A Riemann singularity theorem for integral curves

Sebastian Casalaina-Martin, Jesse Leo Kass

Published 2009-07-01, updated 2010-06-01Version 2

We prove results generalizing the classical Riemann Singularity Theorem to the case of integral, singular curves. The main result is a computation of the multiplicity of the theta divisor of an integral, nodal curve at an arbitrary point. We also conjecture a general formula for the multiplicity of points on the theta divisor of a singular integral curve and present some evidence for this conjecture. Our results give a partial answer to a question posed by Lucia Caporaso in a recent paper.