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15 February 2011 Quiver flag varieties and multigraded linear series
Alastair Craw
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Duke Math. J. 156(3): 469-500 (15 February 2011). DOI: 10.1215/00127094-2010-217


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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Alastair Craw. "Quiver flag varieties and multigraded linear series." Duke Math. J. 156 (3) 469 - 500, 15 February 2011.


Published: 15 February 2011
First available in Project Euclid: 9 February 2011

zbMATH: 1213.14026
MathSciNet: MR2772068
Digital Object Identifier: 10.1215/00127094-2010-217

Primary: 14D22 , 16G20 , 18E30
Secondary: 14M15 , 14M25

Rights: Copyright © 2011 Duke University Press


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Vol.156 • No. 3 • 15 February 2011
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