Jordan bounds for volume-preserving Cremona groups

TL;DR

This paper establishes specific Jordan constants for volume-preserving Cremona groups in two and three dimensions, providing bounds and insights into their algebraic structure.

Contribution

It computes explicit Jordan bounds for volume-preserving Cremona groups in dimensions two and three, advancing understanding of their algebraic properties.

Findings

Jordan constant for $ ext{Bir}( ext{P}^2, ext{Δ})$ is 12

Jordan bound of 144 for $ ext{Bir}( ext{P}^3, ext{Δ})$

Weak geometric Jordan bound of $2^{11} imes 3^2$ for $ ext{Bir}( ext{P}^2)$

Abstract

We show that the Jordan constant for the volume-preserving plane Cremona group $\mathrm{Bir}(\mathbb P^2, \Delta)$ is $12$. We provide a Jordan bound of $144$ for the three-dimensional volume-preserving Cremona group $\mathrm{Bir}(\mathbb P^3,\Delta)$. We also provide a weak geometric Jordan bound of $2^{11} \cdot 3^2$ for $\mathrm{Bir}(\mathbb P^2)$.