# On vector bundles over reducible curves with a node

## TL;DR

This paper studies the moduli space of semistable sheaves on a reducible curve with a node, constructing a projective bundle related to vector bundles on components and proving rationality of certain subsets.

## Contribution

It introduces a birational correspondence between a projective bundle and a component of the moduli space, and establishes rationality results for fixed determinant cases.

## Key findings

- Constructs a projective bundle over product of moduli spaces on components.
- Shows birational equivalence to an irreducible component of the moduli space.
- Proves rationality of subsets with fixed determinant.

## Abstract

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.