Families of affine ruled surfaces: existence of cylinders

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.