# Deformations of $\mathbb{A}^1$-cylindrical varieties

Abstract : An algebraic variety is called $\mathbb{A}^{1}$-cylindrical if it contains an $\mathbb{A}^{1}$-cylinder, i.e. a Zariski open subset of the form $Z\times\mathbb{A}^{1}$ for some algebraic variety $Z$. We show that the generic fiber of a family $f:X\rightarrow S$ of normal $\mathbb{A}^{1}$-cylindrical varieties becomes $\mathbb{A}^{1}$-cylindrical after a finite extension of the base. Our second result is a criterion for existence of an $\mathbb{A}^{1}$-cylinder in $X$ which we derive from a careful inspection of a relative Minimal Model Program ran from a suitable smooth relative projective model of $X$ over $S$.
Keywords :
Type de document :
Pré-publication, Document de travail
2017

Littérature citée [28 références]

https://hal.archives-ouvertes.fr/hal-01622447
Soumis le : mardi 24 octobre 2017 - 16:55:24
Dernière modification le : jeudi 11 janvier 2018 - 06:12:20

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DefA1Cyl.pdf
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### Identifiants

• HAL Id : hal-01622447, version 1
• ARXIV : 1710.09108

### Citation

Adrien Dubouloz, Takashi Kishimoto. Deformations of $\mathbb{A}^1$-cylindrical varieties. 2017. 〈hal-01622447〉

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