On A Relative Mumford-Newstead Theorem

TL;DR

This paper extends the Mumford-Newstead theorem to families of degenerating curves and demonstrates a Torelli-type result for moduli spaces on reducible curves, enabling curve reconstruction from these spaces.

Contribution

It introduces a relative version of the Mumford-Newstead theorem for degenerating curves and establishes a Torelli-type theorem for moduli spaces on reducible curves.

Findings

Proved a relative Mumford-Newstead theorem for degenerating curves.

Established a Torelli-type theorem for moduli spaces on reducible curves.

Showed that certain moduli spaces determine the original curve.

Abstract

In this paper we prove a relative version of the classical Mumford-Newstead theorem for a family of smooth curves degenerating to a reducible curve with a simple node. We also prove a Torelli-type theorem by showing that certain moduli spaces of torsion-free sheaves on a reducible curve allows us to recover the curve from the moduli space.