Rationality of Moduli space over reducible curve

TL;DR

This paper proves that in each component of the moduli space of rank 2 torsion-free sheaves over a reducible nodal curve, the closure of rank 2 vector bundles with fixed determinant is rational, revealing its geometric structure.

Contribution

It establishes the rationality of the closure of rank 2 vector bundles with fixed determinant in the moduli space over reducible nodal curves, a new result in algebraic geometry.

Findings

The moduli space components are rational.

Closure of rank 2 vector bundles with fixed determinant is rational.

Results apply to curves with up to two nodal singularities per component.

Abstract

Let $M(2,\textbf{\underline{w}},\chi)$ be the moduli space of rank $2$ torsion-free sheaves over a reducible nodal curve with each component having utmost two nodal singularities. We show that in each component of $M(2,\textbf{\underline{w}},\chi)$, the closure of rank $2$ vector bundles with fixed determinant is rational.