Mori Dream Spaces as fine moduli of quiver representations

Details Mori Dream Spaces as fine moduli of quiver representations Mori Dream Spaces as fine moduli of quiver representations

2011-04-13

arxiv.org/abs/1104.2490v2

Mathematics

Algebraic Geometry

21 pages, 2 figures; v2 section 3 simplified, typos corrected

Scientific paper

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.

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