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Abstract

Let the special linear group $G := \mathop {\rm SL}\nolimits _{2}$ act regularly on a ${{\mathbb Q}}$-factorial variety $X$. Consider a maximal torus $T \subset G$ and its normalizer $N \subset G$. We prove: If $U \subset X$ is a maximal open $N$-invariant subset admitting a good quotient $U \to U /\! \! /N$ with a divisorial quotient space, then the intersection $W(U)$ of all translates $g \cdot U$ is open in $X$ and admits a good quotient $W(U) \to W(U) /\! \! /G$ with a divisorial quotient space. Conversely, we show that every maximal open $G$-invariant subset $W \subset X$ admitting a good quotient $W \to W /\! \! /G$ with a divisorial quotient space is of the form $W = W(U)$ for some maximal open $N$-invariant $U$ as above.