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

TL;DR

This paper computes motivic measures of moduli stacks of pure sheaves on the projective plane, providing explicit formulas and Betti number information for associated moduli schemes, advancing understanding of their geometric and topological properties.

Contribution

It introduces explicit formulas for motivic measures of moduli stacks of pure sheaves on , linking these to Betti numbers of moduli schemes, with novel calculations for various degrees.

Findings

Explicit motivic measure formulas for  moduli stacks.

Betti number calculations for moduli schemes when degree and Euler characteristic are coprime.

Identification of codimension D related to degree and primality conditions.

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 $\chi$.