Knop, Friedrich; Kraft, Hanspeter; Vust, Thierry 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 Cited in 1 ReviewCited in 50 Documents MSC: 14C22 Picard groups 14L30 Group actions on varieties or schemes (quotients) 14M17 Homogeneous spaces and generalizations Keywords:reductive algebraic quotient; Picard group Citations:Zbl 0682.00008 PDFBibTeX XML