Abstract
It is shown that if a plane of PG(3,q),q even, meets an ovoid in a pointed conic, then eitherq=4 and the ovoid is an elliptic quadric, orq=8 and the ovoid is a Tits ovoid.
Similar content being viewed by others
References
A. Barlotti, Un'estensione del teorema di Segre-Kustaanheimo,Boll. Un. Mat. Ital. 10 (1955), 96–98.
A. Barlotti,Some topics in finite geometrical structures, Institute of Statistics Mimeo Series No. 439, University of North Carolina, North Carolina, 1965.
A. Basile andP. Brutti, Planes and ovals,J. Geom. 13 (1979) no. 2, 101–107.
M.R.Brown, Ovoids of PG(3, q),q even, with a conic section,J. London Math. Soc. (2), to appear.
P.Dembowski,Finite Geometries, Springer Verlag, 1968.
G. Fellegara, Gli ovaloidi di uno spazio tridimensionale de Galois di ordine 8,Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 32 (1962), 170–176.
J.W.P. Hirschfeld,Finite Projective Spaces of Three Dimensions, Oxford University Press, Oxford, 1985.
J.W.P. Hirschfeld,Projective Geometries over Finite Fields, Second Edition, Oxford University Press, Oxford, 1998.
C.M. O'keefe, Ovoids in PG(3, q): a survey,Discrete Math. 151 (1996) No. 1–3, 175–188.
C.M. O'keefe andT. Penttila, Ovoids of PG(3, 16) are elliptic quadrics,J. Geom. 38 (1990), 95–106.
C.M. O'keefe andT. Penttila, Ovoids of PG(3, 16) are elliptic quadrics II,J. Geom. 44 (1992), 140–159.
C.M. O'keefe, T. Penttila andG.F. Royle, Classification of ovoids in PG(3, 32),J. Geom.,50 (1994), 143–150.
G. Panella, Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito,Boll. Un. Mat. Ital. 10 (1955), 507–513.
S.E. Payne, A complete determination of translation ovoids in finite Desarguesian planes,Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 51 (1971), 328–331.
S.E. Payne andJ.A. Thas,Finite Generalized Quadrangles, Pitman, London, 1984.
T. Penttila andC.E. Praeger, Ovoids and translation ovals,J. London Math. Soc. (2)56 (1997), 607–624.
J.A. Thas, Ovoidal translation planes,Arch. Math. 23 (1972), 110–112.
J.A.Thas, Ovoids and spreads of generalized quadrangles, Postgraduate course, Univ. of Ghent, 1981.
J. Tits, Ovoides et groupes du Suzuki,Arch. Math. 13 (1962), 187–198.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Brown, M.R. The determination of ovoids of PG(3,q) containing a pointed conic. J Geom 67, 61–72 (2000). https://doi.org/10.1007/BF01220298
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01220298