Flexible bundles over rigid affine surfaces

TL;DR

This paper constructs a specific affine surface with finite automorphism group that exhibits highly transitive automorphism actions on its cylinder extension, challenging notions of rigidity.

Contribution

It introduces a new example of a rigid affine surface that is not stably rigid, demonstrating complex automorphism behavior.

Findings

Constructed a smooth rational affine surface with finite automorphism group.

Showed the automorphism group of the cylinder acts infinitely transitively.

Provided a counterexample to stable rigidity with respect to the Makar-Limanov invariant.

Abstract

We construct a smooth rational affine surface S with finite automorphism group but with the property that the group of automorphisms of the cylinder SxA^2 acts infinitely transitively on the complement of a closed subset of codimension at least two. Such a surface S is in particular rigid but not stably rigid with respect to the Makar-Limanov invariant.