On vector bundles over reducible curves with a node

Let $C$ be a curve with two smooth components and a single node. Let $\mathcal{U}_C(r,w,χ)$ be the moduli space of $w$-semistable classes of depth one sheaves on $C$ having rank $r$ on both components and Euler characteristic $χ$. In this paper, under suitable assumptions, we produce a projective bundle over the product of the moduli spaces of semistable vector bundles of rank $r$ on each components and we show that it is birational to an irreducible component of $\mathcal{U}_C(r,w,χ)$. Then we prove the rationality of the closed subset containing vector bundles with given fixed determinant.

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