On Generalised Danielewski Surfaces over fields of arbitrary characteristic

TL;DR

This paper investigates exponential maps on certain affine surfaces over arbitrary fields, characterizes their invariants, and explores isomorphisms and counterexamples to the cancellation problem.

Contribution

It provides a comprehensive analysis of $ ext{G}_a$-actions, invariants, and isomorphisms for a family of affine surfaces, including counterexamples to cancellation.

Findings

Characterization of Makar-Limanov and Derksen invariants for these surfaces

Complete classification of isomorphisms between such surfaces

Identification of subfamilies that serve as counterexamples to the cancellation problem

Abstract

In this paper we study exponential maps ($\mathbb{G}_a$-actions) on the family of affine two dimensional surfaces of the form $f(x)y=\phi(x,z)$ over arbitrary fields, describe the Makar-Limanov invariant and Derksen invariant of these surfaces, give a complete characterization of isomorphisms between such surfaces and display a subfamily which provides counterexamples to the cancellation problem.