Stability conditions for line bundles on nodal curves

TL;DR

This paper introduces a new framework for understanding the stability of line bundles on nodal curves, establishing a bijection with smoothable compactified Jacobians and classifying universal Jacobians over moduli spaces.

Contribution

It defines the notion of smoothable fine compactified Jacobians and combinatorial stability conditions, linking them and classifying universal Jacobians over the moduli space of stable curves.

Findings

Smoothable fine compactified Jacobians correspond bijectively to stability conditions.

Every fine compactified universal Jacobian is isomorphic to a known construction by Caporaso, Pandharipande, and Simpson.

No fine compactified universal Jacobian exists unless gcd(d+1-g, 2g-2)=1.

Abstract

We introduce the abstract notion of a \emph{smoothable fine compactified Jacobian} of a nodal curve, and of a family of nodal curves whose general element is smooth. Then we introduce the notion of a combinatorial stability condition for line bundles and their degenerations. We prove that smoothable fine compactified Jacobians are in bijection with these stability conditions. We then turn our attention to \emph{fine compactified universal Jacobians}, that is, fine compactified Jacobians for the moduli space $\overline{\mathcal{M}}_g$ of stable curves (without marked points). We prove that every fine compactified universal Jacobian is isomorphic to the one first constructed by Caporaso, Pandharipande and Simpson in the nineties. In particular, without marked points, there exists no fine compactified universal Jacobian unless $\gcd(d+1-g, 2g-2)=1$.