Theta divisors with curve summands and the Schottky problem

Stefan Schreieder

doi:10.1007/s00208-015-1287-8

Theta divisors with curve summands and the Schottky problem

Stefan Schreieder^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › Research › peer review

Abstract

We prove the following converse of Riemann’s Theorem: let (A, Θ) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety Θ = C+ Y. Then C is smooth, A is the Jacobian of C, and Y is a translate of Wg - 2(C). As applications, we determine all theta divisors that are dominated by a product of curves and characterize Jacobians by the existence of a d-dimensional subvariety with curve summand whose twisted ideal sheaf is a generic vanishing sheaf.

Original language	English
Pages (from-to)	1017-1039
Number of pages	23
Journal	Mathematische Annalen
Volume	365
Issue number	3-4
DOIs	https://doi.org/10.1007/s00208-015-1287-8
Publication status	Published - 1 Aug 2016
Externally published	Yes

Keywords

Schottky Problem
DPC Problem
Theta divisors
Jacobians
generic vanishing

ASJC Scopus subject areas

General Mathematics

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10.1007/s00208-015-1287-8

https://arxiv.org/abs/1409.3134

Cite this

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title = "Theta divisors with curve summands and the Schottky problem",

abstract = "We prove the following converse of Riemann{\textquoteright}s Theorem: let (A, Θ) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety Θ = C+ Y. Then C is smooth, A is the Jacobian of C, and Y is a translate of Wg - 2(C). As applications, we determine all theta divisors that are dominated by a product of curves and characterize Jacobians by the existence of a d-dimensional subvariety with curve summand whose twisted ideal sheaf is a generic vanishing sheaf.",

keywords = "Schottky Problem, DPC Problem, Theta divisors, Jacobians, generic vanishing",

author = "Stefan Schreieder",

note = "Funding information: I would like to thank my advisor D. Huybrechts for constant support, encouragement and discussions about the DPC problem. Thanks go also to C. Schnell for his lectures on generic vanishing theory, held in Bonn during the winter semester 2013/14, where I learned about GV-sheaves and Ein–Lazarsfeld{\textquoteright}s result []. I am grateful to J. Fresan, D. Kotschick, L. Lombardi and M. Popa for useful comments. Special thanks to the anonymous referee for helpful comments and corrections. The author is member of the BIGS and the SFB/TR 45 and supported by an IMPRS Scholarship of the Max Planck Society.",

year = "2016",

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doi = "10.1007/s00208-015-1287-8",

language = "English",

volume = "365",

pages = "1017--1039",

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TY - JOUR

T1 - Theta divisors with curve summands and the Schottky problem

AU - Schreieder, Stefan

N1 - Funding information: I would like to thank my advisor D. Huybrechts for constant support, encouragement and discussions about the DPC problem. Thanks go also to C. Schnell for his lectures on generic vanishing theory, held in Bonn during the winter semester 2013/14, where I learned about GV-sheaves and Ein–Lazarsfeld’s result []. I am grateful to J. Fresan, D. Kotschick, L. Lombardi and M. Popa for useful comments. Special thanks to the anonymous referee for helpful comments and corrections. The author is member of the BIGS and the SFB/TR 45 and supported by an IMPRS Scholarship of the Max Planck Society.

PY - 2016/8/1

Y1 - 2016/8/1

N2 - We prove the following converse of Riemann’s Theorem: let (A, Θ) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety Θ = C+ Y. Then C is smooth, A is the Jacobian of C, and Y is a translate of Wg - 2(C). As applications, we determine all theta divisors that are dominated by a product of curves and characterize Jacobians by the existence of a d-dimensional subvariety with curve summand whose twisted ideal sheaf is a generic vanishing sheaf.

AB - We prove the following converse of Riemann’s Theorem: let (A, Θ) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety Θ = C+ Y. Then C is smooth, A is the Jacobian of C, and Y is a translate of Wg - 2(C). As applications, we determine all theta divisors that are dominated by a product of curves and characterize Jacobians by the existence of a d-dimensional subvariety with curve summand whose twisted ideal sheaf is a generic vanishing sheaf.

KW - Schottky Problem

KW - DPC Problem

KW - Theta divisors

KW - Jacobians

KW - generic vanishing

UR - https://www.scopus.com/pages/publications/84941662456

U2 - 10.1007/s00208-015-1287-8

DO - 10.1007/s00208-015-1287-8

M3 - Article

AN - SCOPUS:84941662456

SN - 0025-5831

VL - 365

SP - 1017

EP - 1039

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 3-4

ER -

Theta divisors with curve summands and the Schottky problem

Abstract

Keywords

ASJC Scopus subject areas

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