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On Ovoids of Parabolic Quadrics

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Abstract

It is known that every ovoid of the parabolic quadric Q(4, q), q=ph, p prime, intersects every three-dimensional elliptic quadric in 1 mod p points. We present a new approach which gives us a second proof of this result and, in the case when p=2, allows us to prove that every ovoid of Q(4, q) either intersects all the three-dimensional elliptic quadrics in 1 mod 4 points or intersects all the three-dimensional elliptic quadrics in 3 mod 4 points.

We also prove that every ovoid of Q(4, q), q prime, is an elliptic quadric. This theorem has several applications, one of which is the non-existence of ovoids of Q(6, q), q prime, q>3.

We conclude with a 1 mod p result for ovoids of Q(6, q), q=ph, p prime.

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Authors and Affiliations

  1. Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Jordi Girona 1–3, Mòdul C3, 08034, Barcelona, Spain

    Simeon Ball

  2. Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281, S22, 9000, Gent, Belgium

    Patrick Govaerts & Leo Storme

Authors
  1. Simeon Ball
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  2. Patrick Govaerts
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  3. Leo Storme
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Corresponding author

Correspondence to Patrick Govaerts.

Additional information

Communicated by: D. Jungnickel

AMS Classification:51E12, 51E20

The first author acknowledges the support of the Ministerio de Ciencia y Tecnología, España.

The third author thanks the Fund for Scientific Research – Flanders (Belgium) for a Research Grant.

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Ball, S., Govaerts, P. & Storme, L. On Ovoids of Parabolic Quadrics. Des Codes Crypt 38, 131–145 (2006). https://doi.org/10.1007/s10623-005-5666-0

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  • DOI: https://doi.org/10.1007/s10623-005-5666-0

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