Counting lines on surfaces

Samuel Boissière; Alessandra Sarti

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 1, page 39-52
  • ISSN: 0391-173X

Abstract

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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.

How to cite

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Boissière, Samuel, and Sarti, Alessandra. "Counting lines on surfaces." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.1 (2007): 39-52. <http://eudml.org/doc/272262>.

@article{Boissière2007,
abstract = {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.},
author = {Boissière, Samuel, Sarti, Alessandra},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {1},
pages = {39-52},
publisher = {Scuola Normale Superiore, Pisa},
title = {Counting lines on surfaces},
url = {http://eudml.org/doc/272262},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Boissière, Samuel
AU - Sarti, Alessandra
TI - Counting lines on surfaces
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 1
SP - 39
EP - 52
AB - 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.
LA - eng
UR - http://eudml.org/doc/272262
ER -

References

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  1. [1] A. B. Altman and S. L. Kleiman, Foundations of the theory of Fano schemes, Compositio Math.34 (1977), 3–47. Zbl0414.14024MR569043
  2. [2] W. Barth and A. Van de Ven, Fano varieties of lines on hypersurfaces, Arch. Math. (Basel) 31 (1978/79), 96–104. Zbl0383.14003MR510081
  3. [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. Zbl0862.14012MR1363052
  4. [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. Zbl0902.14040
  5. [5] R. Hartshorne, “Algebraic geometry”, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977. Zbl0367.14001MR463157
  6. [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. Zbl0803.01037MR1315530JFM16.0061.01
  7. [7] Y. Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann.268 (1984), 159–171. Zbl0521.14013MR744605
  8. [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. Zbl0325.14015MR429917
  9. [9] S. Rams, Three-divisible families of skew lines on a smooth projective quintic, Trans. Amer. Math. Soc. 354 (2002), 2359–2367 (electronic). Zbl0987.14023MR1885656
  10. [10] S. Rams, Projective surfaces with many skew lines, Proc. Amer. Math. Soc. 133 (2005), 11–13 (electronic). Zbl1049.14027MR2085146
  11. [11] A. Sarti, Pencils of symmetric surfaces in ℙ 3 , J. Algebra246 (2001), 429–452. Zbl1064.14038MR1872630
  12. [12] B. Segre, The maximum number of lines lying on a quartic surface, Quart. J. Math., Oxford Ser. 14 (1943), 86–96. Zbl0063.06860MR10431
  13. [13] B. Segre, On arithmetical properties of quartic surfaces, Proc. London Math. Soc. (2) 49 (1947), 353–395. Zbl0034.08603MR21952

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