The Local Structure of Compactified Jacobians

TL;DR

This paper provides an explicit description of the local structure of compactified Jacobians of nodal curves using invariants related to the dual graph, advancing understanding of their geometric and combinatorial properties.

Contribution

It presents a detailed presentation of the completed local ring of the compactified Jacobian as an invariant ring, linking local geometry to the dual graph of the curve.

Findings

Explicit local ring description in terms of invariants

Connections between local geometry and dual graph combinatorics

Results for universal compactified Jacobian over moduli space

Abstract

This paper studies the local geometry of compactified Jacobians constructed by Caporaso, Oda-Seshadri, Pandharipande, and Simpson. The main result is a presentation of the completed local ring of the compactified Jacobian of a nodal curve as an explicit ring of invariants described in terms of the dual graph of the curve. The authors have investigated the geometric and combinatorial properties of these rings in previous work, and consequences for compactified Jacobians are presented in this paper. Similar results are given for the local structure of the universal compactified Jacobian over the moduli space of stable curves.