Geometrically pure Ga-actions

Abstract

We propose the definition of an action σ of the additive group scheme Ga on an affine variety Y to be geometrically pure, which ensures the existence of a geometric quotient of Y by the Ga-action σ if Y is normal. Namely there exists the quotient morphism q:Y→X to a normal affine variety X such that the graph morphism Ψ:Ga×Y→Y×XY is an isomorphism (see Theorem 2.2). Geometric pureness of the given Ga-action is the first criterion ever to guarantee the existence of a geometric quotient Y/Ga. As a consequence, we obtain an algebraic characterization of the affine 3-space A3.