Families of affine ruled surfaces: existence of cylinders

TL;DR

This paper proves that generic fibers of families of smooth affine surfaces with an affine line ruling have an affine line fibration, and describes conditions under which these families factor through such fibrations, especially for irrational surfaces.

Contribution

It establishes the existence of $A^1$-fibrations in families of affine surfaces and describes their structure via MRC and Minimal Model Program techniques.

Findings

Generic fibers carry an $A^1$-fibration after finite base extension.

Families of irrational affine surfaces factor through an $A^1$-fibration over a scheme.

Induced $A^1$-fibrations can be obtained via the Minimal Model Program.

Abstract

We show that the generic fiber of a family of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking the base, such a family actually factors through an $\mathbb{A}^{1}$-fibration over a certain scheme, induced by the MRC-fibration of a relative smooth projective model of the family. For affine threefolds fibered by irrational $\mathbb{A}^{1}$-ruled surfaces, this induced $\mathbb{A}^{1}$-fibration can also be obtained from a relative Minimal Model Program applied to a relative smooth projective model of the family.