Quiver flag varieties and multigraded linear series

TL;DR

This paper studies quiver flag varieties, a class of smooth projective varieties generalizing partial flag varieties, and introduces multigraded linear series and embeddings related to these structures.

Contribution

It defines multigraded linear series for weakly exceptional sheaves and explores their geometric properties, including embeddings and moduli space structures.

Findings

Quiver flag varieties are Mori Dream Spaces.

They are constructed via towers of Grassmann bundles.

Multigraded linear series generalize classical linear series and admit Plucker embeddings.

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_\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