Abstract
We study the 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 point and strata of the moduli space of Abelian differentials of genus three.
Highlights
The purpose of this note is to compute the de Rham cohomology of various moduli spaces of curves of genus 3 with level 2 structure
The cohomology groups become Sp(6, F2)-representations and our goal is to describe these cohomology groups as representations together with their mixed Hodge structure
We have the moduli space Hol 3[2] of genus 3 curves with level 2 structure marked with a holomorphic (i.e. Abelian) differential and some related spaces, e.g. the moduli space of genus 3 curves marked with a canonical divisor
Summary
The purpose of this note is to compute the de Rham cohomology of various moduli spaces of curves of genus 3 with level 2 structure. The group Sp(6, F2) acts on the set of level 2 structures of a curve This action induces actions on the various moduli spaces under consideration which in turn yields actions on the cohomology groups. We have the moduli space M3[2] of genus 3 curves with level 2 structure and some natural loci therein. This will be our main object of study. There are many constructions, some classical and some new, relating the various spaces and which will provide essential information for our cohomological computations
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
