The orbifold cohomology of moduli of genus 3 curves

TL;DR

This paper investigates the orbifold cohomology of the moduli space of genus 3 curves, reducing the problem to cyclic covers and explicitly computing parts of the cohomology for genus 3.

Contribution

It provides a detailed analysis of the orbifold cohomology for genus 3 curves and relates it to the cohomology of cyclic covers, extending previous work on moduli spaces.

Findings

Reduced orbifold cohomology problem to cyclic covers of genus g

Explicit computation of orbifold cohomology for genus 3

Identified cohomology contributions from the inertia stack closure

Abstract

In this work we study the additive orbifold cohomology of the moduli stack of smooth genus g curves. We show that this problem reduces to investigating the rational cohomology of moduli spaces of cyclic covers of curves where the genus of the covering curve is g. Then we work out the case of genus g=3. Furthermore, we determine the part of the orbifold cohomology of the Deligne-Mumford compactification of the moduli space of genus 3 curves that comes from the Zariski closure of the inertia stack of M_3.