The Picard group of a G-variety. (English) Zbl 0705.14005

Algebraische Transformationsgruppen und Invariantentheorie, DMV Semin. 13, 77-87 (1989).
[For the entire collection see Zbl 0682.00008.]
Let G be a reductive algebraic group and X an irreducible G-variety which admits a quotient \(X\to X//G\). The aim of this paper is to describe the link between the groups Pic(X), Pic(X//G), \(Pic_ G(X) (= group\) of isomorphism classes of G-line bundles) on the one hand and the groups \({\mathcal O}(X//G)^*/k^*\), \(H^ 1_{alg}(G,{\mathcal O}(X)^*)\), \(\chi (G_ x) (= character\) groups of isotropy groups) on the other hand. As an application it is proved that if X is the affine space on which G acts via a representation then \(Pic(X//G)=0\) and \(Pic_ G(X)=\chi (G)\).
Reviewer: A.Buium


14C22 Picard groups
14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations


Zbl 0682.00008