Some properties of the group of birational maps generated by the automorphisms of ${\mathbb{P}}_{\mathbb{C}}^n$ and the standard involution

Abstract

We give some properties of the subgroup $G_n(\mathbb{C})$ of the group of birational self-maps of ${\mathbb{P}}_{\mathbb{C}}^n$ generated by the standard involution and the group of automorphisms of ${\mathbb{P}}_{\mathbb{C}}^n$. We prove that there is no nontrivial finite-dimensional linear representation of $G_n(\mathbb{C})$. We also establish that $G_n(\mathbb{C})$ is perfect, and that $G_n(\mathbb{C})$ equipped with the Zariski topology is simple. Furthermore if $\phi$ is an automorphism of $\mathrm{Bir}({\mathbb{P}}_{\mathbb{C}}^n)$, then up to birational conjugacy, and up to the action of a field automorphism ${\phi }_{|G_n(\mathbb{C})}$ is trivial.

References in corpus (5)

• Finite group actions on homology spheres and manifolds with nonzero Euler characteristic
• Cremona maps of de Jonquières type
• The group of Cremona transformations generated by linear maps and the standard involution
• On automorphisms of blowups of ${\mathbb{P}}^3$
• Pseudoautomorphisms with invariant elliptic curves