# zbMATH — the first resource for mathematics

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
##### Keywords:
finite field; del Pezzo surfaces; zeta function; cubic surface
Full Text:
##### References:
 [1] Banwait, B.; Fité, F.; Loughran, D., Del Pezzo surfaces over finite fields and their Frobenius traces, Math. Proc. Camb. Philos. Soc., 167, 1, 35-60 (2019) · Zbl 1442.14087 [2] Carter, R. W., Conjugacy classes in the Weyl group, Compos. Math., 25, 1, 1-59 (1972) · Zbl 0254.17005 [3] Dolgachev, I. V.; Iskovskikh, V. A., Finite subgroups of the plane Cremona group, (Algebra, Arithmetic, and Geometry, Vol. I: In Honor of Yu. I. Manin. Algebra, Arithmetic, and Geometry, Vol. I: In Honor of Yu. I. Manin, Progr. Math., vol. 269 (2009), Birkhäuser: Birkhäuser Basel), 443-548 · Zbl 1219.14015 [4] Dolgachev, I.; Duncan, A., Automorphisms of cubic surfaces in positive characteristic, Izv. Math., 83, 3, 424-500 (2019) · Zbl 1427.14086 [5] Frame, J. S., The classes and representations of the groups of 27 lines and 28 bitangents, Ann. Mat. Pura Appl. (4), 32, 83-119 (1951) · Zbl 0045.00505 [6] Frame, J. S., The characters of the Weyl group $$E_8$$, (Leech, J., Computational Problems in Abstract Algebra. Computational Problems in Abstract Algebra, Oxford (1967)), 111-130 [7] Iskovskikh, V. A., Minimal models of rational surfaces over arbitrary field, Math. USSR, Izv., 43, 19-43 (1979), (in Russian) [8] N. Kaplan, Rational point counts for del Pezzo surfaces over finite fields and coding theory, Harvard Ph.D. Thesis. [9] Knecht, A.; Reyes, K., Full degree two del Pezzo surfaces over small finite fields, (Contemporary Developments in Finite Fields and Applications (2016)), 145-159 · Zbl 1368.14057 [10] Kunyavskij, B. Eh.; Skorobogatov, A. N.; Tsfasman, M. A., Del Pezzo surfaces of degree four, Mém. Soc. Math. Fr., 37, 1-113 (1989) · Zbl 0705.14039 [11] Loughran, D.; Trepalin, A., Inverse Galois problem for del Pezzo surfaces over finite fields, Math. Res. Lett., 27, 3, 845-853 (2020) [12] Manin, Yu. I., Cubic forms: algebra, geometry, arithmetic, North-Holland Mathematical Library, vol. 4 (1974), North-Holland Publishing Co., American Elsevier Publishing Co.: North-Holland Publishing Co., American Elsevier Publishing Co. Amsterdam-London, New York · Zbl 0277.14014 [13] Rybakov, S.; Trepalin, A., Minimal cubic surfaces over finite fields, Mat. Sb.. Mat. Sb., Engl. transl.: Sb. Math., 208, 9, 148-170 (2017) · Zbl 1426.11065 [14] Rybakov, S., Zeta-functions of conic bundles and del Pezzo surfaces of degree 4 over finite fields, Mosc. Math. J., 5, 4, 919-926 (2005) · Zbl 1130.14021 [15] Swinnerton-Dyer, H. P.F., The zeta function of a cubic surface over a finite field, Proc. Camb. Philos. Soc., 63, 55-71 (1967) · Zbl 0201.53702 [16] Swinnerton-Dyer, H. P.F., Cubic surfaces over finite fields, Math. Proc. Camb. Philos. Soc., 149, 385-388 (2010) · Zbl 1222.11081 [17] Trepalin, A., Minimal del Pezzo surfaces of degree 2 over finite fields, Bull. Korean Math. Soc., 54, 5, 1779-1801 (2017) · Zbl 1405.14091 [18] Urabe, T., Calculation of Manin’s invariant for del Pezzo surfaces, Math. Comput., 65, 213, 247-258 (1996), S15-S23 · Zbl 0867.14014 [19] Vlǎduţ, S.; Nogin, D.; Tsfasman, M., Varieties over finite fields: quantitative theory, Usp. Mat. Nauk. Usp. Mat. Nauk, Russ. Math. Surv., 2, 73, 261-322 (2018), Engl. transl.: · Zbl 1427.14052
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.