Skip to main content
SpringerLink
Log in

The cubic Segre variety in PG(5, 2)

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

Abstract

The Segre variety \({\mathcal{S}_{1,2}}\) in PG(5, 2) is a 21-set of points which is shown to have a cubic equation Q(x) = 0. If T(x, y, z) denotes the alternating trilinear form obtained by completely polarizing the cubic polynomial Q, then the associate U # of an r-flat \({U \subset {\rm PG}(5, 2)}\) is defined to be

$$U^{\#} = \{z \in {\rm PG}(5, 2)\,|\,T(u_{1}, u_{2}, z) = 0\, {\rm for \, all}\, u_{1}, u_{2} \in U\},$$

and so is an s-flat for some s. Those lines L of PG(5, 2) which are singular, satisfying that is L # = PG(5.2), are shown to form a complete spread of 21 lines. For each r-flat \({U \subset {\rm PG}(5, 2)}\) its associate U # is determined. Examples are given of four kinds of planes P which are self-associate, P # = P, and three kinds of planes for which P, P #, P ## are disjoint planes such that P ### = P.

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

Access this article

Log in via an institution

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. Shaw R.: A characterization of the primals in PG(m, 2). Des. Codes Cryptogr. 2, 253–256 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Shaw R.: Double-fives and partial spreads in PG(5, 2). In: Hirschfeld J.W.P., Magliveras S.S., de Resmini M.J. (eds.) Geometry, Combinatorial Designs and Related Structures: Proceedings of the First Pythagorean Conference, Spetses, Greece 1996, pp. 201–216. Cambridge University Press (1997).

  3. Shaw R.: Configurations of planes in PG(5, 2). Discrete Math. 208/209, 529–546 (1999)

    Article  Google Scholar 

  4. Shaw R.: Subsets of PG(n, 2) and maximal partial spreads in PG(4, 2). Des. Codes Cryptogr. 21, 209–222 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Shaw R., Gordon N.A.: The quintic Grassmannian \({\mathcal{G}_{1,4,2}}\) in PG(9, 2). Des. Codes Cryptogr. 32, 381–396 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Shaw R.: The ψ-associate X # of a flat X in PG(n, 2). Des. Codes Cryptogr. 21, 229–246 (2007)

    Article  Google Scholar 

  7. Shaw R.: The polynomial degree of Grassmann and Segre varieties over GF(2). Discrete Math. 308, 872–879 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Shaw R., Maks J.G., Gordon N.A.: The classification of flats in PG(9, 2) which are external to the Grassmannian G 1,4,2. Des. Codes Cryptogr. 34, 203–227 (2005)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Mathematics, University of Hull, Hull, HU6 7RX, UK

    Ron Shaw

  2. Department of Computer Science, University of Hull, Hull, HU6 7RX, UK

    Neil A. Gordon

Authors
  1. Ron Shaw
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Neil A. Gordon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ron Shaw.

Additional information

Communicated by J.W.P. Hirschfeld.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shaw, R., Gordon, N.A. The cubic Segre variety in PG(5, 2). Des. Codes Cryptogr. 51, 141–156 (2009). https://doi.org/10.1007/s10623-008-9250-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-008-9250-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation