Abstract
A geometrically rational surface S over a nonclosed field k is k-birational to either a del Pezzo surface of degree \(n\in [1,\ldots , 9]\) or a conic bundle (see [6]). Throughout, we assume that \(S(k)\ne \emptyset \). This implies k-rationality of S when \(n\in [5,\ldots , 9]\) or when the number of degenerate fibers of the conic bundle is at most 3.
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Banwait, B., Fité, F., Loughran, D.: Del Pezzo surfaces over finite fields and their Frobenius traces (2016). arXiv:1606.00300.
Beauville, A., Colliot-Thélène, J.L., Sansuc, J.J., Swinnerton-Dyer, P.: Variétés stablement rationnelles non rationnelles. Ann. of Math. (2) 121(2), 283–318 (1985).
Colliot-Thélène, J.L.: Surfaces stablement rationnelles sur un corps quasi-fini (2017). arXiv:1711.09595.
Colliot-Thélène, J.L., Sansuc, J.J., Swinnerton-Dyer, P.: Intersections of two quadrics and Châtelet surfaces. I. J. Reine Angew. Math. 373, 37–107 (1987).
Colliot-Thélène, J.L., Sansuc, J.J., Swinnerton-Dyer, P.: Intersections of two quadrics and Châtelet surfaces. II. J. Reine Angew. Math. 374, 72–168 (1987).
Iskovskih, V.A.: Minimal models of rational surfaces over arbitrary fields. Izv. Akad. Nauk SSSR Ser. Mat. 43(1), 19–43, 237 (1979).
Kunyavskiĭ, B.E., Skorobogatov, A.N., Tsfasman, M.A.: del Pezzo surfaces of degree four. Mém. Soc. Math. France (N.S.) (37), 113 (1989).
Li, S.: Rational points on del Pezzo surfaces of degree 1 and 2. arxiv.org/0904.3555.
Manin, Y.I.: Rational surfaces over perfect fields. II. Mat. Sb. (N.S.) 72 (114), 161–192 (1967).
Manin, Y.I., Tsfasman, M.A.: Rational varieties: algebra, geometry, arithmetic. Uspekhi Mat. Nauk 41(2(248)), 43–94 (1986).
Swinnerton-Dyer, P.: The zeta function of a cubic surface over a finite field. Proc. Cambridge Philos. Soc. 63, 55–71 (1967).
Urabe, T.: Calculation of Manin’s invariant for Del Pezzo surfaces. Math. Comp. 65(213), 247–258, S15–S23 (1996).
Acknowledgements
We are grateful to J.-L. Colliot-Thélène for helpful comments and suggestions. The first author was partially supported by NSF grant 1601912.
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Tschinkel, Y., Yang, K. (2020). Potentially Stably Rational Del Pezzo Surfaces over Nonclosed Fields. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory III. CANT 2018. Springer Proceedings in Mathematics & Statistics, vol 297. Springer, Cham. https://doi.org/10.1007/978-3-030-31106-3_17
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DOI: https://doi.org/10.1007/978-3-030-31106-3_17
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