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A Hilbert-Mumford criterion for SL₂-actions

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Hausen J.

Colloquium Mathematicum

2003

tom 97

nr 2

151-161

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.

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1 Colloquium Mathematicum

1 Hausen J.

1 2003

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A Hilbert-Mumford criterion for SL₂-actions