Theta divisors with curve summands and the Schottky problem

TL;DR

This paper characterizes Jacobian varieties among indecomposable principally polarized abelian varieties by analyzing theta divisors that decompose into a curve and a subvariety, providing new insights into the Schottky problem.

Contribution

It proves a converse to Riemann's Theorem linking theta divisor decompositions to Jacobians and characterizes Jacobians via subvarieties with curve summands.

Findings

Theta divisors decomposed as a sum of a curve and a subvariety imply the variety is a Jacobian.

Identifies all theta divisors dominated by products of curves.

Characterizes Jacobians through the existence of specific subvarieties with vanishing ideal sheaves.

Abstract

We prove the following converse of Riemann's Theorem: let (A,\Theta) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety \Theta=C+Y. Then C is smooth, A is the Jacobian of C, and Y is a translate of W_{g-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.