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Transactions of the American Mathematical Society

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Three-divisible families of skew lines on a smooth projective quintic

Author: Sławomir Rams
Journal: Trans. Amer. Math. Soc. 354 (2002), 2359-2367
MSC (2000): Primary 14M99; Secondary 14E20.
Published electronically: February 7, 2002
MathSciNet review: 1885656
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Abstract: We give an example of a family of 15 skew lines on a quintic such that its class is divisible by 3. We study properties of the codes given by arrangements of disjoint lines on quintics.

References [Enhancements On Off] (What's this?)

  • W. Barth: Even sets of eight skew lines on a K3 surface, preprint.
  • W. Barth and I. Nieto, Abelian surfaces of type $(1,3)$ and quartic surfaces with $16$ skew lines, J. Algebraic Geom. 3 (1994), no. 2, 173–222. MR 1257320
  • W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574
  • Arnaud Beauville, Sur le nombre maximum de points doubles d’une surface dans ${\bf P}^{3}$ $(\mu (5)=31)$, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 207–215 (French). MR 605342
  • Lucia Caporaso, Joe Harris, and Barry Mazur, How many rational points can a curve have?, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 13–31. MR 1363052, DOI
  • Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
  • Sheng-Li Tan: Cusps on some algebraic surfaces, preprint, 1999.
  • Rick Miranda, Triple covers in algebraic geometry, Amer. J. Math. 107 (1985), no. 5, 1123–1158. MR 805807, DOI
  • Yoichi Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), no. 2, 159–171. MR 744605, DOI
  • V. V. Nikulin, Kummer surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 2, 278–293, 471 (Russian). MR 0429917
  • J. H. van Lint, Introduction to coding theory, 2nd ed., Graduate Texts in Mathematics, vol. 86, Springer-Verlag, Berlin, 1992. MR 1217490
  • D. van Straten: Macaulay script to estimate the number of lines on a surface with some examples of surfaces.
  • B. Segre: The maximum number of lines lying on a quartic surface, Quart. J. Math. Oxford Ser. 14 (1943), 86–96.

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Additional Information

Sławomir Rams
Affiliation: Institute of Mathematics, Jagiellon University, Reymonta 4, PL-30-059 Kraków, Poland
Address at time of publication: Mathematisches Institut, FAU Erlangen-Nürnberg, Bismarckstrasse 1 1/2, D-91054 Erlangen, Germany
Email: and

Keywords: Quintic, cyclic cover, code.
Received by editor(s): December 31, 2000
Published electronically: February 7, 2002
Additional Notes: This research was supported by DFG contract BA 423/8-1
Article copyright: © Copyright 2002 American Mathematical Society