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.
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https://hal.archives-ouvertes.fr/hal-01053674
Submitted on : Thursday, July 31, 2014 - 10:50:58 PM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM
Long-term archiving on : Tuesday, November 25, 2014 - 10:39:08 PM

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• HAL Id : hal-01053674, version 1
• ARXIV : 1408.1328

Citation

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

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