Local properties of algebraic group actions
(joint work with Hanspeter Kraft, Domingo Luna, and Thierry Vust)

Let G be a connected linear algebraic group acting on a normal variety X. This note contains two proofs of a basic theorem of Sumihiro which we hope are more transparent than the original one.

Theorem: 1. Every point of X has a G-stable open quasi-projective neighborhood.
2. If X is quasi-projective then it can be equivariantly embedded into a projective space.

The first proof uses the language of line bundles, the second field and valuation theory. In the last section, the Picard group of G is studied.

Appeared in: Algebraische Transformationsgruppen und Invariantentheorie (H. Kraft, P. Slodowy, T. Springer eds.) DMV-Seminar 13, Birkhäuser Verlag (Basel-Boston) (1989) 63-76

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Last updated: May 24, 2005