {JE}^{ss}$, the sheaf $\rho _*\mathcal {O}_{\mathcal {Z}^{ss}}$ is a sheaf of semi-simple rings on $\mathcal {JE}^{ss}$.

Proof. Essentially, we do this componentwise, using the orthogonal basis of $H^{1,1}_{\rm prim}(X)$ that is provided by the classes $([\eta _a])_{a\in Z}$ described in the previous section.

Let $x\in \mathcal {JE}^{ss}$, and choose an analytic neighbourhood $U$ of $x$ such that $\rho ^{-1}(U)$ is a disjoint union

\[ \rho^{-1}(U)=\bigsqcup_{a\in Z} U_a \]

of copies of $U$.

Suppose $h\in \rho _*\mathcal {O}_{\mathcal {Z}}$. So $h\vert _{U_a}$ is identified with a holomorphic function $h_a$ defined on $U$. If $s\in \Gamma (U,\mathcal {H})$, then we can write $s=\sum f_a[\eta _a]$ where $f_a$ is a function on $U$. Now define $hs$ by the formula

\[ hs=\sum h_af_a[\eta_a]. \]

Proof. We define the pairing $\beta$ in terms of the notation used in the proof of Proposition 5.1, as follows:

\[ \beta(s,t) =\beta\biggl(\sum_a f_a[\eta_a],\sum_a g_a[\eta_a]\biggr) =\sum f_ag_a([\eta_a]\cup [\eta_a]). \]

It is clear that $\beta$ is perfect.

Recall that, tautologically, the tangent bundle $T_{\mathcal {JE}^{\rm gen}}$ is naturally a line bundle $\mathcal {S}$ on $\mathcal {Z}$. That is, $T_{\mathcal {JE}^{\rm gen}}=\rho _*\mathcal {S}$. The derivative $per_*$ of the period map is an $\mathcal {O}_{\mathcal {JE}^{\rm gen}}$-linear map

\[ per_*:\rho_*\mathcal{S} \to \mathcal{H} om_{\mathcal{JE}^{\rm gen}}(\mathcal{H}^{2,0},\rho_*\mathcal{B}). \]

Define

\[ \mathcal{P}=\Omega^1_{\mathcal{C}/\mathcal{JE}^{\rm gen}}\otimes\Phi^*M, \]

where $\mathcal {C}$ is the pull back of the universal curve over $\mathcal {M}_q$ to $\mathcal {JE}^{\rm gen}$ and $\Phi :\mathcal {C}\to \overline {\mathcal {E}\ell \ell }$ is the classifying morphism. Then

\[ \Omega^2_{\mathcal{X}/\mathcal{JE}^{\rm gen}}\cong F^*\mathcal{P}, \]

where $F:\mathcal {X}\to \mathcal {C}$ is the universal Jacobian surface. There is an evaluation map

\[ eval: \rho^*\mathcal{H}^{2,0}\to \mathcal{P}\vert_{\mathcal{Z}} \]

and this map is surjective.

Proposition 5.3 On the locus $\mathcal {JE}^{ss}$ there is a factorization of $per_*$ given by a commutative diagram

where $\widetilde {per_*}$ is $\rho _*\mathcal {O}_{\mathcal {Z}^{ss}}$-linear.

We have proved almost all of the next result. It is a local Schottky theorem in that it gives a precise description of the image of the tangent bundle to moduli under the period map.

Theorem 5.4 (Local Schottky)

There is an isomorphism

\[ {\widetilde{\widetilde{per_*}}}:\mathcal{S} \to \mathcal{H} om_{\mathcal{Z}^{ss}}(\mathcal{P}\vert_{\mathcal{Z}^{ss}},\mathcal{B}) \]

of line bundles on $\mathcal {Z}^{ss}$ such that

\[ \widetilde{per_*}=\rho_*{\widetilde{\widetilde{per_*}}}. \]

6. Recovering $\operatorname {Ram}_\phi$ and $C$ from the infinitesimal period data

For the rest of this paper we make the assumptions (1.3).

Suppose that $f_i:X_i\to C_i$ are two points of $\mathcal {JE}^{ss}$ that have equal geometric genus $h$ and irregularity $q$, and that each classifying morphism $\phi _i:C_i\to \overline {\mathcal {E}\ell \ell }$ is simply ramified. Say $\operatorname {Ram}_{\phi _i}=Z_i$ and $\deg Z_i=N$, so that $N=10h+8(1-q)$. Recall that $L_=\phi _i^*M$. Our assumptions ensure that $Z_i$ is reduced and the linear system $\vert K_{X_i}\vert = f_i^*\vert K_{C_i}+L_i\vert$ has no base points and embeds $C_i$ in ${\mathbb {P}}^{h-1}$ as the canonical model of $X_i$. Note that $\deg C_i = h+q-1$.

