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Families of affine ruled surfaces: existence of cylinders

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Abstract : We 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 $B$.
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https://hal-univ-bourgogne.archives-ouvertes.fr/hal-01446821
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Soumis le : jeudi 26 janvier 2017 - 12:37:42
Dernière modification le : mercredi 3 novembre 2021 - 06:56:15

Citation

Adrien Dubouloz, Takashi Kishimoto. Families of affine ruled surfaces: existence of cylinders. Nagoya Mathematical Journal, Duke University Press, 2016, 223 (1), pp.1-20. ⟨10.1017/nmj.2016.22⟩. ⟨hal-01446821⟩

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