# Brill-Noether theorems and globally generated vector bundles on Hirzebruch surfaces

## TL;DR

This paper investigates the properties of stable vector bundles on Hirzebruch surfaces, establishing cohomology and Betti number formulas, and classifying when these bundles are globally generated, extending classical results to this geometric setting.

## Contribution

It provides new formulas for cohomology and Betti numbers of stable bundles on Hirzebruch surfaces and classifies globally generated cases, generalizing known results from projective planes.

## Key findings

- Cohomology determined by Euler characteristic under intersection conditions
- Betti numbers computed for general stable bundles
- Classification of globally generated stable bundles

## Abstract

In this paper, we show that the cohomology of a general stable bundle on a Hirzebruch surface is determined by the Euler characteristic provided that the first Chern class satisfies necessary intersection conditions. More generally, we compute the Betti numbers of a general stable bundle. We also show that a general stable bundle on a Hirzebruch surface has a special resolution generalizing the Gaeta resolution on the projective plane. As a consequence of these results, we classify Chern characters such that the general stable bundle is globally generated.