The group of automorphisms of a real rational surface is n -transitive

Abstract

Let X be a rational nonsingular compact connected real algebraic surface. Denote by Aut(X) the group of real algebraic automorphisms of X. We show that the group Aut(X) acts n‐transitively on X, for all natural integers n. As an application we give a new and simpler proof of the fact that two rational nonsingular compact connected real algebraic surfaces are isomorphic if and only if they are homeomorphic as topological surfaces.

Cremona transformations and diffeomorphisms of surfaces

“…An indication that Aut(S 2 ) is surprisingly large comes from [1], with a more precise version developed in [9].…”

“…Building on [1], it is proved in [9] that Aut(S 2 ) is n-transitive for any n 1. Using this, it is easy to see (19) that the density property also holds with assigned fixed points.…”

“…Thus Q and σ p,q (Q) are projectively equivalent and the corresponding Cremona transformation σ p,q can be viewed as a diffeomorphism of S 2 to itself, well defined up to left and right multiplication by O (3,1). It is also convenient to allow the points p, q to coincide; see (9) for a precise definition. Let us call these the Cremona transformations with imaginary base points.…”

Birational permutations

“…They proved that the group of birational transformations of the real projective plane P 2 (R) without real indeterminacy points embeds as a dense subgroup into the group of diffeomorphisms of the plane, and acts n-transitively on P 2 (R) for all n 0. The interested reader may consult [2,4,5] and [7].…”

“…(1) Ce résultat et la preuve qui en est donnée sont analogues à plusieurs énoncés obtenus par Biswas, Huisman, Kollár, Lukackiī et Mangolte à propos des transformations birationnelles de P 2 (R) qui n'ont pas de point d'indétermination réel (voir [2,4,5] et [7]). …”