Deformations of A(1)-cylindrical varieties

An algebraic variety is called A1-cylindrical if it contains an A1-cylinder, i.e. a Zariski open subset of the form ZxA1 for some algebraic variety Z. We show that the generic fiber of a family f:XS of normal A1-cylindrical varieties becomes A1-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 A1-cylindrical affine surfaces. Our second result is a criterion for existence of an A1-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.