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

TL;DR

This work proves that for every rational nonsingular compact connected real algebraic surface $X$, the automorphism group Aut$(X)$ acts $n$-transitively on $X$ for all $n$. The authors establish $n$-transitivity for Aut$(S^2)$ by constructing a rich family of automorphisms via algebraic maps $f:I o SO_2(\,\real)$ and then extend the result to general rational surfaces by noting that such $X$ arise from $S^2$ via finitely many blow-ups; Aut$(X)$ thus inherits $(m+n)$-transitivity from Aut$(S^2)$. As an application, they provide a streamlined proof that two rational real algebraic surfaces are isomorphic if and only if they are homeomorphic, by transporting blow-up centers on $S^2$ through an automorphism. The findings reveal that Aut$(X)$ is a large and flexible group on rational real surfaces and lay groundwork for further study of its dynamics and density within the diffeomorphism group.

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.

Figures (1)

• Figure 1: The sphere $S^2$ with the parallels $\Gamma_j$ and $\Gamma_j'$.

Theorems & Definitions (15)

• Conjecture 1.2: BH07
• Theorem 1.3: BH07
• Theorem 1.4
• Theorem 1.6: BH07
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof