Abstract
We determine the cohomology groups of the quartic and hyperelliptic loci inside the moduli space of genus three curves with symplectic level two structure as representations of the symmetric group S_7 together with their mixed Hodge structures by means of making equivariant point counts over finite fields and via purity arguments. This determines the weighted Euler characteristic of the whole moduli space of genus three curves with level two structure.
Highlights
The purpose of this paper is to study the cohomology of the moduli space M3[2] of genus 3 curves with symplectic level 2 structure
In the case of a plane quartic curve, which shall be the case of most importance to us, we point out that each theta characteristic occurring in an Aronhold basis is cut out by a bitangent line
The rows correspond to the cohomology groups and the columns correspond to the irreducible representations of S7
Summary
Let n be a positive integer and let C be a curve. A level n structure on C is a basis for the n-torsion of the Jacobian of C. The purpose of this paper is to study the cohomology of the moduli space M3[2] of genus 3 curves with symplectic level 2 structure. A plane quartic with level 2 structure is specified, up to isomorphism, by an ordered septuple of points in general position in P2, up to the action of PGL(3) It may be of interest to compare the present paper to the work of Bergström [3,4], and Bergström and Tommasi [5] which investigate cohomological questions about moduli spaces of low genus curves via point counts. It should be mentioned that our results are an essential ingredient in the article [7] where further information about the action of Sp(6, Z/2Z) on the cohomology M3[2] and Q[2] is obtained via quite different methods
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