×

Minimal del Pezzo surfaces of degree 2 over finite fields. (English) Zbl 1405.14091

Let \(X\) be a del Pezzo surface of degree 2 over a finite field \(\mathbb{F}_{q}\) and \(\overline{X} = X \otimes \overline{\mathbb{F}_{q}}\). The Galois group \(\mathrm{Gal}(\overline{\mathbb{F}_{q}}/\mathbb{F}_{q})\) acts on \(\mathrm{Pic}(\overline{X})\) and the image of this Galois action, \(\Gamma\) is a finite cyclic subgroup in \(W(E_{7}) = O(\mathrm{Pic}(\overline{X}))\). There are 60 conjugacy classes of finite cyclic subgroups of \(W(E_{7})\). Among them, 18 classes can be realized for minimal del Pezzo surfaces of degree 2 over a finite field. It is also known that, for a sufficiently large \(q\), there exists a minimal del Pezzo surface of degree 2 over \(\mathbb{F}_{q}\) associated to any member in the possible 18 conjugacy classes. [B. Banwait et al., “Del Pezzo surfaces over finite fields and their Frobenius traces”, Preprint, arXiv:1606.00300] This conjugacy class of \(\Gamma\) determines the zeta function of \(X\).
In the paper under review, the author studies detailed conditions of \(q\) for a del Pezzo surface of degree 2 over \(\mathbb{F}_{q}\) associated to each type of conjugacy class to exist. He found the complete conditions for 14 types and some partial results on the remaining 4 types. (Theorem 1.1) The author used the Sarkisov links, which are birational modifications on conic bundles (see section 2) and the Geiser involutions, which are involutions related to the double covering map induced by the anticanonical divisor (see section 3) to prove the main result.

MSC:

14J26 Rational and ruled surfaces
14G15 Finite ground fields in algebraic geometry
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11G25 Varieties over finite and local fields
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] B. Banwait, F. Fit´e, and D. Loughran, Del Pezzo surfaces over finite fields and their Frobenius traces, preprint, see http://arxiv.org/abs/1606.00300.
[2] R. W. Carter, Conjugacy classes in the weyl group, Compositio Math. 25 (1972), 1-59. · Zbl 0254.17005
[3] V. A. Iskovskikh, Minimal models of rational surfaces over arbitrary field, Math. USSR Izv. 43 (1979), 19-43. (in Russian)
[4] , Factorization of birational mappings of rational surfaces from the point of view of Mori theory, Uspekhi Mat. Nauk 51 (1996), 3-72 (in Russian); translation in Russian Math. Surveys 51 (1996), 585-652.
[5] Yu. I. Manin, Cubic forms: algebra, geometry, arithmetic, In: North-Holland Mathematical Library, Vol. 4, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1974.
[6] Yu. G Prokhorov, Rational surfaces, Lection courses of SEC, 24, pp. 3-76, 2015. (in Russian).
[7] S. Rybakov, Zeta-functions of conic bundles and del Pezzo surfaces of degree 4 over finite fields, Mosc. Math. J. 5 (2005), no. 4, 919-926. · Zbl 1130.14021
[8] S. Rybakov and A. Trepalin, Minimal cubic surfaces over finite fields, Sb. Math. 208: 9 (2017), DOI:10.1070/SM8880. · Zbl 1426.11065
[9] H. P. F. Swinnerton-Dyer, The zeta function of a cubic surface over a finite field, Proceedings of the Cambridge Philosophical Soc. 63 (1967), 55-71. · Zbl 0201.53702
[10] , Cubic surfaces over finite fields, Math. Proceedings of the Cambridge Philosophical Society 149 (2010), 385-388. · Zbl 1222.11081
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.