Birational geometry of quiver varieties and other GIT quotients

We introduce a sufficient condition for the Geometric Invariant Theory (GIT) quotient of an affine variety $V$ by the action of a reductive group $G$ to be a relative Mori Dream Space. When the condition holds, we show that the linearisation map identifies a region of the GIT fan with the Mori chamber decomposition of the relative movable cone of $V/\!/_\theta \, G$. If $V/\!/_\theta \, G$ is a crepant resolution of $Y\!\!:= V/\!/_0 \, G$, then every projective crepant resolution of $Y$ is obtained by varying $\theta$. Under suitable conditions, we show that this is the case for Nakajima quiver varieties; in particular, all projective partial crepant resolutions of the affine quiver variety $Y$ are quiver varieties. Similarly, for any finite subgroup $\Gamma\subset \mathrm{SL}(3,\mathbb{k})$ whose nontrivial conjugacy classes are all junior, we obtain a simple geometric proof of the fact that every projective crepant resolution of $\mathbb{A}^3/\Gamma$ is a fine moduli space of $\theta$-stable $\Gamma$-constellations. Our methods apply equally well to nonsingular hypertoric varieties.