On Ovoids of Parabolic Quadrics

Authors:
Simeon Ball;Patrick Govaerts;Leo Storme
Affiliations:
Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Barcelona, Spain 08034;Department of Pure Mathematics and Computer Algebra, Ghent University, Gent, Belgium S22, 9000;Department of Pure Mathematics and Computer Algebra, Ghent University, Gent, Belgium S22, 9000
Venue:
Designs, Codes and Cryptography
Year:
2006

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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, q3.We conclude with a 1 mod p result for ovoids of Q(6, q), q=ph, p prime.