Irreducibility of Moduli Spaces of Vector Bundles on Birationally Ruled Surfaces

TL;DR

This paper proves that moduli spaces of semistable sheaves on birationally ruled surfaces are irreducible under certain conditions, by analyzing the smoothness and irreducibility of stacks of prioritary sheaves.

Contribution

It establishes the irreducibility of moduli schemes of semistable sheaves on birationally ruled surfaces for all relevant invariants under a simple polarization condition.

Findings

Moduli schemes are irreducible for all (r,c_1,c_2) under the polarization condition.

Stacks of prioritary sheaves are smooth and irreducible.

The irreducibility holds for all semistable sheaves with fixed invariants.

Abstract

Let $S$ be a birationally ruled surface. We show that the moduli schemes $M_S(r,c_1,c_2)$ of semistable sheaves on $S$ of rank $r$ and Chern classes $c_1$ and $c_2$ are irreducible for all $(r,c_1,c_2)$ provided the polarization of $S$ used satisfies a simple numerical condition. This is accomplished by proving that the stacks of prioritary sheaves on $S$ of fixed rank and Chern classes are smooth and irreducible.