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

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, 2r ) are elliptic quadrics, this forces that all translation spreads of H(2r ) 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 of H(2r ). Finally, we study the semi-classical locally Hermitian 1-systems of Q(6, q) which are spreads of Q(5, q).