Olof Bergvall: Cohomology of arrangements and moduli spaces

Abstract:
This thesis mainly concerns the cohomology of the moduli spaces \(M_3[2]\) and \(M_{3,1}[2]\) of genus 3 curves with level 2 structure without respectively with a marked point and some of their natural subspaces. A genus 3 curve which is not hyperelliptic can be realized as a plane quartic and the moduli spaces Q[2] and \(Q_1[2]\) of plane quartics without respectively with a marked point are given special attention. The spaces considered come with a natural action of the symplectic group \(Sp(6,2)\) and their cohomology groups thus become \(Sp(6,2)\)-representations. All computations are therefore \(Sp(6,2)\)-equivariant. We also study the mixed Hodge structures of these cohomology groups.

The computations for \(M_3[2]\) are mainly via point counts over finite fields while the computations for \(M_{3,1}[2]\) primarily use a description due to Looijenga in terms of arrangements associated to root systems. This leads us to the computation of the cohomology of complements of toric arrangements associated to root systems. These varieties come with an action of the corresponding Weyl groups and the computations are equivariant with respect to this action.