Abstract
We show that if an ovoid of PG(3, q), where q>2 is even, has a pencil of translation ovals and if the carrier of the pencil is not an axis of at least one of the ovals in the pencil, then the ovoid is a Tits ovoid. It follows, as a corollary of this and a result of Penttila and Praeger, that if an ovoid of PG(3, q), where q>2 is even, has a pencil of translation ovals then the ovoid is either an elliptic quadric or a Tits ovoid.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barlotti, A.: Un'estensione del teorema di Segre-Kustaanheimo, Boll. Un. Mat. Ital. 10 (1955), 498–506.
Barlotti, A.: Some topics in finite geometrical structures, Institute of Statistics Mimeo Series No 439, University of North Carolina, 1965.
Fellegara, G.: Gli ovaloidi di uno spazio tridimensionale di Galois di ordine 8, Atti Accad. Naz. Lincei Rend. 32 (1962), 170–176.
Glynn, D. G.: Two new sequences of ovals in finite Desarguesian planes of even order, in: L.R.A. Casse, (ed.), Combinatorial Mathematics X, Lecture Notes in Mathematics 1036, Springer-Verlag, New York, 1983, pp. 217–229.
Glynn, D. G.: The Hering classification for inversive planes of even order, Simon Stevin 58 (1984), 319–353.
Glynn, D. G., O'Keefe, C. M., Penttila, T. and Praeger, C. E.: Ovoids and monomial ovals, Geom. Dedicata 59 (1996), 223–241.
Glynn D. G. and Penttila, T.: A plane representation of ovoids, in preparation.
Hirschfeld, J. W. P.: Projective Geometries over Finite Fields, Oxford University Press, Oxford 1979.
Hirschfeld, J. W. P.: Finite Projective Spaces of Three Dimensions, Oxford University Press, Oxford 1985.
O'Keefe, C. M. and Penttila, T.: Ovoids of PG(3, 16) are elliptic quadrics, J. Geom. 38 (1990), 95–106.
O'Keefe C. M. and Penttila, T.: Ovoids of PG(3, 16) are elliptic quadrics, II, J. Geom. 44 (1992), 140–159.
O'Keefe, C. M., Penttila, T. and Royle, G. F.: Classification of ovoids in PG(3, 32), J. Geom. 50 (1994), 143–150.
Panella, G.: Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito, Boll. Un. Mat. Ital. 10 (1955) 507–513.
Payne, S. E. and Thas, J. A.: Finite Generalised Quadrangles, Pitman, London, 1984.
Penttila, T. and Praeger, C. E.: Ovoids and translation ovals, Proc. London Math. Soc., to appear.
Prohaska, O. and Walker, M.: A note on the Hering type of inversive planes of even order, Arch. Math. 28 (1977), 431–432.
Segre, B.: On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two, Acta. Arith. 5 (1959), 315–332.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
O'Keefe, C.M., Penttila, T. Ovoids with a pencil of translation ovals. Geom Dedicata 62, 19–34 (1996). https://doi.org/10.1007/BF00239999
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00239999