Codimension two torus actions on the affine space

TL;DR

This paper classifies specific affine varieties with torus actions of complexity two, focusing on those with a unique fixed point and a toric surface quotient, to explore potential counterexamples to the linearization conjecture.

Contribution

It provides a classification of smooth, contractible affine varieties with complexity two torus actions and a unique fixed point, advancing understanding in affine geometry and linearization problems.

Findings

Classified varieties with specified torus actions and properties

Identified candidates for counterexamples to the linearization conjecture

Connected the varieties to toric surface blow-ups

Abstract

In this paper, we classify smooth, contractible affine varieties equipped with faithful torus actions of complexity two, having a unique fixed point and a two-dimensional algebraic quotient isomorphic to a toric blow-up of a toric surface. These varieties are of particular interest as they represent the simplest candidates for potential counterexamples to the linearization conjecture in affine geometry.