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 $\chi$.

References in corpus (1)

• Invariants of moduli spaces of stable sheaves on ruled surfaces