Locally nilpotent derivations on affine surfaces with a $\C^*$-action

TL;DR

This paper classifies normal affine surfaces with a $ ext{C}^*$-action, providing explicit algebraic descriptions and recovering several known classification results in the field.

Contribution

It offers a comprehensive classification of affine surfaces with group actions, including explicit coordinate rings and equations, unifying and extending previous results.

Findings

Explicit algebraic descriptions of affine surfaces with $ ext{C}^*$-actions

Recovery of known classifications such as Gizatullin and Popov

Unified framework for classifying affine surfaces with open orbits

Abstract

We give a classification of normal affine surfaces admitting an algebraic group action with an open orbit. In particular an explicit algebraic description of the affine coordinate rings and the defining equations of such varieties is given. By our methods we recover many known results, e.g. the classification of normal affine surfaces with a `big' open orbit of Gizatullin and Popov or some of the classification results of Danilov-Gizatullin, Bertin and others.