A correspondence of good G-sets under partial geometric quotients

Abstract

For a complex variety $\hat{X}$ with an action of a reductive group $\hat{G}$ and a geometric quotient $\pi :\hat{X}\to X$ by a closed normal subgroup $H\subset \hat{G}$, we show that open sets of $X$ admitting good quotients by $G=\hat{G}/H$ correspond bijectively to open sets in $\hat{X}$ with good $\hat{G}$-quotients. We use this to compute GIT-chambers and their associated quotients for the diagonal action of ${\text{PGL}}_2$ on $({\mathbb{P}}^1{)}^n$ in certain subcones of the ${\text{PGL}}_2$-effective cone via a torus action on affine space. This allows us to represent these quotients as toric varieties with fans determined by convex geometry.

arXiv comment18 pages, comments welcome