Rank-two vector bundles on non-minimal ruled surfaces

TL;DR

The paper advances the study of moduli spaces of stable rank-two vector bundles on non-minimal ruled surfaces by extending the extension-parameter framework from minimal to blown-up bases, introducing the invariants $d_V$ and $r_V$ to express bundles as controlled extensions. It provides a birational description of irreducible components as projective bundles over Hilbert schemes and Pic^0 factors, and shows that for rational surfaces these components are rational or stably rational. For surfaces obtained by blowing up general points on a minimal surface, it gives explicit computation of extension spaces in the cases $c_1\cdot F=0$ and $c_1\cdot F=1$, yielding concrete extension presentations, dominance by extension families, and birational rationality conclusions in terms of products of Jacobians and Hilbert schemes. Overall, the work connects classical extension theory with the birational geometry of moduli spaces in the non-minimal, ruled-surface setting and provides explicit, computable descriptions of dominant extension families.

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.

Theorems & Definitions (21)

• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Remark 2.4
• Theorem 2.5
• Lemma 3.1
• Remark 3.2
• Remark 3.3
• Example 3.4
• Proposition 3.6