Fundamental, Picard, and Class Groups of Rings of Invariants

Abstract

Let G be a n affine algebraic group over the algebraically closed field k, and let V be an affine, normal algebraic variety over k on which G acts. Suppose that the ring of invariants k [F]G is finitely generated over k, and let W be the affine variety with k[W] = k[V]G. The purpose of this paper is to show that the induced homomorphism from the étale fundamental group of V to that of W is surjective, and to examine the consequences of this observation in terms of the relations between the Picard and divisor class groups of k[V] and k[W],