Mathematics – Algebraic Geometry
Scientific paper
2009-07-27
Duke Mathematical Journal 156 (2011) 469-500
Mathematics
Algebraic Geometry
23 pages. Final version includes minor changes, to appear in Duke Math Journal
Scientific paper
This paper introduces a class of smooth projective varieties that generalise and share many properties with partial flag varieties of type A. The quiver flag variety M_\vartheta(Q,r) of a finite acyclic quiver Q (with a unique source) and a dimension vector r 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 E = (O_X,E_1,...,E_\rho) on a projective scheme X to be the quiver flag variety |E| = M_\vartheta(Q,r) of a pair (Q,r) encoded by E. When each E_i is globally generated, we obtain a morphism \phi_|E| : X -> |E| realising each E_i as the pullback of a tautological bundle. As an application we introduce the multigraded Plucker embedding of a quiver flag variety
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