Decomposable theta divisors and generic vanishing

TL;DR

This paper investigates the structure of ample divisors with rational singularities on abelian varieties, establishing bounds on their addition maps and characterizing theta divisors, with implications for generic vanishing subschemes.

Contribution

It provides an optimal lower bound on the degree of the addition map for decomposable divisors and characterizes when such divisors are theta divisors, linking to conjectures about Jacobians.

Findings

Optimal lower bound on addition map degree for decomposable divisors

Characterization of divisors achieving the bound as theta divisors

Generic vanishing subschemes are reduced, irreducible, and have expected genus

Abstract

We study ample divisors X with only rational singularities on abelian varieties that decompose into a sum of two lower dimensional subvarieties, X=V+W. For instance, we prove an optimal lower bound on the degree of the corresponding addition map, and show that the minimum can only be achieved if X is a theta divisor. Conjecturally, the latter happens only on Jacobians of curves and intermediate Jacobians of cubic threefolds. As an application, we prove that nondegenerate generic vanishing subschemes of indecomposable principally polarized abelian varieties are automatically reduced and irreducible, have the expected geometric genus, and property (P) with respect to their theta duals.