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We prove the following converse of Riemann's Theorem: let (A,\Theta) be anindecomposable principally polarized abelian variety whose theta divisor can bewritten as a sum of a curve and a codimension two subvariety \Theta=C+Y. Then Cis smooth, A is the Jacobian of C, and Y is a translate of W_{g-2}(C). Asapplications, we determine all theta divisors that are dominated by a productof curves and characterize Jacobians by the existence of a d-dimensionalsubvariety with curve summand whose twisted ideal sheaf is a generic vanishingsheaf.

Theta divisors with curve summands and the Schottky problem