Duke Mathematical Journal

Quiver flag varieties and multigraded linear series

Alastair Craw

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Abstract

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.

Article information

Source
Duke Math. J., Volume 156, Number 3 (2011), 469-500.

Dates
First available in Project Euclid: 9 February 2011

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1297258907

Digital Object Identifier
doi:10.1215/00127094-2010-217

Mathematical Reviews number (MathSciNet)
MR2772068

Zentralblatt MATH identifier
1213.14026

Subjects
Primary: 14D22: Fine and coarse moduli spaces 16G20: Representations of quivers and partially ordered sets 18E30: Derived categories, triangulated categories
Secondary: 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 14M25: Toric varieties, Newton polyhedra [See also 52B20]

Citation

Craw, Alastair. Quiver flag varieties and multigraded linear series. Duke Math. J. 156 (2011), no. 3, 469--500. doi:10.1215/00127094-2010-217. https://projecteuclid.org/euclid.dmj/1297258907


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