Rigid affine surfaces with isomorphic A2-cylinders

TL;DR

This paper constructs families of smooth affine surfaces that have non-isomorphic A1-cylinders but share isomorphic A2-cylinders, revealing interesting properties of affine surface cylinders.

Contribution

It introduces new examples of affine surfaces with distinct A1-cylinder structures but identical A2-cylinder structures, derived from cubic surface complements.

Findings

A1-cylinders are pairwise non-isomorphic.

A2-cylinders are all isomorphic.

Examples are complements of cuspidal hyperplane sections of cubic surfaces.

Abstract

We construct families of smooth affine surfaces with pairwise non isomorphic A 1-cylinders but whose A 2-cylinders are all isomorphic. These arise as complements of cuspidal hyperplane sections of smooth projective cubic surfaces.