A Hilbert-Mumford criterion for SL_2-actions

Details A Hilbert-Mumford criterion for SL_2-actions A Hilbert-Mumford criterion for SL_2-actions

2002-11-26

arxiv.org/abs/math/0211412v1

Coll. Math. Vol. 97, No. 2, 151-161 (2003)

Mathematics

Algebraic Geometry

9 pages, to appear in Coll. Math

Scientific paper

Let the special linear group $G := SL_{2}$ 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.

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