Title: A Riemann Singularity Theorem for Singular Curves

The Jacobian of a non-singular curve contains a
distinguished divsior known as the theta divisor. A classical result
of Riemann computes the multiplicity of this divisor  at a point. Namely, if
x is a point of Theta that corresponds to a line bundle L, then the
Riemann singularity theorem states that:
mult_x( Theta )=h^1(X,L). 

I will talk about extending this result to singular curves. In
particular, I will prove a generalization of Riemann's theorem to
integral, nodal curves. This result yields a partial answer to a
recent question of Lucia Caporaso.

All of this work is joint with Sebastian (Yano) Casalaina-Martin.