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Spreads and ovoids in finite generalized quadrangles

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Abstract

This paper is a survey on the existence and non-existence of ovoids and spreads in the known finite generalized quadrangles. It also contains the following new results. We prove that translation generalized quadrangles of order (s,s 2), satisfying certain properties, have a spread. This applies to three known infinite classes of translation generalized quadrangles. Further a new class of ovoids in the classical generalized quadranglesQ(4, 3e),e≥3, is constructed. Then, by the duality betweenQ(4, 3e) and the classical generalized quadrangleW (3e), we get line spreads of PG(3, 3e) and hence translation planes of order 32e. These planes appear to be new. Note also that only a few classes of ovoids ofQ(4,q) are known. Next we prove that each generalized quadrangle of order (q 2,q) arising from a flock of a quadratic cone has an ovoid. Finally, we give the following characterization of the classical generalized quadranglesQ(5,q): IfS is a generalized quadrangle of order (q,q 2),q even, having a subquadrangleS′ isomorphic toQ(4,q) and if inS′ each ovoid consisting of all points collinear with a given pointx ofS\S′ is an elliptic quadric, thenS is isomorphic toQ(5,q).

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References

  1. Brouwer, A. E.: Private communication, 1981.

  2. Brouwer, A. E. and Wilbrink, H. A.: Ovoids and fans in the generalized quadrangleQ(4, 2),Geom. Dedicata 36 (1990), 121–124.

    Google Scholar 

  3. De Clerck, F. and Herssens, G.: Flocks of the quadratic cone in PG(3,q) forq small,Reports of the CAGe Project 8 (1993), 1–75.

    Google Scholar 

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

    Google Scholar 

  5. Hirschfeld, J. W. P.:Finite Projective Spaces of Three Dimensions, Clarendon Press, Oxford, 1985.

    Google Scholar 

  6. Kantor, W. M.: Generalized quadrangles associated withG 2(q),J. Combin. Theory Ser. A 29 (1980), 212–219.

    Google Scholar 

  7. Kantor, W. M.: Ovoids and translation planes,Canad. J. Math. 34 (1982), 1195–1203.

    Google Scholar 

  8. Kantor, W. M.: Some generalized quadrangles with parameters (q 2,q),Math. Z. 192 (1986), 45–50.

    Google Scholar 

  9. Knarr, N.: A geometric construction of generalized quadrangles from polar spaces of rank three,Result. Math. 21 (1992), 332–344.

    Google Scholar 

  10. Lunardon, G.: Private communication, 1993.

  11. O'Keefe, C. M. and Penttila, T.: Ovoids of PG(3, 16) are elliptic quadrics II,J. Geom. 44 (1992), 140–159.

    Google Scholar 

  12. O'Keefe, C. M., Penttila, T. and Royle, G. F.: Classification of ovoids in PG(3, 32), Preprint, 1992.

  13. Payne, S. E.: A new infinite family of generalized quadrangles,Congr. Numer. 49 (1985), 115–128.

    Google Scholar 

  14. Payne, S. E.: An essay on skew translation generalized quadrangles,Geom. Dedicata 32 (1989), 93–118.

    Google Scholar 

  15. Payne, S. E.: The generalized quadrangle with (s, t) = (3, 5),Congr. Numer. 77 (1990), 5–29.

    Google Scholar 

  16. Payne, S. E.: A census of finite generalized quadrangles, in W. M. Kantor, R. A. Liebler, S. E. Payne, E. E. Shult (eds),Finite Geometries, Buildings and Related Topics, Clarendon Press, Oxford, 1990, pp. 29–36.

    Google Scholar 

  17. Payne, S. E. and Maneri, C. C.: A family of skew-translation generalized quadrangles of even order,Congr. Numer. 36 (1982), 127–135.

    Google Scholar 

  18. Payne, S. E. and Thas, J. A.:Finite Generalized Quadrangles, Pitman, London, 1984.

    Google Scholar 

  19. Payne, S. E. and Thas, J. A.: Generalized quadrangles, BLT-sets, and Fisher flocks,Congr. Numer. 84 (1991), 161–192.

    Google Scholar 

  20. Penttila, T. and Royle, G. F.: Private communication, 1993.

  21. Segre, B.: Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane,Ann. Mat. Pura Appl. 64 (1964), 1–76.

    Google Scholar 

  22. Thas, J. A.: Ovoidal translation planes,Arch. Math. 23 (1972), 110–112.

    Google Scholar 

  23. Thas, J. A.: Semi-partial geometries and spreads of classical polar spaces,J. Combin. Th. Ser. A 35 (1983), 58–66.

    Google Scholar 

  24. Thas, J. A.: Old and new results on spreads and ovoids of finite classical polar spaces, in A. Barlottiet al. (eds),Combinatorics '90, Ann. Discrete Math., Elsevier, 1992, pp. 529–544.

    Google Scholar 

  25. Thas, J. A.: Generalized quadrangles of order (s,s 2). Part I,J. Combin. Th. Ser. A (to appear).

  26. Thas, J. A.: Projective geometry over a finite field, Chapter 7 of F. Buekenhout (ed.),Handbook of Incidence Geometry, North-Holland, Amsterdam.

  27. Tits, J.: Ovoides et groupes de Suzuki,Arch. Math. 13 (1962), 187–198.

    Google Scholar 

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Authors and Affiliations

  1. Dept of Pure Mathematics and Computer Algebra, University of Gent, Krijgslaan 281, B-9000, Gent, Belgium

    J. A. Thas

  2. Dept of Mathematics, Campus Box 170, CU-Denver, PO Box 173364, 80217-3364, Denver, CO, U.S.A.

    S. E. Payne

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  1. J. A. Thas
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Thas, J.A., Payne, S.E. Spreads and ovoids in finite generalized quadrangles. Geom Dedicata 52, 227–253 (1994). https://doi.org/10.1007/BF01278475

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