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Deformations of \(\mathbb {A}^1\)-cylindrical varieties

  • Adrien Dubouloz
  • Takashi Kishimoto
Article
  • 23 Downloads

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. This generalizes the main result of Dubouloz and Kishimoto (Nagoya Math J 223:1–20, 2016) which established this property for families of smooth \(\mathbb {A}^{1}\)-cylindrical affine surfaces. 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 run from a suitable smooth relative projective model of X over S.

Mathematics Subject Classification

14R25 14D06 14M20 14E30 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IMB UMR5584, CNRS, Univ. Bourgogne Franche-ComtéDijonFrance
  2. 2.Department of Mathematics, Faculty of ScienceSaitama UniversitySaitamaJapan

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