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(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 = (OX,E1,...,E) on a projective scheme X to be the quiver flag variety |E| = M(Q,r) of a pair (Q,r) encoded by E. When each Ei is globally generated, we obtain a morphism φ|E| : X -> |E| realising each Ei as the pullback of a tautological bundle. As an application we introduce the multigraded Plucker embedding of a quiver flag variety