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.Comment: 23 pages; final version, to appear in Mathematische Annale
