Ample stable vector bundles on rational surfaces

TL;DR

This paper classifies moduli spaces of ample stable vector bundles on rational surfaces, showing large sums of stable bundles can be deformed to ample bundles, using modern moduli theory beyond classical rank-two cases.

Contribution

It provides a complete classification of certain moduli spaces and demonstrates that large sums of stable bundles typically admit ample deformations, extending results to arbitrary rank.

Findings

Complete classification of moduli spaces with ample, globally generated stable bundles

Large sums of stable bundles can be deformed to ample bundles unless restricted by numerical conditions

Utilizes recent advances in moduli theory beyond classical constructions

Abstract

We study ample stable vector bundles on minimal rational surfaces. We give a complete classification of those moduli spaces for which the general stable bundle is both ample and globally generated. We also prove that if $V$ is any stable bundle, then a large enough direct sum $V^{\oplus n}$ has ample deformations unless there is an obvious numerical reason why it cannot. Previous work in this area has mostly focused on rank two bundles and relied primarily on classical constructions such as the Serre construction. In contrast, we use recent advances in moduli of vector bundles to obtain strong results for vector bundles of any rank.