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Pré-Publication, Document De Travail Année : 2014

Families of affine ruled surfaces: existence of cylinders

Résumé

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.
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Dates et versions

hal-01053674 , version 1 (31-07-2014)

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Adrien Dubouloz, Takashi Kishimoto. Families of affine ruled surfaces: existence of cylinders. 2014. ⟨hal-01053674⟩
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