Extensions of the universal theta divisor

TL;DR

This paper extends the universal theta divisor over the moduli space of stable marked curves, analyzes its variation with stability parameters, and relates it to recent divisors studied in algebraic geometry.

Contribution

It introduces a method to extend the universal theta divisor over stable curves and derives a wall-crossing formula for its variation.

Findings

Derived a wall-crossing formula for the theta divisor

Computed the pullback of the theta divisor in various contexts

Analyzed a divisor on the moduli space related to recent studies

Abstract

The Jacobian varieties of smooth curves fit together to form a family, the universal Jacobian, over the moduli space of smooth marked curves, and the theta divisors of these curves form a divisor in the universal Jacobian. In this paper we describe how to extend these families over the moduli space of stable marked curves (or rather an open subset thereof) using a stability parameter. We then prove a wall-crossing formula describing how the theta divisor varies with the stability parameter. We use that result to analyze a divisor on the moduli space of smooth marked curves that has recently been studied by Grushevsky-Zakharov, Hain and M\"uller. In particular, we compute the pullback of the theta divisor studied in Alexeev's work on stable abelic varieties and in Caporaso's work on theta divisors of compactified Jacobians.