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

TL;DR

This paper studies the structure of quotient morphisms arising from $G_a$-actions on factorial affine varieties, providing criteria for when these quotients are trivial bundles or affine spaces, especially in the context of smooth acyclic fourfolds.

Contribution

It introduces new criteria for the triviality of $G_a$-quotients and characterizes when such quotients are isomorphic to affine spaces, advancing understanding of $G_a$-actions on factorial varieties.

Findings

Criteria for $ abla$-flat quotients to be trivial $ abla$-bundles.

Conditions under which the algebraic quotient $Y$ is isomorphic to $A^3$.

Sufficient conditions for $X$ to be isomorphic to $Y imes A^1$, i.e., $A^4$.

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 $X\cong Y\times \mathbb A^1\cong \mathbb A^4$.