On vector bundles over reducible curves with a node

Let $C$ be a curve with two smooth components and a single node. Let ${\mathcal{U}}_C(r,w,\chi )$ be the moduli space of $w$-semistable classes of depth one sheaves on $C$ having rank $r$ on both components and Euler characteristic $\chi$. 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 ${\mathcal{U}}_C(r,w,\chi )$. Then we prove the rationality of the closed subset containing vector bundles with given fixed determinant.