# Equivariant cohomology of moduli spaces of genus three curves with level two structure

## TL;DR

This paper computes the equivariant cohomology of the moduli space of genus three curves with level two structure, revealing detailed algebraic and geometric properties of these spaces and their symmetries.

## Contribution

It provides explicit cohomology computations for genus three moduli spaces with level two structure, including representations of the symplectic group, and extends to related moduli spaces.

## Key findings

- Cohomology groups of the moduli space of plane quartics with level two structure are determined.
- Representation of the symplectic group on cohomology is explicitly described.
- Analogous computations are performed for related moduli spaces such as marked curves and Abelian differentials.

## Abstract

We compute cohomology of the moduli space of genus three curves with level two structure and some related spaces. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as representations of the symplectic group on a six dimensional vector space over the field of two elements. We also make the analogous computations for some related spaces such as moduli spaces of genus three curves with a marked points and strata of the moduli space of Abelian differentials of genus three.