Fibrations by affine lines on rational affine surfaces with irreducible boundaries

TL;DR

This paper studies fibrations by affine lines on certain rational affine surfaces, revealing conditions for the finiteness of such fibrations and describing their structure, including exceptions and the role of boundary self-intersection.

Contribution

It provides a new proof characterizing when the set of affine line fibrations is finite based on the boundary's self-intersection number.

Findings

Most surfaces admit infinitely many affine line fibrations with a unique large multiplicity singular fiber.

Finiteness of fibrations is characterized by the boundary's self-intersection number being at most 6.

Exceptions to the general pattern are explicitly identified.

Abstract

We consider fibrations by affine lines on smooth affine surfaces obtained as complements of smooth rational curves $B$ in smooth projective surfaces $X$ defined over an algebraically closed field of characteristic zero. We observe that except for two exceptions, these surfaces $X \setminus B$ admit infinitely many families of $\mathbb{A}^1$-fibrations over the projective line with irreducible fibers and a unique singular fiber of arbitrarily large multiplicity. For $\mathbb{A}^1$-fibrations over the affine line, we give a new and essentially self-contained proof that the set of equivalence classes of such fibrations up to composition by automorphisms at the source and target is finite if and only if the self-intersection number of $B$ in $X$ is less than or equal to 6.