Translation Ovoids of Generalized Quadrangles and Hexagnos

We define the notion of a translation ovoid in the classical generalized quadrangles and hexagons of order q, and we enumerate all known examples; translation spreads are defined dually. A modification of the known ovoids in the generalized hexagon H(q), q=3^2h+1, yields new ovoids of that hexagon. Dualizing and projecting along reguli, we obtain an alternative construction of the Roman ovoids due to Thas and Payne. Also, we construct a new translation spread in H(q) for any 1 mod 3, q odd, with the property that any projection along reguli yields the classical ovoid in the generalized quadrangle Q(4,q). Finally, we prove that for q odd, the new example is the only non-Hermitian translation spread in H(q) with the property that any projection along reguli yields the classical ovoid in Q(4,q).