Abstract
A connection between locally hermitian 1-systems of Q(6, q) and a generalization of flocks of quadratic cones in PG(3, q) is given. This connection will be used to describe geometrically the semi-classical non-hermitian spread S [9] of the hexagon H(q). The construction yields several examples of locally hermitian 1-systems of Q(6, q) which are shown to be pairwise non-isomorphic. Finally, all semi-classical non-hermitian 1-systems of Q(6, q), q odd, are determined and classified.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. Bader, G. Lunardon and J.A. Thas, Derivation of flocks of quadratic cones, Forum Math. 2 (1990), 163–174.
I. Bloemen, J.A. Thas and H. Van Maldeghem, Translation ovoids of generalized quadrangles and hexagons, Geom. Dedicata 72 (1998), 19–62.
V. De Smet and H. Van Maldeghem, The finite Moufang hexagons coordinatized, Beiträge Algebra Geom. 34 (1993), 217–232.
J.C. Fisher and J.A. Thas, Flocks in PG(3, q), Math. Z. 169 (1979), 1–11.
H. Gevaert, N.L. Johnson and J.A. Thas, Spreads covered by reguli, Simon Stevin 62 (1988), 51–62.
J.W.P. Hirschfeld, Projective Geometries over Finite Fields, Second edition, Oxford University Press, Oxford, 1998.
J.W.P. Hirschfeld and J.A. Thas, General Galois Geometries, Oxford University Press, Oxford, 1991.
S.E. Payne and J.A. Thas, Conical flocks, partial flocks, derivation and generalized quadrangles, Geom. Dedicata 38 (1991), 229–243.
J.G. Semple and L. Roth, Introduction to Algebraic Geometry, Oxford University Press, Oxford, 1949.
E.E. Shult and J.A. Thas, m-systems of polar spaces, J. Combin. Theory Ser. A 68 (1994), 184–204.
J.A. Thas, Generalized quadrangles and flocks of cones, European J. Combin. 8 (1987), 441–452.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Kluwer Academic Publishers
About this chapter
Cite this chapter
Luyckx, D., Thas, J.A. (2001). Flocks and Locally Hermitian 1-Systems of Q(6, q). In: Blokhuis, A., Hirschfeld, J.W.P., Jungnickel, D., Thas, J.A. (eds) Finite Geometries. Developments in Mathematics, vol 3. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0283-4_15
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0283-4_15
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7977-5
Online ISBN: 978-1-4613-0283-4
eBook Packages: Springer Book Archive