Factorial Affine $$G_a$$ -Varieties with Height One Plinth Ideals

Abstract

Let \(X={\text {Spec}}\;B\) 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. In this article, we study X of dimension \(\ge 3\) under the assumption that the plinth ideal \(\text {pl}(\delta )=\delta (B)\cap A\) is contained in an ideal \(\alpha A\) generated by a prime element \(\alpha \in A={\text {Ker}}\,\delta \). Suppose that \(A={\text {Ker}}\,\delta \) is an affine k-domain. The quotient morphism \(\pi : X \rightarrow Y={\text {Spec}}\;A\) splits to a composite \(\textrm{pr} \circ p\) of the projection \(\textrm{pr}: Y\times \mathbb A^1 \rightarrow Y\) and a \(G_a\)-equivariant birational morphism \(p: X \rightarrow Y\times \mathbb A^1\) where \(G_a\) acts on \(\mathbb A^1\) by translation. By decomposing \(p: X \rightarrow Y\times \mathbb A^1\) to a sequence of \(G_a\)-equivariant affine modifications, we investigate the structure of X. We also show that the general closed fiber of \(\pi \) over the closed set \(V(\alpha )={\text {Spec}}\;A/\alpha A\) consists of a disjoint union of m affine lines where \(m\ge 2\).