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At most 64 lines on smooth quartic surfaces (characteristic 2). (English) Zbl 1409.14067

This paper proves the following theorem.
Theorem: Let \(k\) be a field of characteristic \(2\). Then any smooth quartic surface over \(k\) contains at most \(64\) lines.
The field \(k\) is assumed to be algebraically closed, as the upper bound is not affected by the base change. The proof rests on geometric arguments, e.g., elliptic (quasi-elliptic) fibrations in characteristic \(2\), and ramification types and the Hessian of a cubic in characteristic \(2\), lines of the first kind and the lines of the second kind on a smooth quartic surface.
A. Degtyarev [“Lines in supersingular quartics”, Preprint, arXiv:1604.05836] has improved the bound to \(60\) in characteristic \(2\).
This article gives an explicit example of a smooth quartic surface defined over the finite field \(\mathbb{F}_4\) which contains \(60\) lines over \(\mathbb{F}_{16}\)i. Such a surface is given by a quartic surface with \(S_5\)-action.

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
14G17 Positive characteristic ground fields in algebraic geometry
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14J70 Hypersurfaces and algebraic geometry
14N25 Varieties of low degree
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References:

[1] M.Artin, Supersingular K3 surfaces, Ann. Sci. Éc. Norm. Supér. (4)7 (1974), 543-568.10.24033/asens.1279 · Zbl 0322.14014
[2] A.Degtyarev, Lines in supersingular quartics, preprint, 2016, arXiv:1604.05836. · Zbl 1496.14040
[3] A.Degtyarev, I.Itenberg and A. S.Sertöz, Lines on quartic surfaces, Math. Ann.368 (2017), 753-809. doi:10.1007/s00208-016-1484-0.10.1007/s00208-016-1484-0 · Zbl 1368.14052
[4] N. D.Elkies and M.Schütt, Genus 1 fibrations on the supersingular K3 surface in characteristic 2 with Artin invariant 1, Asian J. Math.19 (2015), 555-582.10.4310/AJM.2015.v19.n3.a7 · Zbl 1319.14043
[5] K.Kodaira, On compact analytic surfaces I-III, Ann. of Math. (2)71 (1960), 111-152; 77 (1963), 563-626; 78 (1963), 1-40.10.2307/1969881 · Zbl 0098.13004
[6] W. E.Lang, Configurations of singular fibers on rational elliptic surfaces in characteristic two, Comm. Algebra28 (2000), 5813-5836.10.1080/00927870008827190 · Zbl 1017.14014
[7] A.Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Publ. Math. Inst. Hautes Études Sci.21 (1964), 5-128.10.1007/BF02684271 · Zbl 0132.41403
[8] S.Rams and M.Schütt, The Barth quintic surface has Picard number 41, Ann. Sci. Éc. Norm. Supér. Pisa Cl. Sci. (5)XIII (2014), 533-549. · Zbl 1300.14039
[9] S.Rams and M.Schütt, 64 lines on smooth quartic surfaces, Math. Ann.362 (2015), 679-698. · Zbl 1319.14042
[10] S.Rams and M.Schütt, 112 lines on smooth quartic surfaces (characteristic 3), Q. J. Math.66 (2015), 941-951.10.1093/qmath/hav018 · Zbl 1330.14064
[11] A. N.Rudakov and I. R.Shafarevich, Surfaces of type K3 over fields of finite characteristic, J. Sov. Math.22(4) (1983), 1476-1533. · Zbl 0518.14015
[12] F.Schur, Ueber eine besondre Classe von Flächen vierter Ordnung, Math. Ann.20 (1882), 254-296.10.1007/BF01446525 · JFM 14.0564.01
[13] M.Schütt, The maximal singular fibres of elliptic K3 surfaces, Arch. Math. (Basel)87(4) (2006), 309-319.10.1007/s00013-006-1734-6 · Zbl 1111.14034
[14] M.Schütt and A.Schweizer, On the uniqueness of elliptic K3 surfaces with maximal singular fibre, Ann. Inst. Fourier (Grenoble)63 (2013), 689-713.10.5802/aif.2773 · Zbl 1273.14078
[15] A.Schweizer, Extremal elliptic surfaces in characteristic 2 and 3, Manuscripta Math.102 (2000), 505-521.10.1007/s002290070039 · Zbl 0989.14013
[16] B.Segre, The maximum number of lines lying on a quartic surface, Q. J. Math., Oxford Ser.14 (1943), 86-96.10.1093/qmath/os-14.1.86 · Zbl 0063.06860
[17] T.Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Pauli39 (1990), 211-240. · Zbl 0725.14017
[18] J.Tate, “Algorithm for determining the type of a singular fibre in an elliptic pencil”, in Modular Functions of One Variable IV (Antwerpen 1972), Lecture Notes in Mathematics 476, Springer-Verlag Berlin Heidelberg, 1975, 33-52.10.1007/BFb0097582 · Zbl 1214.14020
[19] D. C.Veniani, The maximum number of lines lying on a K3 quartic surface, Math. Z.285 (2017), 1141-1166.10.1007/s00209-016-1742-6 · Zbl 1387.14107
[20] D. C.Veniani, Lines on K3 quartic surfaces in characteristic 2, Q. J. Math., to appear, doi:10.1093/qmath/haw055. · Zbl 1376.14037
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