A Hilbert-Mumford criterion for SL_2-actions

Abstract

Let the special linear group $G:=S{L}_2$ act regularly on a $Q$-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 $g\dot{U}$ is open in $X$ and admits a good quotient $W(U)\to W(U)//G$ with a divisorial quotient space. Conversely, we obtain 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.

9 pages, to appear in Coll. Math

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