Quiver flag varieties and multigraded linear series
Alastair Craw
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Source: Crossref
Published: Feb 15, 2011
DOI: 10.1215/00127094-2010-217
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This paper introduces a class of smooth projective varieties that generalize and share many properties with partial flag varieties of type A. The quiver flag variety Mϑ(Q,r̲) of a finite acyclic quiver Q (with a unique source) and a dimension vector r̲ is a fine moduli space of stable representations of Q. Quiver flag varieties are Mori dream spaces, they are obtained via a tower of Grassmann bundles, and their bounded derived category of coherent sheaves is generated by a tilting bundle. We define the multigraded linear series of a weakly exceptional sequence of locally free sheaves E̲=(OX,E1,…,Eρ) on a projective scheme X to be the quiver flag variety |E̲|:=Mϑ(Q,r̲) of a pair (Q,r̲) encoded by E̲. When each Ei is globally generated, we obtain a morphism ϕ|E̲|:X→|E̲|, realizing each Ei as the pullback of a tautological bundle. As an application, we introduce the multigraded Plücker embedding of a quiver flag variety.
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