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 $f:X\to S$ 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.

References in corpus (3)

• Affine cones over Fano threefolds and additive group actions
• Cylinders in del Pezzo fibrations
• New examples of cylindrical Fano fourfolds