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Minimal cubic surfaces over finite fields

Research paper by Sergey Rybakov, Andrey Trepalin

Indexed on: 08 Nov '16Published on: 08 Nov '16Published in: arXiv - Mathematics - Algebraic Geometry



Abstract

Let $X$ be a minimal cubic surface 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 $\operatorname{Aut}(\operatorname{Pic}(\overline{X}))$ 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.