A Torelli Type theorem for Nodal curves

TL;DR

This paper proves a Torelli type theorem for nodal curves by utilizing the moduli space of stable Gieseker vector bundles, revealing new insights into the curve's structure through its vector bundle moduli.

Contribution

It establishes a Torelli theorem for nodal curves using the moduli space of Gieseker vector bundles, extending classical results to singular curves.

Findings

Proves a Torelli type theorem for nodal curves.

Shows the moduli space has normal crossing singularities.

Demonstrates the moduli space provides a flat degeneration.

Abstract

The moduli space of Gieseker vector bundles is a compactification of moduli of vector bundles on a nodal curve. This moduli space has only normal crossing singularity and it provides a flat degeneration. We prove a Torelli type theorem for a nodal curve using the moduli space of stable Gieseker vector bundles of fixed rank (strictly greater than $1$) and fixed degree such that rank and degree are co-prime.