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 ${\mathfrak{n}}_{\phi }$ of involutions we need to write a birational self map $\phi$ of ${\mathbb{P}}_{\mathbb{C}}^2$. We prove that de Jonquières maps of ${\mathbb{P}}_{\mathbb{C}}^3$ and maps of small bidegree of ${\mathbb{P}}_{\mathbb{C}}^3$ can be written as a composition of involutions of ${\mathbb{P}}_{\mathbb{C}}^3$ and give an upper bound of ${\mathfrak{n}}_{\phi }$ for such maps $\phi$. We get similar results in particular for automorphisms of $({\mathbb{P}}_{\mathbb{C}}^1{)}^n$, automorphisms of ${\mathbb{P}}_{\mathbb{C}}^n$, tame automorphisms of ${\mathbb{C}}^n$, monomial maps of ${\mathbb{P}}_{\mathbb{C}}^n$, and elements of the subgroup generated by the standard involution of ${\mathbb{P}}_{\mathbb{C}}^n$ and $\mathrm{PGL}(n+1,\mathbb{C})$.

Cited by in corpus (1)

• Generators of the plane Cremona group over the field with two elements