Quiver flag varieties and multigraded linear series

Abstract

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 $M_{\vartheta }(Q,\underline{r})$ of a finite acyclic quiver $Q$ (with a unique source) and a dimension vector $\underline{r}$ 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 $\underline{E}=(O_X,E_1,\dots ,E_{\rho })$ on a projective scheme $X$ to be the quiver flag variety $\left|\underline{E}\right|:=M_{\vartheta }(Q,\underline{r})$ of a pair $(Q,\underline{r})$ encoded by $\underline{E}$. When each $E_i$ is globally generated, we obtain a morphism ${\varphi }_{\left|\underline{E}\right|}:X\to \left|\underline{E}\right|$, realizing each $E_i$ as the pullback of a tautological bundle. As an application, we introduce the multigraded Plücker embedding of a quiver flag variety.