Spreads in H(q) and 1-systems of Q(6,q)

Authors: Cardinali I.; Lunardon G.; Polverino O.; Trombetti R.

Source: European Journal of Combinatorics, Volume 23, Number 4, May 2002 , pp. 367-376(10)

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Abstract:

In this paper we prove that the projections along reguli of a translation spread of the classical generalized hexagon H(q) are translation ovoids of Q(4, q). As translation ovoids of Q(4, 2^r) are elliptic quadrics, this forces that all translation spreads ofH(2^r) are semi-classical. By representing H(q) as a coset geometry, we obtain a characterization of a translation spread in terms of a set of points of PG(3, q)which belong to imaginary chords of a twisted cubic and we construct a new example of a semi-classical spread ofH(2^r). Finally, we study the semi-classical locally Hermitian 1-systems ofQ(6, q) which are spreads of Q^-(5, q).Copyright 2002 Elsevier Science Ltd.