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. 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.