# Theta divisors with curve summands and the Schottky problem

Stefan Schreieder

Number | 8 |
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Author | Stefan Schreieder |

Year | 2014 |

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 solve the DPC Problem for theta divisors and characterize Jacobians by the existence of a d-dimensional subvariety with curve summand whose twisted ideal sheaf is a GV-sheaf.