Skip to main content
SpringerLink
Log in

Affine ovoids and extensions of generalized quadrangles

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

A set Δ of vertices of a generalized quadrangle of order (s, t) is said to be a hyperoval if any line intersects Δ in either 0, or 2 points. A hyperoval Δ is called an affine ovoid if |Δ|=2st. It is well known that μ-subgraphs in triangular extensions of generalized quadrangles are hyperovals. In the present paper we prove that ifS is a triangular extension forGQ(s, t) with totally regular point graph Γ such that μ=2st, thens is even, Γ is an τ-antipodal graph of diameter 3 with τ=1+s/2, and eithers=2, ort=s+2.

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. P. Cameron, D. R. Hughes, and A. Pasini, “Extended generalized quadrangles,”Geom. Dedicata,35, 193–228 (1990).

    Article  MATH  Google Scholar 

  2. P. Fisher and S. Hobart, “Triangular extended generalized quadrangles,”Geom. Dedicata,37, 339–344 (1991).

    Article  MATH  Google Scholar 

  3. A. Del Fra, D. Ghinelli, T. Meixner, and A. Pasini, “Flag-transitive extensions ofC n geometries,”Geom. Dedicata,37, 253–273 (1992).

    Google Scholar 

  4. A. Pasini, “Remarks on double ovoids in finite classical generalized quadrangles, with an application to extended generalized quadrangles,”Quart. J. Pure Appl. Math.,66, 41–68 (1992).

    MATH  Google Scholar 

  5. C. D. Godsil, “Covers of complete graphs,”Adv. Stud. Pure Math.,24, 137–163 (1996).

    Google Scholar 

  6. A. A. Makhnev, “LocallyGQ(3,5)-graphs and geometries with short lines,”Diskret. Mat. [Discrete Math. Appl.],10, 72–86 (1998).

    Google Scholar 

  7. S. Yoshiara, “On some flag-transitive non-classicalc.C 2 -geometries,”Europ. J. Combin.,14, 59–77 (1993).

    Article  MATH  Google Scholar 

  8. A. Del Fra, D. V. Pasechnik, and A. Pasini, “A new family of extended generalized quadrangles,”Europ. J. Combin.,18, 155–169 (1997).

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Ural Division of the Russian Academy of Sciences, Ekaterinburg

    A. A. Makhnev

Authors
  1. A. A. Makhnev
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 266–271, August, 2000.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Makhnev, A.A. Affine ovoids and extensions of generalized quadrangles. Math Notes 68, 232–236 (2000). https://doi.org/10.1007/BF02675348

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02675348

Key words

Search

Navigation