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FAMILIES OF AFFINE RULED SURFACES: EXISTENCE OF CYLINDERS

  • ADRIEN DUBOULOZ (a1) and TAKASHI KISHIMOTO (a2)

Abstract

We show that the generic fiber of a family $f:X\rightarrow S$ 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 $\unicode[STIX]{x1D70C}:X\rightarrow Y$ over a certain $S$ -scheme $Y\rightarrow S$ induced by the MRC-fibration of a relative smooth projective model of $X$ over $S$ . For affine threefolds $X$ equipped with a fibration $f:X\rightarrow B$ by irrational $\mathbb{A}^{1}$ -ruled surfaces over a smooth curve $B$ , the induced $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ can also be recovered from a relative minimal model program applied to a smooth projective model of $X$ over $B$ .

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