Rank-two vector bundles on non-minimal ruled surfaces

TL;DR

This paper investigates the birational geometry of moduli spaces of stable rank-two vector bundles on non-minimal ruled surfaces, expressing bundles as extensions and analyzing their rationality properties.

Contribution

It introduces a new approach to describe vector bundles via natural extensions using specific invariants, extending previous work to non-minimal surfaces.

Findings

Explicit computation of natural extensions on blowups of minimal surfaces

Proving irreducible components of moduli spaces are rational or stably rational for rational surfaces

Extension of invariants to non-minimal ruled surfaces

Abstract

We continue previous works by various authors and study the birational geometry of moduli spaces of stable rank-two vector bundles on surfaces with Kodaira dimension $-\infty$. To this end, we express vector bundles as natural extensions, by using two numerical invariants associated to vector bundles, similar to the invariants defined by Brinzanescu and Stoia in the case of minimal surfaces. We compute explicitly these natural extensions on blowups of general points on a minimal surface. In the case of rational surfaces, we prove that any irreducible component of a moduli space is either rational or stably rational.