Z. Reichstein, N. Vonessen , Stable affine models for algebraic group actions , Journal of Lie Theory, no. 2 (2004), 563--568.

Abstract: Let G be a reductive linear algebraic group defined over an algebraically closed base field k of characteristic zero. A G-variety X is an algebraic variety with a regular action of G, defined over k. An affine G-variety Y is called stable if points of Y in general position have closed G-orbits. Our main result is a necessary and sufficient condition for a G-variety X to have a stable affine birational model.

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