Geometry of the theta divisor of a compactified jacobian

TL;DR

This paper investigates the structure of the theta divisor on the compactified Jacobian of nodal curves, revealing its components and geometric properties, with applications to hyperelliptic stable curves.

Contribution

It provides a detailed description of the irreducible components of the theta divisor and offers a geometric interpretation aligned with classical Brill-Noether theory for singular curves.

Findings

Identified the irreducible components of the theta divisor.

Provided a geometric interpretation consistent with classical theory.

Applied results to hyperelliptic stable curves.

Abstract

We study the theta divisor of the compactified jacobian of a nodal, possibly reducible, curve. We compute its irreducible components and give it a geometric interpretation consistent with the classical Brill-Noether theory of smooth curves. Some applications on hyperelliptic stable curves are appended.