Del Pezzo surfaces over finite fields

TL;DR

This paper classifies the possible Galois group conjugacy classes, called types, of del Pezzo surfaces over finite fields of degree 2 or higher, and investigates which types can occur for a given finite field size.

Contribution

It systematically studies the realizability of Galois group types for del Pezzo surfaces over finite fields, filling gaps in existing classifications.

Findings

Identifies which Galois types occur over specific finite fields.

Provides a comprehensive overview of known realizability results.

Completes the classification by filling previously unknown cases.

Abstract

Let $X$ be a del Pezzo surface of degree $2$ or greater 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 $\operatorname{Aut}(\mathrm{Pic}(\overline{X}))$ is a cyclic subgroup preserving the anticanonical class and the intersection form. The conjugacy class of $\Gamma$ in the subgroup of $\operatorname{Aut}(\mathrm{Pic}(\overline{X}))$ preserving the anticanonical class and the intersection form is a natural invariant of $X$. We say that the conjugacy class of $\Gamma$ in $\operatorname{Aut}(\mathrm{Pic}(\overline{X}))$ is the \textit{type} of a del Pezzo surface. In this paper we study which types of del Pezzo surfaces of degree $2$ or greater can be realized for given $q$. We collect known results about this problem and fill the gaps.