no. 1
Counting lines on surfaces
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, pp. 39-52

This paper deals with surfaces with many lines. It is well-known that a cubic contains 27 of them and that the maximal number for a quartic is 64. In higher degree the question remains open. Here we study classical and new constructions of surfaces with high number of lines. We obtain a symmetric octic with 352 lines, and give examples of surfaces of degree d containing a sequence of d(d-2)+4 skew lines.

Classification : 14N10, 14Q10 
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     title = {Counting lines on surfaces},
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Boissière, Samuel; Sarti, Alessandra. Counting lines on surfaces. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, pp. 39-52. https://www.numdam.org/item/ASNSP_2007_5_6_1_39_0/

[1] A. B. Altman and S. L. Kleiman, Foundations of the theory of Fano schemes, Compositio Math. 34 (1977), 3-47. | Zbl | MR | Numdam

[2] W. Barth and A. Van De Ven, Fano varieties of lines on hypersurfaces, Arch. Math. (Basel) 31 (1978/79), 96-104. | Zbl | MR

[3] L. Caporaso, J. Harris and B. Mazur, How many rational points can a curve have?, In: “The moduli space of curves” (Texel Island, 1994), Progr. Math., Vol. 129, Birkhäuser Boston, Boston, MA, 1995, 13-31. | Zbl | MR

[4] G.-M. Greuel, G. Pfister and H. Schönemann, Singular 2.0, A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, 2001, http://www.singular.uni-kl.de. | Zbl

[5] R. Hartshorne, “Algebraic geometry”, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977. | Zbl | MR

[6] Felix Klein, “Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade”, Birkhäuser Verlag, Basel, 1993, Reprint of the 1884 original, Edited, with an introduction and commentary by Peter Slodowy. | Zbl | MR | JFM

[7] Y. Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), 159-171. | Zbl | MR

[8] V. V. Nikulin, Kummer surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 278-293, 471. English translation: Math. USSR Izv. 9 (1975), 261-275. | Zbl | MR

[9] S. Rams, Three-divisible families of skew lines on a smooth projective quintic, Trans. Amer. Math. Soc. 354 (2002), 2359-2367 (electronic). | Zbl | MR

[10] S. Rams, Projective surfaces with many skew lines, Proc. Amer. Math. Soc. 133 (2005), 11-13 (electronic). | Zbl | MR

[11] A. Sarti, Pencils of symmetric surfaces in ℙ 3 , J. Algebra 246 (2001), 429-452. | Zbl | MR

[12] B. Segre, The maximum number of lines lying on a quartic surface, Quart. J. Math., Oxford Ser. 14 (1943), 86-96. | Zbl | MR

[13] B. Segre, On arithmetical properties of quartic surfaces, Proc. London Math. Soc. (2) 49 (1947), 353-395. | Zbl | MR