## Local properties of algebraic group actions.(English)Zbl 0722.14032

Algebraische Transformationsgruppen und Invariantentheorie, DMV Semin. 13, 63-75 (1989).
[For the entire collection see Zbl 0682.00008.]
Let G be a connected linear algebraic group and X be a normal G-variety over an algebraically closed field of characteristic zero. The authors give a new proof of the following result of Sumihiro:
Let $$Y\subset X$$ be an orbit in X. There is a finite-dimensional rational representation $$G\to GL(V)$$ and a G-stable open neighborhood U of Y in X which is G-equivariantly isomorphic to a G-stable locally closed subvariety of the projective space P(V).
The main technical ingredients are G-linearizations of line bundles and the study of the Picard group of a linear algebraic group.

### MSC:

 14L30 Group actions on varieties or schemes (quotients) 20G05 Representation theory for linear algebraic groups 14C22 Picard groups

### Keywords:

Picard group of a linear algebraic group

Zbl 0682.00008