Minimal del Pezzo surfaces of degree 2 over finite fields
Bull. Korean Math. Soc. 2017 Vol. 54, No. 5, 1779-1801
https://doi.org/10.4134/BKMS.b160757
Published online September 30, 2017
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
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