Olof Bergvall: Moduli of plane quartics with level 2 structure

A plane quartic curve is a subvariety of the projective plane defined by a homogeneous polynomial of degree 4. A bitangent to a curve is a line which is tangent to the curve at two points and already in 1834 Plücker showed that a plane quartic has 28 bitangents. From a more modern perspective, bitangents can be used to define so called "level structures" which have many interesting properties and applications. Most notably they were used by Mumford in order to construct moduli spaces of curves.

In this talk we shall study the moduli space of plane quartics with level 2 structure and in particular investigate its cohomology. On one hand, we shall use a description due to Looijenga in terms of arrangements of tori, and on the other we shall use point counts over finite fields and Lefschetz trace formula. The spaces we shall encounter will have groups acting naturally on them and their cohomology groups will therefore become representations of these groups. We will discuss these representations and we will also consider the mixed Hodge structures of the cohomology groups.