Minimal cubic surfaces over finite fields

Abstract

Let $X$ be a minimal cubic surface 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 $W(E_6)$. There are $25$ conjugacy classes of cyclic subgroups in $W(E_6)$, and $5$ of them correspond to minimal cubic surfaces. It is natural to ask which conjugacy classes come from minimal cubic surfaces over a given finite field. In this paper we give a partial answer to this question and present many explicit examples.

arXiv commentFinal version, published in Math. Sb. 20 pages, 1 table