Factorial affine $G_a$-varieties with principal plinth ideals

Abstract

Let $X={\rm Spec}\: B$ be a factorial affine variety defined over an algebraically closed field $k$ of characteristic zero with a nontrivial action of the additive group $G_a$ associated to a locally nilpotent derivation $\delta$ on $B$. Suppose that $A={\rm Ker}\: \delta$ is an affine $k$-domain. The quotient morphism $\pi : X \to Y={\rm \Spec}\: A$ splits to a composite ${\rm pr} \circ p$ of the projection ${\rm pr} : Y\times \mathbb A^1 \to Y$ and a $G_a$-equivariant birational morphism $p : X \to Y\times \mathbb A^1$ where $G_a$ acts on $\mathbb A^1$ by translation. In this article, we study $X$ of dimension $\ge 3$ under the assumption that the plinth ideal $\delta(B)\cap A$ is a principal ideal generated by a non-unit element $a$ of $A$. By decomposing $p : X \to Y\times \mathbb A^1$ to a sequence of $G_a$-equivariant affine modifications, we investigate the structure of $X$. We show in algebraic way that the general closed fiber of $\pi$ over the closed set $V(a)$ of $Y$ consists of a disjoint union of affine lines. The $G_a$-action on $X$ and the fixed-point locus $X^{G_a}$ are studied with particular interest.