# Topological Formulae for the Zeroth Cohomology of Line Bundles on del Pezzo and Hirzebruch Surfaces

## TL;DR

This paper demonstrates that the zeroth cohomology of effective line bundles on del Pezzo and Hirzebruch surfaces can be calculated using a topological index, simplifying cohomology computations on these surfaces.

## Contribution

It introduces a topological formula for computing the zeroth cohomology of line bundles on specific algebraic surfaces, providing a new computational approach.

## Key findings

- Cohomology can be derived from topological indices on these surfaces.
- The method simplifies calculations compared to traditional algebraic techniques.
- Applicable to effective line bundles on del Pezzo and Hirzebruch surfaces.

## Abstract

We show that the zeroth cohomology of effective line bundles on del Pezzo and Hirzebruch surfaces can always be computed in terms of a topological index.