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).