Duke Mathematical Journal

Quiver flag varieties and multigraded linear series

Alastair Craw

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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.

Article information

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

First available in Project Euclid: 9 February 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


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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  • K. Altmann and L. Hille, Strong exceptional sequences provided by quivers, Algebr. Represent. Theory 2 (1999), 1–17.
  • D. Baer, Tilting sheaves in representation theory of algebras, Manuscripta Math. 60 (1988), 323–347.
  • A. A. Beĭlinson, Coherent sheaves on $\mathbb{P}^{n}$ and problems in linear algebra, Funktsional. Anal. Prilozhen. 12 (1978), 68–69.
  • A. I. Bondal, “Helices, representations of quivers and Koszul algebras” in Helices and Vector Bundles, London Math. Soc. Lecture Note Ser. 148, Cambridge Univ. Press, Cambridge, 1990, 75–95.
  • N. Broomhead, Cohomology of line bundles on a toric variety and constructible sheaves on its polytope, preprint.
  • L. Costa and R. M. Miró-roig, Tilting sheaves on toric varieties, Math. Z. 248 (2004), 849–865.
  • A. Craw and G. G. Smith, Projective toric varieties as fine moduli spaces of quiver representations, Amer. J. Math. 130 (2008), 1509–1534.
  • W. Crawley-Boevey, Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001), 257–293.
  • J.-P. Demailly, Vanishing theorems for tensor powers of an ample vector bundle, Invent. Math. 91 (1988), 203–220.
  • H. Derksen and J. Weyman, The combinatorics of quiver representations, preprint.
  • D. Eisenbud, M. Mustaţǎ, and M. Stillman, Cohomology on toric varieties and local cohomology with monomial supports, J. Symbolic Comput. 29 (2000), 583–600.
  • W. Fulton, “Young tableaux” in With Applications to Representation Theory and Geometry, London Math. Soc. Student Texts 138, Cambridge Univ. Press, Cambridge, 1997.
  • V. Ginzburg, Lectures on Nakajima's quiver varieties, preprint.
  • D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, preprint.
  • A. Grothendieck, Éléments de géométrie algébrique, II: Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961).
  • A. Grothendieck and J. Dieudonné, Éléments de Géométrie Algébrique, I, 2nd ed. Springer, New York, 1971.
  • M. Halic, Strong exceptional sequences of vector bundles on certain Fano varieties, preprint.
  • M. Hering, M. Mustaţǎ, and S. Payne, Positivity properties for toric vector bundles, Ann. Inst. Fourier 60 (2010), 607–640.
  • L. Hille, “Toric quiver varieties” in Algebras and Modules, II (Geiranger, 1996), CMS Conf. Proc. 24, Amer. Math. Soc., Providence, 1998, 311–325.
  • Y. Hu and S. Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348.
  • M. M. Kapranov, Derived category of coherent sheaves on Grassmann manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), 192–202.
  • –-, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), 479–508.
  • A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. 45 (1994), 515–530.
  • –-, Tilting bundles on some rational surfaces, preprint, 1997.
  • D. O. Orlov, Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), 852–862.