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 $\mathrm{Gal}({\overline{\mathbb{F}}}_q/{\mathbb{F}}_q)$ in the group $\mathrm{Aut}(\mathrm{Pic}(\overline{X}))$ is a cyclic subgroup preserving the anticanonical class and the intersection form. The conjugacy class of $\Gamma$ in the subgroup of $\mathrm{Aut}(\mathrm{Pic}(\overline{X}))$ preserving the anticanonical class and the intersection form is a natural invariant of $X$. We say that the conjugacy class of $\Gamma$ in $\mathrm{Aut}(\mathrm{Pic}(\overline{X}))$ 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.

28 pages, 4 tables. arXiv admin note: text overlap with arXiv:1611.02832