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**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.

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.

Reviewer: Junmyeong Jang (Ulsan)

### 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 |

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\textit{A. Trepalin}, Bull. Korean Math. Soc. 54, No. 5, 1779--1801 (2017; Zbl 1405.14091)

### 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 |

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