Cremona maps and involutions

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 $\mathrm{PGL}(n+1,\mathbb{C})$.