# Cohomology of moduli spaces of Del Pezzo surfaces

## TL;DR

This paper computes the rational Betti cohomology groups of moduli spaces of Del Pezzo surfaces of degrees three and four, revealing their structure as Weyl group representations.

## Contribution

It provides explicit cohomology computations for these moduli spaces using novel methods combining point counting and arrangement complement techniques.

## Key findings

- Cohomology groups are explicitly computed for degree 3 and 4 Del Pezzo surfaces.
- Results describe the cohomology as Weyl group representations.
- Methodology blends algebraic geometry with combinatorial techniques.

## Abstract

We compute the rational Betti cohomology groups of the coarse moduli spaces of geometrically marked Del Pezzo surfaces of degree three and four as representations of the Weyl groups of the corresponding root systems. The proof uses a blend of methods from point counting over finite fields and techniques from arrangement complements.