Affine open subsets in A^3 without the cancellation property

TL;DR

This paper presents examples of affine open subsets in three-dimensional space that lack the cancellation property, and demonstrates that certain cylinders over specific threefolds have a trivial Makar-Limanov invariant.

Contribution

It provides new examples of affine subsets without the cancellation property and analyzes the Makar-Limanov invariant of cylinders over Koras-Russell threefolds.

Findings

Examples of affine open subsets without cancellation property

Cylinders over Koras-Russell threefolds have trivial Makar-Limanov invariant

Advances understanding of cancellation problem in affine algebraic geometry

Abstract

We give families of examples of principal open subsets of the affine space \mathbb{A}^{3} which do not have the cancellation property. We show as a by-product that the cylinders over Koras-Russell threefolds of the first kind have a trivial Makar-Limanov invariant.