Minimal del Pezzo surfaces of degree $2$ over finite fields

TL;DR

This paper investigates the possible Galois group actions on the Picard group of minimal degree 2 del Pezzo surfaces over finite fields, classifying which cyclic subgroups occur for different field sizes.

Contribution

It classifies achievable Galois subgroup actions on minimal degree 2 del Pezzo surfaces over finite fields, linking algebraic group theory with geometric classification.

Findings

Identifies which of the 18 conjugacy classes occur for various finite fields.

Provides explicit conditions on the finite field size for each subgroup.

Enhances understanding of the arithmetic of del Pezzo surfaces over finite fields.

Abstract

Let $X$ be a minimal del Pezzo surface of degree $2$ over a finite field $\mathbb{F}_q$. The image $\Gamma$ of the Galois group $\operatorname{Gal}(\overline{\mathbb{F}}_q / \mathbb{F}_q)$ in the group $\operatorname{Aut}(\operatorname{Pic}(\overline{X}))$ is a cyclic subgroup of the Weyl group $W(E_7)$. There are $60$ conjugacy classes of cyclic subgroups in $W(E_7)$ and $18$ of them correspond to minimal del Pezzo surfaces. In this paper we study which possibilities of these subgroups for minimal del Pezzo surfaces of degree $2$ can be achieved for given $q$.