The orbifold cohomology of moduli of hyperelliptic curves

TL;DR

This paper analyzes the inertia stack of moduli spaces of hyperelliptic curves to compute their orbifold cohomology, providing new insights into the geometric structure of these moduli stacks.

Contribution

It offers a novel description of the inertia stack of hyperelliptic moduli spaces and computes their orbifold cohomology additively.

Findings

Description of the inertia stack of H_g

Additive computation of Chen-Ruan cohomology of H_g

Connections between moduli of genus 0 and hyperelliptic curves

Abstract

We study the inertia stack of [M_{0,n}/S_n], the quotient stack of the moduli space of smooth genus 0 curves with n marked points via the action of the symmetric group S_n. Then we see how from this analysis we can obtain a description of the inertia stack of H_g, the moduli stack of hyperelliptic curves of genus g. From this, we can compute additively the Chen-Ruan (or orbifold) cohomology of H_g.