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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 havingutmost two nodal singularities. We show that in each component of$M(2,\textbf{\underline{w}},\chi)$, the closure of rank $2$ vector bundles withfixed determinant is rational.

Rationality of Moduli space over reducible curve