Automorphism groups of real rational quartic del Pezzo surfaces

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 $4$.