Two ways to degenerate the Jacobian are the same

TL;DR

This paper compares two methods of degenerating Jacobian varieties by relating degenerate fibers of group varieties and moduli spaces of sheaves, providing conditions for their equivalence in certain families.

Contribution

It establishes conditions under which the line bundle locus in moduli spaces of sheaves is isomorphic to the Néron model, linking two approaches to degeneration.

Findings

Conditions for isomorphism between line bundle locus and Néron model.

Extension of previous results to new classes of moduli spaces.

Application to families over smooth curves.

Abstract

A basic technique for studying a family of Jacobian varieties is to extend the family by adding degenerate fibers. Constructing an extension requires a choice of fibers, and one typically chooses to include either degenerate group varieties or degenerate moduli spaces of sheaves. Here we relate these two different approaches when the base of the family is a regular, 1-dimensional scheme such as a smooth curve. Specifically, we provide sufficient conditions for the line bundle locus in a family of compact moduli spaces of pure sheaves to be isomorphic to the N\'eron model. The result applies to moduli spaces constructed by Eduardo Esteves and Carlos Simpson, extending results of Busonero, Caporaso, Melo, Oda, Seshadri, and Viviani.