Minimal del Pezzo surfaces of degree $2$ 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 $\mathrm{Gal}({\overline{\mathbb{F}}}_q/{\mathbb{F}}_q)$ in the group $\mathrm{Aut}(\mathrm{Pic}(\overline{X}))$ 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 $q$.

arXiv commentFinal version, published in BKMS. 21 pages, 1 table