A Riemann singularity theorem for integral curves

TL;DR

This paper extends the classical Riemann Singularity Theorem to integral, singular curves by computing the multiplicity of the theta divisor at arbitrary points and proposing a conjecture for a general formula.

Contribution

It generalizes the Riemann Singularity Theorem to singular curves and introduces a conjecture for the multiplicity of points on the theta divisor.

Findings

Computed the multiplicity of the theta divisor at arbitrary points for integral, nodal curves.

Proposed a conjecture for the general multiplicity formula on the theta divisor.

Provided evidence supporting the conjecture and addressed a question by Lucia Caporaso.

Abstract

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.