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 $\mathrm{Gal}(\bar{\mathbf{k}}\mathrm{/}\mathbf{k})$-action on the $(-1)$-curves of $X$ and we describe explicitly its group of automorphisms, ${\mathrm{Aut}}_{\mathbf{k}}(X)$. 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 ${\mathrm{Aut}}_{\mathbf{k}}(X)$ are described geometrically.