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.