Semistable modifications of families of curves and compactified Jacobians

TL;DR

This paper studies how semistable modifications of families of nodal curves affect their Jacobians, establishing functorial isomorphisms between different compactification methods for the relative Jacobian.

Contribution

It introduces comparison theorems linking torsion-free sheaves and invertible sheaves in semistably modified families, unifying two approaches to compactified Jacobians.

Findings

Comparison theorems between sheaves in original and modified families

Functorial isomorphisms between Caporaso's and Pandharipande's Jacobian compactifications

Unification of different Jacobian compactification methods

Abstract

Given a family of nodal curves, a semistable modification of it is another family made up of curves obtained by inserting chains of rational curves of any given length at certain nodes of certain curves of the original family. We give comparison theorems between torsion-free, rank-1 sheaves in the former family and invertible sheaves in the latter. We apply them to show that there are functorial isomorphisms between the compactifications of relative Jacobians of families of nodal curves constructed through Caporaso's approach and those constructed through Pandharipande's approach.