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- W4387406763 abstract "Suppose that $f:Xto C$ 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 $(v_a)_{ain Z}$ , where $Zsubset C$ is the ramification locus of the classifying morphism $phi :Cto overline {mathcal {E}ell ell }$ that corresponds to $Xto C$ . (2) For each $ain Z$ 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 $hge q+3$ . (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.)" @default.
- W4387406763 created "2023-10-07" @default.
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- W4387406763 date "2023-10-06" @default.
- W4387406763 modified "2023-10-08" @default.
- W4387406763 title "Generic Torelli and local Schottky theorems for Jacobian elliptic surfaces" @default.
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- W4387406763 doi "https://doi.org/10.1112/s0010437x23007443" @default.
- W4387406763 hasPublicationYear "2023" @default.
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