The Orbifold Cohomology of Moduli of Hyperelliptic Curves

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