Rationality of Moduli space over reducible curve

Abstract

Let $M(2,\text{w},\chi )$ 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 $M(2,\text{w},\chi )$, the closure of rank $2$ vector bundles with fixed determinant is rational.