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

TL;DR

The paper investigates how $A^1$-cylindrical structures in algebraic varieties behave in families, showing that generic fibers become cylindrical after base extension and providing a criterion for their existence via the Minimal Model Program.

Contribution

It proves that the generic fiber of a family of $A^1$-cylindrical varieties becomes cylindrical after a finite base extension and offers a new criterion for the existence of $A^1$-cylinders using the Minimal Model Program.

Findings

Generic fibers become $A^1$-cylindrical after finite base extension.

A criterion for the existence of $A^1$-cylinders is established.

The approach involves analyzing a relative Minimal Model Program.

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. 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.