Abstract
This paper is a contribution to the classification of ovoids. We show, under some rather technical assumptions, that if an ovoid of PG(3, q) has a pencil of monomial ovals, then it is either an elliptic quadric or a Tits ovoid. Further, we show that if an ovoid of PG(3, q) has a bundle of translation ovals, again under some extra assumptions, then the ovoid is 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. (3) 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.: A condition for the existence of ovoids in PG(3, q), q even, in G. Faina, G. Tallini (eds), Giornate di Geometria Combinatoria, Perugia, 1993, pp. 213–225.
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, 1983, pp. 217–229.
Glynn, D. G.: The Hering classification for inversive planes of even order, Simon Stevin 58 (1984), 319–353.
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. and Penttila, T.: Ovoids with a pencil of translation ovals, (submitted).
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. (3) 10 (1955), 507–513.
Payne, S. E.: A complete determination of translation ovoids in finite Desarguesian planes, Atti Accad. Naz. Lincei Rend. 51 (1971), 328–331.
Penttila, T. and Praeger, C. E.: Ovoids and translation ovals, 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.
Segre, B. and Bartocci, U.: Ovali ed altre curve nei piani di Galois di caratteristica due, Acta Arith. 18 (1971), 423–449.
Tits, J.: Ovoïdes à translations, Rend. Mat. V 21 (1962), 37–59.
Tits, J.: Ovoïdes et groupes de Suzuki, Arch. Math. 13 (1962), 187–198.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Glynn, D.G., O'Keefe, C.M., Penttila, T. et al. Ovoids and monomial ovals. Geom Dedicata 59, 223–241 (1996). https://doi.org/10.1007/BF00181693
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00181693