Faithfully flat quotient morphisms by $G_a$-actions on factorial affine varieties

Abstract

Let $X$ be a factorial complex affine variety of dimension $\ge 3$ with an algebraic action of the additive group $G_a$. Let $\pi : X \to Y$ be the algebraic quotient morphism where we assume $Y$ is an affine variety. When $\pi$ is faithfully flat, we investigate $\pi$ by $G_a$-equivariant affine modifications and give criteria for $\pi$ to be a trivial $\mathbb A^1$-bundle. For a smooth acyclic fourfold $X$ with a free $G_a$-action and a $G_a$-equivariant $\mathbb A^3$-fibration $f : X \to \mathbb A^1$ where $G_a$ acts trivially on $\mathbb A^1$, we give a criterion for the algebraic quotient $Y$ to be isomorphic to $\mathbb A^3$ with $f$ as a coordinate. Together with a criterion for $\pi : X \to Y$ to be a trivial $\mathbb A^1$-bundle, we obtain a sufficient condition for $X\cong Y\times \mathbb A^1\cong \mathbb A^4$.