Let the special linear group G : = SL₂ act regularly on a ℚ-factorial variety X. Consider a maximal torus T ⊂ G and its normalizer N ⊂ G. We prove: If U ⊂ X is a maximal open N-invariant subset admitting a good quotient U → U ⃫N with a divisorial quotient space, then the intersection W(U) of all translates g · U is open in X and admits a good quotient W(U) → W(U) ⃫G with a divisorial quotient space. Conversely, we show that every maximal open G-invariant subset W ⊂ X admitting a good quotient W → W ⃫G with a divisorial quotient space is of the form W = W(U) for some maximal open N-invariant U as above.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm97-2-2,
author = {J\"urgen Hausen},
title = {A Hilbert-Mumford criterion for SL2-actions},
journal = {Colloquium Mathematicae},
volume = {96},
year = {2003},
pages = {151-161},
zbl = {1054.14060},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm97-2-2}
}
Jürgen Hausen. A Hilbert-Mumford criterion for SL₂-actions. Colloquium Mathematicae, Tome 96 (2003) pp. 151-161. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm97-2-2/