FAMILIES OF AFFINE RULED SURFACES: EXISTENCE OF CYLINDERS

We show that the generic fiber of a family f: X -> S of smooth A(1)-ruled affine surfaces always carries an A(1)-fibration, possibly after a finite extension of the base S. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking S, such a family actually factors through an A(1)-fibration rho: X -> Y over a certain S-scheme Y -> S induced by the MRC-fibration of a relative smooth projective model of X over S. For affine threefolds X equipped with a fibration f : X -> B by irrational A(1)-ruled surfaces over a smooth curve B, the induced A(1)-fibration rho: X -> Y can also be recovered from a relative minimal model program applied to a smooth projective model of X over B.