Moduli spaces of vector bundles on a singular rational ruled surface

TL;DR

This paper investigates the geometric properties of moduli spaces of vector bundles on a singular rational ruled surface, establishing conditions for their irreducibility, smoothness, rationality, and non-emptiness, including over real numbers.

Contribution

It provides new results on the structure and rationality of moduli spaces of vector bundles on singular rational ruled surfaces, extending understanding to cases with singular base curves and real structures.

Findings

Moduli spaces are irreducible, smooth, and rational under certain conditions.

Non-emptiness of these moduli spaces is established in specific cases.

Rationality over the real numbers is proven for the moduli spaces on real rational ruled surfaces.

Abstract

We study moduli spaces $M_X(r,c_1,c_2)$ parametrizing slope semistable vector bundles of rank $r$ and fixed Chern classes $c_1, c_2$ on a ruled surface whose base is a rational nodal curve. We show that under certain conditions, these moduli spaces are irreducible, smooth and rational (when non-empty). We also prove that they are non-empty in some cases. We show that for a rational ruled surface defined over real numbers, the moduli space $M_X(r,c_1,c_2)$ is rational as a variety defined over $\mathbb R$.