TY - JOUR

T1 - Generic Torelli and local Schottky theorems for Jacobian elliptic surfaces

AU - Shepherd-Barron, N. I.

N1 - Publisher Copyright:
© 2023 The Author(s).

PY - 2023/10/6

Y1 - 2023/10/6

N2 - Suppose that is a general Jacobian elliptic surface over of irregularity and positive geometric genus. Assume that 12(q-1)$]]>, that 0$]]> and let denote the stack of generalized elliptic curves. (1) The moduli stack of such surfaces is smooth at the point and its tangent space there is naturally a direct sum of lines, where is the ramification locus of the classifying morphism that corresponds to. (2) For each the map defined by the derivative of the period map is of rank one. Its image is a line and its kernel is, where. (3) The classes form an orthogonal basis of and is represented by a meromorphic -form in of the second kind. (4) We prove a local Schottky theorem; that is, we give a description of in terms of a certain additional structure on the vector bundles that are involved. Assume further that 10(q-1)$]]> and that. (5) Given the period point of that classifies the Hodge structure on the primitive cohomology and the image of under we recover as a subset of and then, by quadratic interpolation, the curve. (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 for which can be calculated. (In an earlier version of this paper we used variations constructed by Fay. However, Schiffer variations are slightly more powerful.)

AB - Suppose that is a general Jacobian elliptic surface over of irregularity and positive geometric genus. Assume that 12(q-1)$]]>, that 0$]]> and let denote the stack of generalized elliptic curves. (1) The moduli stack of such surfaces is smooth at the point and its tangent space there is naturally a direct sum of lines, where is the ramification locus of the classifying morphism that corresponds to. (2) For each the map defined by the derivative of the period map is of rank one. Its image is a line and its kernel is, where. (3) The classes form an orthogonal basis of and is represented by a meromorphic -form in of the second kind. (4) We prove a local Schottky theorem; that is, we give a description of in terms of a certain additional structure on the vector bundles that are involved. Assume further that 10(q-1)$]]> and that. (5) Given the period point of that classifies the Hodge structure on the primitive cohomology and the image of under we recover as a subset of and then, by quadratic interpolation, the curve. (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 for which can be calculated. (In an earlier version of this paper we used variations constructed by Fay. However, Schiffer variations are slightly more powerful.)

KW - elliptic surface

KW - forms of the second kind

KW - moduli

KW - periods

UR - http://www.scopus.com/inward/record.url?scp=85175250603&partnerID=8YFLogxK

U2 - 10.1112/S0010437X23007443

DO - 10.1112/S0010437X23007443

M3 - Article

AN - SCOPUS:85175250603

SN - 0010-437X

VL - 159

SP - 2521

EP - 2550

JO - COMPOSITIO MATHEMATICA

JF - COMPOSITIO MATHEMATICA

IS - 12

ER -