A Hilbert-Mumford criterion for SL_2-actions

TL;DR

This paper establishes a criterion for SL_2-actions on factorial varieties, linking maximal open invariant subsets under the normalizer to those under the whole group, with implications for good quotients and divisorial spaces.

Contribution

It provides a Hilbert-Mumford type criterion for SL_2-actions, characterizing maximal invariant open subsets with good quotients in terms of the normalizer's action.

Findings

Characterization of maximal open N-invariant subsets with good quotients

Equivalence between N-invariant and G-invariant maximal open subsets with divisorial quotients

Extension of the Hilbert-Mumford criterion to SL_2-actions on factorial varieties

Abstract

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.