Del Pezzo surfaces of degree $5$ over perfect fields

Abstract

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.

Figures (11)

• Figure 1: Representation of the ten $(-1)$-curves on $X_{\overline{\mathbf{k}}}$.
• Figure 2: The $\mathop{\mathrm{Gal}}\nolimits(\overline{\mathbf{k}}/\mathbf{k})$-orbits of $(-1)$-curves on the Petersen diagram of a del Pezzo surface of degree $5$; where the curves in one $\mathop{\mathrm{Gal}}\nolimits(\overline{\mathbf{k}}/\mathbf{k})$-orbit are indicated by the same color and where the ones that are defined over $\mathbf{k}$ are in particular represented in red.
• Figure 3: Blow-up model for a del Pezzo surface as in Proposition \ref{['Prop:Proposition1_Z/2Z_transposition']}.
• Figure 4: Blow-up model for a del Pezzo surface as in Proposition \ref{['Prop:Proposition5_Z2xZ2_1']}.
• Figure 5: Blow-up model for a del Pezzo surface as in Proposition \ref{['Prop:Proposition_3_Z/3Z']}.

Theorems & Definitions (46)

• Theorem 1.1
• Lemma 2.1: man86
• Definition 2.2
• Proposition 2.3: [ enr97, man66, sd72, isk96, see also valv13]
• Lemma 2.4: sch19
• Lemma 2.5: sz21
• Lemma 2.6
• proof
• Lemma 2.7: sz21
• Lemma 3.1