Separated 𝐺ₐ-actions

Abstract

Let X be an open subvariety of an affine variety, i.e. a quasi-affine variety, over an algebraically closed field, and suppose the additive algebraic group G a {G_a} acts on X. Then a geometric quotient of X by G a {G_a} exists if and only if every point x of X has a G a {G_a} -stable open neighborhood U such that the morphism G a × U → U × U {G_a} \times U \to U \times U which sends (t, u) to (tu, u) has closed image and finite fibres.