The stability space of compactified universal Jacobians

TL;DR

This paper characterizes the stability conditions for compactified universal Jacobians, providing a combinatorial description of the stability space, and applies it to classify these Jacobians and resolve indeterminacies in Abel-Jacobi sections.

Contribution

It offers a detailed combinatorial description of the stability space for compactified universal Jacobians, linking polytopes to moduli space compactifications and addressing open classification questions.

Findings

Explicit description of the stability space as an affine space with polytope decomposition.

Classification of isomorphism classes of compactified universal Jacobians.

Resolution of indeterminacy in Abel-Jacobi sections.

Abstract

In this paper we describe compactified universal Jacobians, i.e. compactifications of the moduli space of line bundles on smooth curves obtained as moduli spaces of rank 1 torsion-free sheaves on stable curves, using an approach due to Oda-Seshadri. We focus on the combinatorics of the stability conditions used to define compactified universal Jacobians. We explicitly describe an affine space, the stability space, with a decomposition into polytopes such that each polytope corresponds to a proper Deligne-Mumford stack that compactifies the moduli space of line bundles. We apply this description to describe the set of isomorphism classes of compactified universal Jacobians (answering a question of Melo), and to resolve the indeterminacy of the Abel-Jacobi sections (addressing a problem raised by Grushevsky-Zakharov).