The effective cone of the moduli space of sheaves on the plane

Abstract

Let $\xi$ be the Chern character of a stable coherent sheaf on ${\mathbb{P}}^2$. We compute the cone of effective divisors on the moduli space $M(\xi )$ of semistable sheaves on ${\mathbb{P}}^2$ with Chern character $\xi$. The computation hinges on finding a good resolution of the general sheaf in $M(\xi )$. This resolution is determined by Bridgeland stability and arises from a well-chosen Beilinson spectral sequence. The existence of a good choice of spectral sequence depends on remarkable number-theoretic properties of the slopes of exceptional bundles.