A Note On the moduli space of rank two vector bundles on Hirzebruch surfaces

Abstract

The aim of this paper is to study the Segre invariant for rank $2$ vector bundles on Hirzebruch surfaces ${\mathbb{F}}_e$, $e\ge 0$. We give necessary conditions in order to determine what numbers can appear as the Segre Invariant of a rank $2$ vector bundle on ${\mathbb{F}}_e$ with fixed Chern classes. We compute a bound of the dimensions of the strata whenever are non-empty. Finally, we present applications to Brill-Noether Theory and describe differences between moduli spaces under change of polarization for rank $2$ vector bundles on ${\mathbb{F}}_e$.