Remark on Deformations of Affine Surfaces with $$\mathbb{A}^{1}$$ -Fibrations

Abstract

The possible structure of singular fibers of an \(\mathbb{A}^{1}\)-fibration on a smooth affine surface is well understood, in particular, any such fiber is a disjoint union of affine lines (possibly with multiplicities). This paper lies in a three-dimensional generalization of this fact, i.e., properties concerning a fiber component of a given fibration f: X → B from a smooth affine algebraic threefold X onto a smooth algebraic curve B whose general fibers are affine surfaces admitting \(\mathbb{A}^{1}\)-fibrations. The phenomena differ according to the type of \(\mathbb{A}^{1}\)-fibrations on general fibers of f (namely, of affine type, or of complete type). More precisely, in case of affine type, each irreducible component of every fiber of f: X → B admits an effective \(\mathbb{G}_{a}\)-action provided Pic(X) = (0) with some additional conditions concerning a compactification, whereas for the complete type, there exists an example in which a special fiber of \(f: \mathbb{A}^{3} \rightarrow \mathbb{A}^{1}\) possesses no longer an \(\mathbb{A}^{1}\)-fibration.