Abstract
We prove that if S is a smooth reflexive surface in \(\mathbb {P}^3\) defined over a finite field \(\mathbb {F}_q\), then there exists an \(\mathbb {F}_q\)-line meeting S transversely provided that \(q\ge c\deg (S)\), where \(c=\frac{3+\sqrt{17}}{4}\approx 1.7808\). Without the reflexivity hypothesis, we prove the existence of a transverse \(\mathbb {F}_q\)-line for \(q\ge \deg (S)^2\).
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16 January 2023
An Erratum to this paper has been published: https://doi.org/10.1007/s00229-023-01456-9
References
Asgarli, S.: Sharp Bertini theorem for plane curves over finite fields. Can. Math. Bull. 62(2), 223–230 (2019)
Ballico, E.: An effective Bertini theorem over finite fields. Adv. Geom. 3(4), 361–363 (2003)
Fukasawa, S.: On Kleiman-Piene’s question for Gauss maps. Compos. Math. 142(5), 1305–1307 (2006)
Fukasawa, S., Kaji, H.: The separability of the Gauss map and the reflexivity for a projective surface. Math. Z. 256(4), 699–703 (2007)
Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)
Hefez, A., Voloch, J.F.: Frobenius nonclassical curves. Arch. Math. (Basel) 54(3), 263–273 (1990)
Homma, M.: Numbers of points of hypersurfaces without lines over finite fields. In: Topics in Finite Fields, Volume 632 of Contemp. Math., pp. 151–156. Amer. Math. Soc., Providence (2015)
Homma, M., Kim, S.J.: An elementary bound for the number of points of a hypersurface over a finite field. Finite Fields Appl. 20, 76–83 (2013)
Katz, N.M.: Space filling curves over finite fields. Math. Res. Lett. 6(5–6), 613–624 (1999)
Kleiman, S., Piene, R.: On the inseparability of the Gauss map. In: Enumerative Algebraic Geometry (Copenhagen, 1989), Volume 123 of Contemp. Math., pp. 107–129. Amer. Math. Soc., Providence (1991)
Kleiman, S.L.: Tangency and duality. In: Proceedings of the 1984 Vancouver Conference in Algebraic Geometry, Volume 6 of CMS Conf. Proc., pp. 163–225. Amer. Math. Soc., Providence (1986)
Poonen, B.: Bertini theorems over finite fields. Ann. Math. (2) 160(3), 1099–1127 (2004)
Stöhr, K.-O., Voloch, J.F.: Weierstrass points and curves over finite fields. Proc. Lond. Math. Soc. (3) 52(1), 1–19 (1986)
Voloch, J.F.: Surfaces in \({\bf P}^{3} \) over finite fields. In Topics in Algebraic and Noncommutative Geometry (Luminy/Annapolis, MD, 2001), Volume 324 of Contemp. Math., pp. 219–226. Amer. Math. Soc., Providence (2003)
Zak, F.L.: Tangents and Secants of Algebraic Varieties, Volume 127 of Translations of Mathematical Monographs. American Mathematical Society, Providence. Translated from the Russian manuscript by the author (1993)
Acknowledgements
We thank Brendan Hassett for the support and valuable suggestions. We thank the referee for the comments on the manuscript. Research by the first author was partially supported by funds from NSF Grant DMS-1701659.
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Asgarli, S., Duan, L. & Lai, KW. Transverse lines to surfaces over finite fields. manuscripta math. 165, 135–157 (2021). https://doi.org/10.1007/s00229-020-01200-7
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DOI: https://doi.org/10.1007/s00229-020-01200-7