The Jordan constant for Cremona group of rank 2

TL;DR

This paper calculates the Jordan constant for the group of birational automorphisms of the projective plane over various fields, providing key insights into the structure of these automorphism groups.

Contribution

It determines the Jordan constant for the Cremona group of rank 2 over different fields, a previously unresolved problem.

Findings

Jordan constant computed for algebraically closed fields of characteristic 0

Jordan constant computed for the real numbers

Jordan constant computed for the rational numbers

Abstract

We compute the Jordan constant for the group of birational automorphisms of a projective plane $\mathbb{P}^2_{\mathbb k}$, where ${\mathbb k}$ is either an algebraically closed field of characteristic 0, or the field of real numbers, or the field of rational numbers.