# Variations on the theme of Zariski's Cancellation Problem

## TL;DR

This paper explores local variants of Zariski's Cancellation Problem, examining special classes of varieties and algebraic groups, and their implications for conjugacy problems in Cremona groups.

## Contribution

It introduces new perspectives on local versions of the Zariski Cancellation Problem and investigates related classes of algebraic varieties and groups.

## Key findings

- Identification of specific classes of varieties relevant to the problem
- Insights into algebraic group classifications in the context of Cremona groups
- Connections between local cancellation properties and conjugacy problems

## Abstract

This is an expanded version of the talk by the author at the conference Polynomial Rings and Affine Algebraic Geometry, February 12--16, 2018, Tokyo Metropolitan University, Tokyo, Japan. Considering a local version of the Zariski Cancellation Problem naturally leads to exploration of some classes of varieties of special kind and their equivariant versions. We discuss several topics inspired by this exploration, including the problem of classifying a class of affine algebraic groups that are naturally singled out in studying the conjugacy problem for algebraic subgroups of the Cremona groups.