The integral $K_0$-ring of the stack of smooth hyperelliptic curves

Abstract

We compute the integral $K_0$-ring of the moduli stack ${\mathcal{H}}_g$ of smooth hyperelliptic curves of genus $g\ge 2$ over a field of characteristic zero. We show that it is a free abelian group of rank $8g+2$ for even $g$ and $8g+4$ for odd $g$. We use the isotypic decomposition of the $K_0$ of a related ${\mu }_{g+1}$-gerbe to obtain presentations for both even and odd $g$'s from the standard quotient presentation of ${\mathcal{H}}_g$. Thus, our technique differs from the $C{H}^{\bullet }({\mathcal{H}}_g)$ computations of Edidin--Fulghesu, Fulghesu--Viviani, Di Lorenzo, and the $K_0({\mathcal{M}}_2)$ computation of Edidin--Hu. Using Pagani's description of the inertia stack of ${\mathcal{H}}_g$, we apply Riemann--Roch to interpret the rank of $K_0({\mathcal{H}}_g)$ as the number of inertia components. Finally, we compare the $K_0$ and Chow rings of ${\mathcal{H}}_g$ through the gamma-filtration and give presentations to the associated graded rings ${\mathrm{gr}}_{\gamma }{K}_0({\mathcal{H}}_g)\otimes {\mathbb{F}}_p$ for every prime $p$.

arXiv comment30 pages, 1 figure. Comments welcome!