Automorphism groups of real rational quartic del Pezzo surfaces

TL;DR

This paper classifies all automorphism groups of real rational degree 4 del Pezzo surfaces by analyzing their geometric structure as blow-ups of real quadrics, detailing the Galois actions and automorphism generators.

Contribution

It provides a complete geometric and group-theoretic description of automorphisms of real degree 4 del Pezzo surfaces, including subgroup classifications.

Findings

Classification of all automorphism groups for these surfaces.

Description of Galois actions on conic bundle structures.

Identification of finite subgroups acting faithfully.

Abstract

In this paper we give a complete description of all possible automorphism groups of real $\mathbb{R}$-rational del Pezzo surfaces $X$ of degree $4$, using the description of $X$ as the blow-up of some smooth real quadric surface $Q$ in $\mathbb{P}^{3}_{\mathbb{R}}$. We examine all possible ways to blow up $4$ geometric points on $Q$, illustrate in each case the $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$-action on the conic bundle structures on $X_{\mathbb{C}}$, and use it to give a geometric description of the real automorphism group $\operatorname{Aut}_{\mathbb{R}}(X)$ by generators in terms of automorphisms and birational automorphisms of $Q$. As a consequence, we get which finite subgroups of $\operatorname{Bir}_{\mathbb{C}}(\mathbb{P}^{2})$ can act faithfully by automorphisms on real $\mathbb{R}$-rational del Pezzo surfaces of degree $4$.