On stable conjugacy of finite subgroups of the plane Cremona group, I

TL;DR

This paper investigates the stable conjugacy of finite subgroups within the plane Cremona group, establishing a key invariant and computing it in specific cases to understand their classification.

Contribution

It introduces the group $H^1(G,Pic(X))$ as a stable birational invariant and provides computations for particular instances, advancing the understanding of subgroup conjugacy.

Findings

$H^1(G,Pic(X))$ is a stable birational invariant

Computed the invariant in specific cases

Enhanced classification of finite subgroups in Cremona groups

Abstract

We discuss the problem of stable conjugacy of finite subgroups of Cremona groups. We show that the group $H^1(G,Pic(X))$ is a stable birational invariant and compute this group in some cases.