Quiver flag varieties and multigraded linear series

Craw, A. (2011) Quiver flag varieties and multigraded linear series. Duke Mathematical Journal, 156(3), pp. 469-500. (doi:10.1215/00127094-2010-217)

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Abstract

This paper introduces a class of smooth projective varieties that generalise and share many properties with partial flag varieties of type <i><b>A</b></i>. The quiver flag variety <i><b>M<sub>θ</sub>(Q,<u>r</u></b></i>) of a finite acyclic quiver <b><i>Q</i></b> (with a unique source) and a dimension vector <i><b><u>r</u></b></i> is a fine moduli space of stable representations of <b><i>Q</i></b>. 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 <b><i><u>E</u> = (O<sub>X</sub>,E_1,...,E<sub>ρ</sub>)</i></b> on a projective scheme <b>X</b> to be the quiver flag variety <b><i>|<u>E</u>|= Mθ(Q,<u>r</u>)</i></b> of a pair <b><i>(Q,<u>r</u></i></b>) encoded by <b><i><u>E</u></i></b>. When each <i><b>E<sub>i</sub></b></i> is globally generated, we obtain a morphism <i><b>φ<sub>|<u>E</u>|</sub> : X -> |<u>E</u>|</b></i> realising each <i><b>E<sub>i</sub></b></i> as the pullback of a tautological bundle. As an application we introduce the multigraded Plucker embedding of a quiver flag variety

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Craw, Dr Alastair
Authors: Craw, A.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics
Journal Name:Duke Mathematical Journal
Publisher:Duke University Press
ISSN:0012-7094
ISSN (Online):1547-7398
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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
480011Noncommutative toric geometry and multilinear seriesAlastair CrawEngineering & Physical Sciences Research Council (EPSRC)EP/G004048/1Mathematics