Cremona groups of real surfaces
Jérémy Blanc, Frédéric Mangolte
TL;DR
This work addresses the structure of real Cremona groups by employing the Sarkisov program to decompose birational maps between real Mori fiber spaces. It delivers explicit generating sets: the real projective automorphisms, the standard quadratic involution $\sigma_0$, the real quadratic involution $\sigma_1$, and a family of degree-$5$ maps with only nonreal base-points, which generate $\operatorname{Bir}_{\mathbb R}(\mathbb P^2)$; analogous generator results are established for the subgroups $\operatorname{Aut}(\mathbb P^2(\mathbb R))$, $\operatorname{Aut}(Q_{3,1}(\mathbb R))$, and $\operatorname{Aut}(\mathbb F_0(\mathbb R))$. The paper also classifies and utilizes standard vs special links and degree constraints for quintic and cubic transformations, providing a unifying framework that recovers and extends prior results. Additionally, it explores infinite transitivity and density phenomena for automorphism groups on real loci of rational surfaces, illuminating the dynamics of real algebraic surfaces under birational and biregular actions. The findings have implications for explicit birational classification and for understanding symmetry groups of real rational surfaces via Mori theory and Sarkisov links.
Abstract
We give an explicit set of generators for various natural subgroups of the real Cremona group Bir_R(P^2). This completes and unifies former results by several authors.