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.