Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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


Author: Slawomir Rams
Journal: Trans. Amer. Math. Soc. 354 (2002), 2359-2367
MSC (2000): Primary 14M99; Secondary 14E20.
DOI: https://doi.org/10.1090/S0002-9947-02-02979-3
Published electronically: February 7, 2002
MathSciNet review: 1885656
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)

  • 1. W. Barth: Even sets of eight skew lines on a K3 surface, preprint.
  • 2. 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
  • 3. 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
  • 4. Arnaud Beauville, Sur le nombre maximum de points doubles d’une surface dans 𝑃³ (𝜇(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
  • 5. 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, https://doi.org/10.1007/978-1-4612-4264-2_2
  • 6. Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
  • 7. Sheng-Li Tan: Cusps on some algebraic surfaces, preprint, 1999.
  • 8. Rick Miranda, Triple covers in algebraic geometry, Amer. J. Math. 107 (1985), no. 5, 1123–1158. MR 805807, https://doi.org/10.2307/2374349
  • 9. Yoichi Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), no. 2, 159–171. MR 744605, https://doi.org/10.1007/BF01456083
  • 10. V. V. Nikulin, Kummer surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 2, 278–293, 471 (Russian). MR 0429917
  • 11. J. H. van Lint, Introduction to coding theory, 2nd ed., Graduate Texts in Mathematics, vol. 86, Springer-Verlag, Berlin, 1992. MR 1217490
  • 12. D. van Straten: Macaulay script to estimate the number of lines on a surface with some examples of surfaces.
  • 13. B. Segre: The maximum number of lines lying on a quartic surface, Quart. J. Math. Oxford Ser. 14 (1943), 86-96. MR 6:16g

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14M99, 14E20.

Retrieve articles in all journals with MSC (2000): 14M99, 14E20.


Additional Information

Slawomir 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: rams@mi.uni-erlangen.de and rams@im.uj.edu.pl

DOI: https://doi.org/10.1090/S0002-9947-02-02979-3
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