The twisted cubic in PG(3,q) and translation spreads in H(q)

Using the connection between translation spreads of the classical generalized hexagon H(q) and the twisted cubic of PG(3,q), established in [European J. Combin. 23 (2002) 367-376], we prove that if qn≡1(mod3), q odd, q⩾4n2-8n+2 and n>2, then H(qn) does not admit an Fq-translation spread.