Abstract
We show that if an ovoid of Q (4,q),q even, admits a flock of conics then that flock must be linear. It follows that an ovoid of PG (3,q),q even, which admits a flock of conics must be an elliptic quadric. This latter result is used to give a characterisation of the classical example Q -(5,q) among the generalized quadrangles T 3(\(O\)), where \(O\) is an ovoid of PG (3q) and q is even, in terms of the geometric configuration of the centres of certain triads.
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References
A. Barlottti, Un'estensione del teorema di Segre-Kustaanheimo, Boll. Un. Mat. Ital. Vol. 10 (1955) pp. 96-98.
R. C. Bose and S. S. Shrikhande, Geometric and pseudo-geometric graphs (q 2 + 1, q + 1, 1), J. Geom., Vol. 2 (1972) pp. 75-94.
G. Fellegara, Gli ovaloidi di uno spazio tridimensionale di Galois di ordine 8, Atti. Accad. Naz. Lincei Rend., Vol. 32 (1962) pp. 170-176.
H. Gevaert, N. L. Johnson and J. A. Thas, Spreads covered by reguli, Simon Stevin, Vol. 62 (1988) pp. 51-62.
D. G. Glynn, The Hering classification for inversive planes of even order, Simon Stevin, Vol. 58 (1984) pp. 319-353.
J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Oxford University Press, Oxford (1991).
C. M. O'Keefe and T. Penttila, Ovoids of PG(3, 16) are elliptic quadrics, J. Geom., Vol. 38 (1990) pp. 95-106.
C. M. O'Keefe and T. Penttila, Ovoids of P G (3, 16) are elliptic quadrics, II, J. Geom., Vol. 44 (1992) pp. 140-159.
C. M. O'Keefe and T. Penttila, Elation generalized quadrangles of order (q 2, q), Geometry, Combinatorial Designs and Related Structures, Proceedings of the First Pythagorean Conference, (J. W. P. Hirschfeld, S. S. Magliveras, M. J. de Resmini, eds.) LMS Lecture Note Series 245, Cambridge University Press (1997) pp. 181-192.
C. M. O'Keefe, T. Penttila and G.F. Royle, Classification of ovoids in P G (3, 32), J. Geom., Vol. 50 (1994) pp. 143-150.
S. E. Payne and J. A. Thas, Finite Generalized Quadrangles, Pitman, London (1984).
T. Penttila and C. E. Praeger, Ovoids and translation ovals, Proc. Lon. Math. Soc., Vol. 56 (1997) pp. 607-624.
O. Prohaska and M. Walker, A note on the Hering type of inversive planes of even order, Arch. Math., Vol. 28 (1977) pp. 431-432.
B. Segre, On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two, Acta Arith., Vol. 5 (1959) pp. 315-332.
J. A. Thas, Flocks of finite egglike inversive planes, Finite Geometric structures and their applications. Cremonese, Rome (CIME, Bressanone 1972), (1973) pp. 189-191.
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Brown, M.R., O'Keefe, C.M. & Penttila, T. Triads, Flocks of Conics and Q -(5,q). Designs, Codes and Cryptography 18, 63–70 (1999). https://doi.org/10.1023/A:1008376900914
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DOI: https://doi.org/10.1023/A:1008376900914