On the stabilization of the Betti numbers of the moduli space of sheaves on ${\mathbb{P}}^2$

Abstract

Let $r\ge 2$ be an integer, and let $a$ be an integer coprime to $r$. We show that if $c_2\ge n+\lfloor \frac{r-1}{2r}{a}^2+\frac{1}{2}(r^2+1)\rfloor$, then the $2n$th Betti number of the moduli space $M_{{\mathbb{P}}^2,H}(r,aH,c_2)$ stabilizes, where $H=c_1({\mathcal{O}}_{{\mathbb{P}}^2}(1))$.

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