Skip to main content
SpringerLink
Log in

A characterization of the line set of an odd-dimensional Baer subspace

  • Published:
Journal of Geometry Aims and scope Submit manuscript

Abstract

Generalizing a theorem of Beutelspacher and Seeger, we consider line sets\(\mathcal{L}\) inP=PG(2t + 1,q),t ∈ IN, with the following properties: (1) any (t + 1)-dimensional subspace ofP contains at least one line of\(\mathcal{L}\), (2) if a pointx ofP is incident with at least two lines of\(\mathcal{L}\) then the points in the factor geometryP/x which are induced by the lines of\(\mathcal{L}\) throughx form a blocking set of type (t, 1) inP/x, (3) any line of\(\mathcal{L}\) is coplanar with at least one further line of\(\mathcal{L}\). We will show that the examples of minimal cardinality are exactly the line sets of Baer subspaces ofP.

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. A. Beutelspacher: On Baer Subspaces of Finite Projective Spaces,Math. Z. 184 (1983), 301 - 319.

    Google Scholar 

  2. A.Beutelspacher: Geometry — from B. Segre's point of View. Variations on a classical theme,Proc. Combinatorics '88 (to appear).

  3. A. Beutelspacher, D. Seeger: Characterizing the Line Set of a Baer Subspace,J. Combin. Theory, Series A 45 (1987), 152 - 156.

    Google Scholar 

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

    Google Scholar 

  5. R. H. Bruck: Difference sets in a finite group,Trans. Amer. Math. Soc. 78 (1955), 464 - 481.

    Google Scholar 

  6. A. A. Bruen: Baer subplanes and blocking sets,Bull. Amer. Math. Soc. 76 (1970), 342 - 344.

    Google Scholar 

  7. A. A. Bruen: Blocking sets in finite projective spaces,SIAM J. Appl. Math. 21, 3 (1971), 380 - 392.

    Google Scholar 

  8. A. A. Bruen: Blocking sets and skew subspaces of a projective space,Can. J. Math. 32, (1980), 628 - 630.

    Google Scholar 

  9. A. A. Bruen, J. A. Thas: Blocking Sets,Geom. Dedicata 6 (1977), 193 - 203.

    Google Scholar 

  10. N. G. de Bruijn, P. Erdös: On a combinatorial problem,Indag. Math. 10 (1948), 421 - 423.

    Google Scholar 

  11. B. Segre: Ovals in a finite projective plane,Can. J. Math. 7 (1955), 414 - 416.

    Google Scholar 

  12. G. Tallini: Sulle k-calotte di uno spazio lineare finito,Ann. Mat. 42 (1956), 119 - 164.

    Google Scholar 

  13. M. Tallini Scafati: Caratterizazzione grafica delle forme hermitane di unS r,q ,Rend. Mat. Roma 26 (1967), 273 - 303.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Universität Mainz, Saarstraße 21, D-6500, Mainz 1

    Johannes Ueberberg

Authors
  1. Johannes Ueberberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ueberberg, J. A characterization of the line set of an odd-dimensional Baer subspace. J Geom 37, 171–180 (1990). https://doi.org/10.1007/BF01230370

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01230370

Keywords

Search

Navigation