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 $H^2(X_i,{\mathbb {Z}})$ has been appropriately normalized. In particular, such a normalization gives a normalized basis $\underline {\omega }_i=[\omega ^{(1)}_i,\ldots,\omega ^{(h)}_i]$ of $H^0(X_i,\Omega ^2_{X_i})$ for each $i$. We can therefore regard the curves $C_i$ as embedded in the same projective space ${\mathbb {P}}^{h-1}$. That is, any point $P$ of $C_i$ is identified with the point $\underline {\omega }_i(\widetilde P)$ in ${\mathbb {P}}^{h-1}$, where $\widetilde P\in X$ is any point of $X$ that lies over $P$. The basis $\underline {\omega }_i$ also gives an identification of $H^0(X_i,\Omega ^2_{X_i})$ with its dual.

We show first that $Z_1=Z_2$. We then deduce, by quadratic interpolation, that $C_1=C_2$.

By the results of § 4, especially Theorem 4.10, the tangent space $T_{\mathcal {JE}}X_1$ is of dimension $N$ and its image in $H^0(X_1,\Omega ^2_{X_1})^\vee \otimes H^{1,1}_{\rm prim}(X_1)$ is a direct sum $\bigoplus _{a\in Z_1} L_{a}$ where $L_{a}$ is a line described by Theorem 4.10. We make the identifications

\[ H^0(X_1,\Omega^2_{X_1})= H^0(X_2,\Omega^2_{X_2})=U\quad \textrm{and}\quad H^{1,1}_{\rm prim}(X_1)=H^{1,1}_{\rm prim}(X_2)=V \]

and assume that the tangent spaces $T_{\mathcal {JE}}X_1$ and $T_{\mathcal {JE}}X_2=\bigoplus _{b\in Z_2}M_{b}$ are equal as subspaces of $U\otimes V$. We know that $L_{a}$ is spanned by the rank $1$ tensor $u^{(i)}_0\otimes v^{(i)}_0$ (the image of $\overline {\nabla }_{\partial /\partial t_2}$). Similarly, $M_{b}$ is spanned by a rank 1 tensor $p^{(j)}_0\otimes q^{(j)}_0$.

Proof. Let $(v_i^\vee )$ be the dual basis of $V^\vee$. There is an index $j$ such that $\langle y, v_j^\vee \rangle \ne 0$. Then

\[ \langle y, v_j^\vee\rangle x= \sum \lambda_k\delta_{jk}u_k=\lambda_ju_j, \]

so $x$ is a multiple of $u_j$. Since no two of the $u_i$ are linearly dependent, this index $j$ is unique, so $y=\mu _jv_j$ and then $x\otimes v_j=\sum \lambda '_ku_k\otimes v_k.$

Say $Z_1=\{a_1,\ldots,a_N\}$ and $Z_2=\{a'_1,\ldots,a'_N\}$. Take $U= H^{2,0}(X_1)$ and $V=H^{1,1}_{\rm prim}(X_1)$ and write

(6.3)\begin{equation} {x_k={\underline{\omega}}_1(a_k),}\ {x'_k={\underline{\omega}}_2(a'_k),}\quad {y_k=[\eta_{1,a_k}]}\quad {\textrm{and}}\quad {y'_k=[\eta_{2,a'_k}].} \end{equation}

From Theorem 4.10 and the assumption that the IVHS of the two surfaces $X_1$ and $X_2$ are isomorphic, the tensors $x'_l\otimes y'_l$ span the same $N$-dimensional subspace of $U\otimes V$ as do the tensors $x_k\otimes y_k$. In particular, each $x'_l\otimes y'_l$ is a linear combination of the tensors $x_k\otimes y_k$. 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 ${\mathbb {P}}^{h-1}$ by a complete linear system, and $\deg C_i\ge 2q+2$, so that, by the results of [Reference MumfordMum70] and [Reference Saint-DonatSai72], $C_i$ is an intersection of quadrics. Since

\[ \deg Z >2\deg C_i, \]

again by (1.3), each $C_i$ equals the intersection of the quadrics through $Z$, so that $C_1=C_2$.

We can reformulate this as follows.

