Del Pezzo surfaces over finite fields

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.