On the Stabilization of the Betti Numbers of the Moduli Space of Sheaves on ℙ2

Abstract

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