Quiver flag varieties and multigraded linear series

Abstract

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_ρ) 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 ϕ_|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

23 pages. Final version includes minor changes, to appear in Duke Math Journal

References in corpus (4)

• Lectures on Nakajima's Quiver Varieties
• Cohomology of line bundles on a toric variety and constructible sheaves on its polytope
• Strong exceptional sequences of vector bundles on certain Fano varieties
• Stability islands in domains of separatrix crossings in slow-fast Hamiltonian systems

Cited by in corpus (8)

• Tilting objects on twisted forms of some relative flag varieties
• A rim-hook rule for quiver flag varieties
• Semi-invariants of filtered quiver representations with at most two pathways
• Blow ups of ${\mathbb{P}}^n$ as quiver moduli for exceptional collections
• Quotients of spectra of almost factorial domains and Mori dream spaces
• Mori Dream Spaces as fine moduli of quiver representations
• Reconstructing the Grassmannian of lines from Kapranov's tilting bundle
• Suggestions to study affine and GIT quotients of the extended Grothendieck--Springer resolution