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MINIMAL DEL PEZZO SURFACES OF DEGREE 2 OVER FINITE FIELDS

  • Received : 2016.09.19
  • Accepted : 2016.12.26
  • Published : 2017.09.30

Abstract

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

Acknowledgement

Supported by : Russian Foundation for Sciences

References

  1. B. Banwait, F. Fite, 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.
  3. V. A. Iskovskikh, Minimal models of rational surfaces over arbitrary field, Math. USSR Izv. 43 (1979), 19-43. (in Russian)
  4. V. A. Iskovskikh, 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.
  8. S. Rybakov and A. Trepalin, Minimal cubic surfaces over finite fields, Sb. Math. 208: 9 (2017), DOI:10.1070/SM8880. https://doi.org/10.1070/SM8880
  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. https://doi.org/10.1017/S0305004100040895
  10. H. P. F. Swinnerton-Dyer, Cubic surfaces over finite fields, Math. Proceedings of the Cambridge Philosophical Society 149 (2010), 385--388. https://doi.org/10.1017/S0305004110000320