Rees algebras of additive group actions

We establish basic properties of a sheaf of graded algebras canonically associated to every relative affine scheme f : X -> S endowed with an action of the additive group scheme G(a,S) over a base scheme or algebraic space S, which we call the (relative) Rees algebra of the G(a,S)-action. In the case of affine algebraic varieties defined over a field of characteristic zero, we establish further properties of the Rees algebra of a G(a)-action in terms of its associated locally nilpotent derivation. We give an algebro-geometric characterization of pairs consisting of an affine algebraic variety and a G(a)-action on it whose associated Rees algebras are finitely generated and provide an algorithm extending van den Essen's kernel algorithm for locally nilpotent derivations to compute generators of these Rees algebras. We illustrate these properties on several examples which played important historical roles in the development of the algebraic theory of locally nilpotent derivations and give applications to the construction of new families of affine threefolds with G(a)-actions.