Mori Dream Spaces as fine moduli of quiver representations

Abstract

We construct Mori Dream Spaces as fine moduli spaces of representations of bound quivers, thereby extending results of Craw–Smith [6] beyond the toric case. Any collection of effective line bundles ℒ=(풪X,L1,…,Lr) on a Mori Dream Space X defines a bound quiver of sections and a map from X to a toric quiver variety |ℒ| called the multigraded linear series. We provide necessary and sufficient conditions for this map to be a closed immersion and, under additional assumptions on ℒ, the image realises X as the fine moduli space of ϑ-stable representations of the bound quiver. As an application, we show how to reconstruct del Pezzo surfaces from a full, strongly exceptional collection of line bundles.