Abstract
In this paper, a method is developed to study locally hermitian 1-systems of Q(6, q), q even, by associating a kind of flock in PG(4, q) to them. This method is applied to a known locally hermitian 1-system of Q(6, 22e), which was discovered by Offer as a spread of the hexagon H(22e). The results concerning this spread appear to be suitable for generalization and enable us to find new classes of 1-systems of Q(6, q), q even. We also prove that a locally hermitian 1-system of Q(6, q), q even, which is not contained in a 5-dimensional subspace, is semi-classical if and only if it belongs to the new classes we describe. Finally, from the new classes of 1-systems arise new classes of semipartial geometries.
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References
I. Bloemen, J. A. Thas and H. Van Maldeghem, Translation ovoids of generalized quadrangles and hexagons, Geom. Dedicata, Vol. 72, No. 1 (1998) pp. 19–62.
I. Cardinali, G. Lunardon, O. Polverino and R. Trombetti, Spreads in H(q) and 1–systems of Q(6, q), European J. Combin., Vol. 23, No. 4 (2002) pp. 367–376.
N. Hamilton and R. Mathon, Existence and non-existence ofm-systems of polar spaces, European J. Combin., Vol. 22, No. 1 (2001) pp. 51–61.
J. W. P. Hirschfeld, Projective Geometries over Finite Fields, 2nd ed., The Clarendon Press, Oxford University Press, Oxford (1998).
A. F. Horadam, A Guide to Undergraduate Projective Geometry, Pergamon Press, Australia (1970).
D. R. Hughes and F. C. Piper, Projective Planes, Springer-Verlag, New York (1973); Graduate Texts in Mathematics, Vol. 6.
D. Luyckx, m-systems of polar spaces and SPG reguli, Bull. Belg. Math. Soc., Vol. 9, No. 2 (2002) pp. 177–183.
D. Luyckx and J. A. Thas, Flocks and locally hermitian 1–systems of Q(6, q). In Finite Geometries—Proceedings of the Fourth Isle of Thorns Conference, Developments in Mathematics, Vol. 3 (2001) pp. 257–275.
A. Offer, Spreads and Ovoids of the Split Cayley Hexagon, Ph.D. Thesis, University of Adelaide (2000), Second edition available at http://cage.rug.ac.be/~nick/Theses/theses.html.
A. Offer, Translation spreads of the Split CayleyHexagon, Advances in Geometry,Vol. 3 (2003) pp. 105–121.
J. G. Semple and L. Roth, Introduction to Algebraic Geometry, The Clarendon Press, Oxford University Press, New York (1985), reprint of the 1949 original.
E. E. Shult and J. A. Thas, m-Systems of Polar Spaces, J. Combin. Theory Ser. A, Vol. 68, No. 1 (1994) pp. 184–204.
H. Van Maldeghem, Generalized Polygons, Birkhäuser Verlag, Basel (1998).
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Luyckx, D., Thas, J.A. On 1-Systems of Q(6, q), q Even. Designs, Codes and Cryptography 29, 179–197 (2003). https://doi.org/10.1023/A:1024112710780
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DOI: https://doi.org/10.1023/A:1024112710780