Motivic measures of the moduli spaces of pure sheaves on $\mathbb{P}^2$ with all degrees

Abstract

Let $\mathcal{M}(d,\chi)$ be the moduli stack of stable sheaves of rank 0, Euler characteristic $\chi$ and first Chern class $dH~(d>0)$, with $H$ the hyperplane class in $\mathbb{P}^2$. We compute the $A$-valued motivic measure $\mu_A(\mathcal{M}(d,\chi))$ of $\mathcal{M}(d,\chi)$ and get explicit formula in codimension $D:=\rho_d-1$, where $\rho_d$ is $d-1$ for $d=p$ or $2p$ with $p$ prime, and $7$ otherwise. As a corollary, we get the last $2(D+1)$ Betti numbers of the moduli scheme $M(d,\chi)$ when $d$ is coprime to $\chi$.