A Hilbert–Mumford criterion for $SL_2$-actions

Jürgen Hausen¹

¹ Mathematisches Forschungsinstitut Oberwolfach Lorenzenhof 77709 Oberwolfach-Walke, Germany

Colloquium Mathematicum, Tome 97 (2003) no. 2, pp. 151-161.

Résumé

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.

Détail

DOI : 10.4064/cm97-2-2

Keywords: special linear group mathop nolimits act regularly mathbb factorial variety consider maximal torus subset its normalizer subset prove subset maximal n invariant subset admitting quotient divisorial quotient space intersection translates cdot admits quotient divisorial quotient space conversely every maximal g invariant subset subset admitting quotient divisorial quotient space form maximal n invariant above