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Classification of ovoids inPG(3, 32)

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Abstract

A classification of the ovoids inPG(3, 32) is completed with the aid of a computer. The ovoids are examined in terms of which ovals can possibly appear as secant plane sections. A weak necessary condition for two ovals to appear together as plane sections of an ovoid surprisingly turns out to be sufficient to demonstrate that the only possible secant plane sections are translation ovals. A known result regarding ovoids with such plane sections then identifies the ovoids as either elliptic quadrics or Tits ovoids.

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References

  1. Barlotti, A. Un'estensione del teorema di Segre-Kustaanheimo.Boll. Un. Mat. Ital.,10, 1955, 498–506.

    Google Scholar 

  2. Dembowski, P. Inversive planes of even order.Bull. Amer. Math. Soc. 69, 1963, 850–854.

    Google Scholar 

  3. Dembowski, P. Finite Geometries. Springer-Verlag, Berlin, 1968.

    Google Scholar 

  4. Fellegara, G. Gli ovaloidi di uno spazio tridimensionale di Galois di ordine 8.Atti. Accad. Naz. Lincei Rend. 32, 1962, 170–176.

    Google Scholar 

  5. Glynn, D.G and T. Penttila. A plane representation of ovoids.Submitted.

  6. Hirschfeld, J.W.P. Finite Projective Spaces of Three Dimensions. Oxford University Press, Oxford, 1985.

    Google Scholar 

  7. O'Keefe, C.M. and T. Penttila. Ovoids ofPG(3, 16) are elliptic quadrics.Journal of Geometry 38, 1990, 95–106.

    Google Scholar 

  8. O'Keefe, C.M and T. Penttila. Ovoids ofPG(3,16) are elliptic quadrics II. To appear inJournal of Geometry

  9. Panella, G. Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito.Boll. Un. Mat. Ital.,10, 1955, 507–513.

    Google Scholar 

  10. Payne, S.E and J.A Thas. Finite generalized quadrangles. Research Notes in Mathematics 110, Pitman, 1984.

  11. Penttila, T and C.E. Praeger. Ovoids and translation ovals.Submitted.

  12. Penttila, T and G. F. Royle. Classification of hyperovals inPG(2, 32).University of Western Australia Pure Mathematics Research Report 1992/2, 1992. To appear inJournal of Geometry.

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Authors and Affiliations

  1. Department of Pure Mathematics, The University of Adelaide, GPO Box 498, 5001, Adelaide, Australia

    Christine M. O'Keefe

  2. Department of Computer Science, University of Western Australia, 6009, Nedlands, Australia

    Tim Penttila

  3. Department of Mathematics, University of Western Australia, 6009, Nedlands, Australia

    Gordon F. Royle

Authors
  1. Christine M. O'Keefe
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  2. Tim Penttila
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  3. Gordon F. Royle
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O'Keefe, C.M., Penttila, T. & Royle, G.F. Classification of ovoids inPG(3, 32). J Geom 50, 143–150 (1994). https://doi.org/10.1007/BF01222671

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  • DOI: https://doi.org/10.1007/BF01222671

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