7. The structure of the tangent bundle to $\mathcal {SE}^{\rm gen}$

Recall from § 3 that we have a morphism $\pi :\overline {\mathcal {M}}_1\to \overline {\mathcal {E}\ell \ell }$ that gives, locally on $\overline {\mathcal {E}\ell \ell }$, an isomorphism $\overline {\mathcal {M}}_1\to B\mathcal {G}$, so that the tangent complex $T^\bullet _{\overline {\mathcal {M}}_1}$ is locally isomorphic to the $2$-term complex

\[ 0\to \pi^*M^\vee[1] \to \pi^*T_{\overline{\mathcal{E}\ell\ell}}[0]\to 0 \]

whose differential is zero. The morphism $\pi$ determines a morphism

\[ \Pi:\mathcal{SE}^{\rm gen} \to\mathcal{JE}^{\rm gen}. \]

Suppose that $f:X\to C$ corresponds to $\psi :C\to \overline {\mathcal {M}}_1$ and maps under $\Pi$ to $Y\to C$. Say $\phi =\pi \circ \psi$. Observe that $\operatorname {Ram}_\phi =\operatorname {Ram}_\psi =Z,$ say. The description just given of $T^\bullet _{\overline {\mathcal {M}}_1}$ leads to a short exact sequence

\[ 0 \to H^1(C, -L) \to T_{\mathcal{SE}}X \to T_{\mathcal{JE}}Y \to 0, \]

where $L=\phi ^*M$. The subobject $H^1(C,-L)$ in this sequence is identified with the tangent space $T_{\Pi ^{-1}(Y)}X$ to the fibre of $\Pi$ though $X$. Thus,

\[ T_{\Pi^{-1}(Y)}(X)\cong H^1(C,-L)\cong H^0(C,K_C+L)^\vee, \]

which is, in turn, naturally isomorphic to both $H^{2,0}(X)^\vee$ and to $H^{2,0}(Y)^\vee$. Let $\xi$ denote the class of a fibre of $X\to C$; then the period map gives the following commutative diagram with exact rows.

Proof. Put $H^{2,0}(X)^\vee =H^{2,0}(Y)^\vee =U$ and $\xi ^\perp =V$, so that $\dim U=h$ and $\dim V=N+1$. The image of $T_{\mathcal {SE}}(X)$ under $per_{X,*}$ is an $(N+h)$-dimensional subspace $W$ of $U\otimes V$ such that $W$ contains a subspace $U\otimes \xi$ and $W/U\otimes \xi$, which is the image of $T_{\mathcal {JE}}(Y)$ under $per_{Y,*}$, is spanned by rank 1 tensors

\[ x_1\otimes y_1,\ldots,x_N\otimes y_N \]

where the vectors $y_1,\ldots,y_N$ form a basis of $V/\xi =H^{1,1}_{\rm prim}(Y)$. Let $\pi :V\to V/\xi$ denote the projection.

Suppose that the same IVHS arises also from another surface $X'$. Then there is a vector $\xi '\in V$ that arises from $X$ such that $(1_U\otimes \pi )(U\otimes \xi ')$ is a subspace of $U\otimes V/\xi$ and lies in the subspace of $U\otimes V/\xi$ that is spanned by the $x_i\otimes y_i$. However, by Lemma 6.2, the $x_i\otimes y_i$ are the only rank 1 tensors in $U\otimes V/\xi$, so that $(1_U\otimes \pi )(U\otimes \xi ')=0$ and, therefore, $\xi '$ is proportional to $\xi$. Consider the vectors $x_i',y_i'$ that arise from $X'$; then the tensors $x_i\otimes y_i$ and $x_i'\otimes y_i'$ lie in the same vector space $U\otimes V/\xi$, and then we can use Lemma 6.2 again to conclude the proof.

Compare the case of $\mathcal {M}_q$: over the non-hyperelliptic locus Schiffer variations give a cone structure in the tangent bundle, where at each point $C$ of $\mathcal {M}_q$ 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 $\phi _1:C\to \overline {\mathcal {E}\ell \ell }$ is generic and if $\phi _2:C\to \overline {\mathcal {E}\ell \ell }$ is another morphism such that $\operatorname {Ram}_{\phi _1}=\operatorname {Ram}_{\phi _2}$ and $\phi _1^*M\cong \phi _2^*M$, then $\phi _1$ and $\phi _2$ are equivalent modulo the action of $\operatorname {Aut}_{\overline {\mathcal {E}\ell \ell }}={\mathbb {G}}_m$ provided that also there are sufficiently many points $a_i\in Z$ such that $\phi _1(a_i)$ is isomorphic to $\phi _2(a_i)$. Therefore an effective form of generic Torelli holds for Jacobian elliptic surfaces modulo this action of ${\mathbb {G}}_m$.

