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

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 $Ztimesmathbb{A}^{1}$ for some algebraic variety Z. We show that the generic fiber of a family $f:Xrightarrow S$ 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.