Invariants and separating morphisms for algebraic group actions

The first part of this paper is a refinement of Winkelmann's work on invariant rings and quotients of algebraic groups actions an affine varieties where we take a more geometric point of view. We show that the (schematic) quotient X//G given by the possibly not finitely generated ring of invariants is "almost" an algebraic variety, and that the quotient morphism π : X → X//G has a number of nice properties. One of the main difficulties comes from the fact that the quotient morphism is not surjective. These general results are then refined for actions of the additive group Ga where we can say much more. We get a rather explicit description of the so-called plinth variety and of the separation variety which measures how much orbits are separated by invariants. The most complete results are obtained for representations. We also give a complete and detailed analysis of Roberts' famous example of a 7-dimensional representation of Ga with a non-finitely generated ring of invariants.