Rank-two vector bundles on non-minimal ruled surfaces

Abstract

We continue previous work 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 Brînzănescu 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.