Isomorphisms between cylinders over Danielewski surfaces

Abstract

A special Danielewski surface is an affine surface which is the total space of a principal $$({{\mathbb {C}}},+)$$ -bundle over an affine line with a multiple origin. Using a fiber product trick introduced by Danielewski, it is known that cylinders over two such surfaces are always isomorphic provided that both bases have the same number of origins. The goal of this note is to give an explicit method to find isomorphisms between cylinders over special Danielewski surfaces. The method is based on the construction of appropriate locally nilpotent derivations.