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Del Pezzo surfaces over finite fields. (English) Zbl 1455.14046
Summary: Let \(X\) be a del Pezzo surface of degree 2 or greater 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 preserving the anticanonical class and the intersection form. The conjugacy class of \(\Gamma\) in the subgroup of \(\operatorname{Aut}(\operatorname{Pic}(\overline{X}))\) preserving the anticanonical class and the intersection form is a natural invariant of \(X\). We say that the conjugacy class of \(\Gamma\) in \(\operatorname{Aut}(\operatorname{Pic}(\overline{X}))\) is the type of a del Pezzo surface. In this paper we study which types of del Pezzo surfaces of degree 2 or greater can be realized for given \(q\). We collect known results about this problem and fill the gaps.
MSC:
14G15 Finite ground fields in algebraic geometry
14J26 Rational and ruled surfaces
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11G25 Varieties over finite and local fields
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
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