Del Pezzo surfaces of degree $5$ over perfect fields

In this paper we study the classification of del Pezzo surfaces $X$ of degree $5$ over any perfect field $\mathbf{k}$ in explicit geometric terms. More precisely, in each case we use the Petersen graph to illustrate the $\operatorname{Gal}(\overline{\mathbf{k}}/\mathbf{k})$-action on the $(-1)$-curves of $X$ and we describe explicitly its group of automorphisms, $\operatorname{Aut}_{\mathbf{k}}(X)$. For the cases when $X$ is not minimal, we describe how to realize it as the blow-up of $\mathbb{P}^{2}$, or of a (minimal) quadric in $\mathbb{P}^{3}$, and classify them up to $\mathbf{k}$-isomorphism. In all cases, the elements of the group $\operatorname{Aut}_{\mathbf{k}}(X)$ are described geometrically.