To begin, we rewrite some results of Tannenbaum [Reference TannenbaumTan84] in the context of Deligne–Mumford stacks. Assume that $\mathcal {S}$ is a smooth $2$-dimensional Deligne–Mumford stack (the relevant example will be $\mathcal {S}=\overline {\mathcal {E}\ell \ell }\times \overline {\mathcal {E}\ell \ell }$), that $C$ is a smooth projective curve and that $\pi :C\to \mathcal {S}$ is a morphism that factors as

where $D$ is a projective curve with only cusps, $\pi '$ is birational and $i$ induces surjections of henselian local rings at all points. (We shall say that ‘$\pi$ is birational onto its image’.) Then there is a conormal sheaf $\mathcal {N}^\vee _D$, a line bundle on $D$, which is generated by the pull back under $i$ of the kernel $\mathcal {I}_D$ of $\mathcal {O}_{\mathcal {S}}\to i_*\mathcal {O}_D$. It fits into a short exact sequence

\[ 0\to \mathcal{N}^\vee_D\to i^*\Omega^1_{\mathcal{S}}\to\Omega^1_D\to 0. \]

Thus, the adjunction formula (or duality for the morphism $i:D\to \mathcal {S}$) gives an isomorphism

\[ \mathcal{N}_D \mathop{\to}\limits^{\cong} \omega_D\otimes i^*\omega_{\mathcal{S}}^\vee. \]

There is also a homomorphism $\mathcal {N}_D\to T^1(D,\mathcal {O}_D)$; let $\mathcal {N}'_D$ denote its kernel. The space $H^0(D,\mathcal {N}'_D)$ is the tangent space to the functor that classifies deformations of the morphism $i:D\to \mathcal {S}$ 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 $T_C(R)=T_{C}\otimes \mathcal {O}_C(R)$ is the saturation of $T_{C}$ in $\pi ^*T_{\mathcal {S}}$. Define $\mathcal {N}'_\pi$ by

\[ \mathcal{N}'_\pi=\mathrm{coker}(T_C(R)\to\pi^*T_{\mathcal{S}}); \]

this is a line bundle on $C$.

Define $\mathcal {J}\subset \mathcal {O}_{\mathcal {S}}$ to be the Jacobian ideal of the ideal $\mathcal {I}_D$. The next result is a slight variant of Lemma 1.5 of [Reference TannenbaumTan84].

Proof. We have $\Omega ^1_D=\mathcal {J}.\omega _D$, by direct calculation, and, from the definition of $\mathcal {N}'_D$, we have $\mathcal {N}'_D=\mathcal {J}.\mathcal {N}_D$.

From the definition, $\omega ^\vee _{C}(R)\otimes \mathcal {N}'_\pi \cong \pi ^*\omega _{\mathcal {S}}^\vee$. Let $\mathcal {C}$ denote the conductor ideal. Then $\omega _{C}\cong \mathcal {C}\otimes \pi '^*\omega _D$, so that

\[ \mathcal{N}'_\pi\cong \pi'^*\omega_D\otimes\mathcal{C}\otimes\mathcal{O}_{C}(-R-K_{\mathcal{S}}). \]

Now $\mathcal {C}=\mathcal {O}_{C}(-R)$, by the nature of a cusp, and $\mathcal {J}.\mathcal {O}_{C}=\mathcal {O}_{C}(-2R)$ for the same reason. Thus,

\[ \mathcal{N}'_\pi\cong\pi'^*(\omega_D\otimes i^*\omega_S^\vee)\otimes\mathcal{O}_{C}(-2R) \]

and, therefore,

\[ \pi'_*\mathcal{N}'_\pi\cong \mathcal{J}.(\omega_D\otimes\omega_{\mathcal{S}}^\vee)\cong\mathcal{J}.\mathcal{N}_D=\mathcal{N}'_D. \]

Proof. Assume that $\phi _2$ exists. Take $\overline {\mathcal {E}\ell \ell }\times \overline {\mathcal {E}\ell \ell }=\mathcal {S}$ and $(\phi _1,\phi _2)=\pi$. Then there is a factorization

as before, and the divisor $R$ is $R=\operatorname {Ram}_{\phi _1}=\operatorname {Ram}_{\phi _2}$. We get

\begin{align*} \deg \mathcal{N}'_\pi&=2\deg\phi_1^*T_{\overline{\mathcal{E}\ell\ell}}+2q-2-N\\ &= 2.24(h+1-q).\frac{5}{12}+2q-2-N\\ &=10h+10(1-q), \end{align*}

so that

\[ h^0(C,\mathcal{N}'_\pi)=10h+11(1-q)=N-(3q-3). \]

However, $\dim \mathcal {JE}=N,$ and the proposition is proved.