Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2017; 54(5): 1779-1801

Online first article July 28, 2017      Printed September 30, 2017

https://doi.org/10.4134/BKMS.b160757

Copyright © The Korean Mathematical Society.

Minimal del Pezzo surfaces of degree 2 over finite fields

Andrey Trepalin

Institute for Information Transmission Problems

Abstract

Let $X$ be a minimal del Pezzo surface of degree $2$ over a finite field $\F_q$. The image $\Gamma$ of the Galois group $\Gal(\overline{\F}_q / \F_q)$ in the group $\Aut(\Pic(\XX))$ 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 $q$.

Keywords: del Pezzo surfaces, finite field, zeta function

MSC numbers: 14J26, 14G15, 14G10, 11G25