ABSTRACT
This chapter considers an affine algebraic group G acting regularly on an algebraic variety X and study once again the existence and properties of the different kinds of "quotient varieties" associated to this action. Even though the Rosenlicht's theorem is quite general and holds for quotients by general affine algebraic groups, the main inconvenient for the application of such a general statement is that it has not been easy to produce explicit descriptions of these open sets. The chapter presents the concepts of stable and semistable points and depict the role of one parameter subgroups, which are two of the main ingredients in the Hilbert-Mumford criterion. In order to study actions of reductive groups on varieties, it is convenient to consider first the case of the one-dimensional reductive group Gm, where we can profit from the fact that the dimensions of the orbits have to be zero or one.
