Mori Dream Spaces as fine moduli of quiver representations

TL;DR

This paper constructs Mori Dream Spaces as fine moduli spaces of quiver representations, extending previous results beyond toric varieties and providing conditions for embeddings and reconstructions of surfaces.

Contribution

It introduces a method to realize Mori Dream Spaces as moduli spaces of quiver representations, generalizing prior work to non-toric cases and applying it to reconstruct del Pezzo surfaces.

Findings

Conditions for the multigraded linear series to be a closed immersion

Realization of Mori Dream Spaces as moduli spaces of stable quiver representations

Reconstruction of del Pezzo surfaces from exceptional collections

Abstract

We construct Mori Dream Spaces as fine moduli spaces of representations of bound quivers, thereby extending results of Craw--Smith \cite{CrawSmith} beyond the toric case. Any collection of effective line bundles $\mathscr{L}=(\mathscr{O}_X, L_1,..., L_r)$ on a Mori Dream Space $X$ defines a bound quiver of sections and a map from $X$ to a toric quiver variety $|\mathscr{L}|$ called the multigraded linear series. We provide necessary and sufficient conditions for this map to be a closed immersion and, under additional assumptions on $\mathscr{L}$, the image realises $X$ as the fine moduli space of $\vartheta$-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.