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

Source abstract

We compute the integral $K_0$-ring of the moduli stack $\mathscr H_g$ of smooth hyperelliptic curves of genus $g\ge2$ 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 $μ_{g+1}$-gerbe to obtain presentations for both even and odd $g$'s from the standard quotient presentation of $\mathscr H_g$. Thus, our technique differs from the $CH^\bullet(\mathscr H_g)$ computations of Edidin--Fulghesu, Fulghesu--Viviani, Di Lorenzo, and the $K_0(\mathscr M_2)$ computation of Edidin--Hu. Using Pagani's description of the inertia stack of $\mathscr H_g$, we apply Riemann--Roch to interpret the rank of $K_0(\mathscr H_g)$ as the number of inertia components. Finally, we compare the $K_0$ and Chow rings of $\mathscr H_g$ through the gamma-filtration and give presentations to the associated graded rings $\operatorname{gr}_γK_0(\mathscr H_g)\otimes\mathbb F_p$ for every prime $p$.