# Rees algebras of additive group actions

## TL;DR

This paper introduces the concept of the relative Rees algebra associated with additive group actions on schemes, exploring its properties and applications in algebraic geometry, especially in constructing affine threefolds with Ga-actions.

## Contribution

It defines and studies the properties of the relative Rees algebra for additive group actions, providing new tools for the algebraic theory of locally nilpotent derivations.

## Key findings

- Established basic properties of the relative Rees algebra.
- Illustrated properties with key examples in algebraic geometry.
- Applied the theory to construct families of affine threefolds with Ga-actions.

## Abstract

We establish basic properties of a sheaf of graded algebras canonically associated to every relative affine scheme $f : X \rightarrow S$ endowed with an action of the additive group scheme $\mathbb{G}_{ a,S}$ over a base scheme or algebraic space $S$, which we call the (relative) Rees algebra of the $\mathbb{G}_{ a,S}$-action. We illustrate these properties on several examples which played important roles in the development of the algebraic theory of locally nilpotent derivations and give some applications to the construction of families of affine threefolds with Ga-actions.