TY - JOUR
T1 - Mori Dream Spaces as fine moduli of quiver representations
AU - Craw, Alastair
AU - Winn, Dorothy
PY - 2013/1
Y1 - 2013/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84865974840&partnerID=8YFLogxK
UR - http://arxiv.org/abs/1104.2490
UR - http://dx.doi.org/10.1016/j.jpaa.2012.06.014
U2 - 10.1016/j.jpaa.2012.06.014
DO - 10.1016/j.jpaa.2012.06.014
M3 - Article
SN - 0022-4049
VL - 217
SP - 172
EP - 189
JO - Journal of Pure and Applied Algebra
JF - Journal of Pure and Applied Algebra
IS - 1
ER -