Cremona maps and involutions

TL;DR

This paper investigates whether the Cremona group is generated by involutions, providing bounds on the number of involutions needed to express various classes of birational maps and automorphisms in projective spaces.

Contribution

It establishes that the Cremona group in dimension 2 is generated by involutions and extends bounds for expressing maps in higher dimensions using involutions.

Findings

Cremona group in dimension 2 is generated by involutions.

Provides upper bounds for the number of involutions needed for certain birational maps.

Extends results to automorphisms and specific classes of maps in higher dimensions.

Abstract

We deal with the following question of Dolgachev : is the Cremona group generated by involutions ? Answer is yes in dimension $2$ (Cerveau-Deserti). We give an upper bound of the minimal number $\mathfrak{n}_\varphi$ of involutions we need to write a birational self map $\varphi$ of $\mathbb{P}^2_\mathbb{C}$. We prove that de Jonqui\`eres maps of $\mathbb{P}^3_\mathbb{C}$ and maps of small bidegree of $\mathbb{P}^3_\mathbb{C}$ can be written as a composition of involutions of $\mathbb{P}^3_\mathbb{C}$ and give an upper bound of $\mathfrak{n}_\varphi$ for such maps $\varphi$. We get similar results in particular for automorphisms of $(\mathbb{P}^1_\mathbb{C})^n$, automorphisms of $\mathbb{P}^n_\mathbb{C}$, tame automorphisms of $\mathbb{C}^n$, monomial maps of $\mathbb{P}^n_\mathbb{C}$, and elements of the subgroup generated by the standard involution of $\mathbb{P}^n_\mathbb{C}$ and $\mathrm{PGL}(n+1,\mathbb{C})$.