Theta divisors with curve summands and the Schottky problem

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.