Skip to main content
Springer Nature Link
Log in

Characterizations of Hermitian varieties by intersection numbers

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

In this paper, we give characterizations of the classical generalized quadrangles H(3, q 2) and H(4, q 2), embedded in PG(3, q 2) and PG(4, q 2), respectively. The intersection numbers with lines and planes characterize H(3, q 2), and H(4, q 2) is characterized by its intersection numbers with planes and solids. This result is then extended to characterize all Hermitian varieties in dimension at least 4 by their intersection numbers with planes and solids.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ball S.: Multiple blocking sets and arcs in finite planes. J. Lond. Math. Soc. (2) 54(3), 581–593 (1996)

    MATH  Google Scholar 

  2. Blokhuis A., Storme L., Szőnyi T.: Lacunary polynomials, multiple blocking sets and Baer subplanes. J. Lond. Math. Soc. (2) 60(2), 321–332 (1999)

    Article  Google Scholar 

  3. Bose R.C., Burton R.C.: A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes. J. Comb. Theory 1, 96–104 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bruen A.: Baer subplanes and blocking sets. Bull. Am. Math. Soc 76, 342–344 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bruen A.A.: Polynomial multiplicities over finite fields and intersection sets. J. Comb. Theory Ser. A 60(1), 19–33 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bruen A.A., Thas J.A.: Blocking sets. Geom. Dedicata 6(2), 193–203 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  7. Buekenhout F., Lefèvre C.: Generalized quadrangles in projective spaces. Arch. Math. (Basel) 25, 540–552 (1974)

    MATH  MathSciNet  Google Scholar 

  8. Buekenhout F., Shult E.E.: On the foundations of polar geometry. Geom. Dedicata 3, 155–170 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ferri O., Tallini G.: A characterization of nonsingular quadrics in PG(4, q). Rend. Mat. Appl. (7) 11(1), 15–21 (1991)

    MATH  MathSciNet  Google Scholar 

  10. Hirschfeld J.W.P.: Projective Geometries Over Finite Fields, 2nd edn, 555 pp. Oxford University Press, Oxford (1998).

  11. Hirschfeld J.W.P., Thas J.A.: General Galois Geometries, 407 pp. Oxford University Press, Oxford (1991).

  12. Payne S.E., Thas J.A.: Finite Generalized Quadrangles, 312 pp. Pitman, London (1984).

  13. Tits J.: Buildings of spherical type and finite BN-pairs. Lecture Notes in Mathematics, vol. 386, 299 pp. Springer, Berlin (1974).

  14. Veldkamp F.D.: Polar geometry, i-v. Indag. Math. 21, 512–551 (1959)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281 S 22, 9000, Gent, Belgium

    J. Schillewaert & J. A. Thas

Authors
  1. J. Schillewaert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. A. Thas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Schillewaert.

Additional information

Communicated by J. W. P. Hirschfeld.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schillewaert, J., Thas, J.A. Characterizations of Hermitian varieties by intersection numbers. Des. Codes Cryptogr. 50, 41–60 (2009). https://doi.org/10.1007/s10623-008-9213-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-008-9213-7

Keywords

AMS Classifications