Quotients of spectra of almost factorial domains and Mori dream spaces

TL;DR

This paper shows that GIT quotients of affine varieties with almost factorial domains are Mori dream spaces if projective, providing explicit descriptions of their geometric cones and applying results to quiver moduli and geometric quotients.

Contribution

It establishes that certain GIT quotients are Mori dream spaces with explicit geometric descriptions, extending to quiver moduli and geometric quotients.

Findings

GIT chamber quotients of almost factorial domains are Mori dream spaces if projective.

Explicit descriptions of Picard number, pseudoeffective cone, and Mori chambers are provided.

Quiver moduli without oriented cycles are Mori dream spaces, with a formula for Picard number.

Abstract

We prove that a GIT chamber quotient of an affine variety $X=Spec(A)$ by a reductive group $G$, where $A$ is an almost factorial domain, is a Mori dream space if it is projective, regardless of the codimension of the unstable locus. This includes an explicit description of the Picard number, the pseudoeffective cone, and the Mori chambers in terms of GIT. We apply the results to quiver moduli, to show that they are Mori dream spaces if the quiver contains no oriented cycles, and if stability and semistability coincide. We give a formula for the Picard number in quiver terms. As a second application we prove that geometric quotients of Mori dream spaces are Mori dream spaces as well, which again includes a description of the Picard number and the Mori chambers. Some examples are given to illustrate the results.