Moduli of quiver representations for exceptional collections on surfaces

Abstract

Suppose $S$ is a smooth projective surface over an algebraically closed field $k$, $\mathcal{L}=\{L_1,\dots ,L_n\}$ is a full strong exceptional collection of line bundles on $S$. Let $Q$ be the quiver associated to this collection. One might hope that $S$ is the moduli space of representations of $Q$ with dimension vector $(1,\dots ,1)$ for a suitably chosen stability condition $\theta$: $S\cong M_{\theta }$. In this paper, we show that this is the case for del Pezzo surfaces. Furthermore, we show the blow-up at a point can be recovered from an augmentation of exceptional collections (in the sense of L. Hille and M.Perling) via morphism between moduli of quiver representations.

34 pages. New version deals only with exceptional collections on del Pezzo surfaces with cleaner presentation. For general cases, please see the old version