Theta divisors with curve summands and the Schottky problem

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 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.

23 pages; final version, to appear in Mathematische Annalen

Cited by in corpus (3)

• Characteristic cycles and the microlocal geometry of the Gauss map, II
• Decomposable theta divisors and generic vanishing
• Generic vanishing and minimal cohomology classes on abelian fivefolds