Families of affine ruled surfaces: existence of cylinders

International audienceWe show that the generic fiber of a family $f: X → S$ of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-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 $\mathbb{A}^{1}$-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 $\mathbb{A}^{1}$-ruled surfaces over a smooth curve $B$, the induced $\mathbb{A}^{1}$-fibration $\rho : X → Y$ can also be recovered from a relative minimal model program applied to a smooth projective model of $X$ over $B$