Ovoids and spreads of the generalized hexagon H(3)

In this paper, we construct and classify all ovoids and spreads of the known generalized hexagon of order 3 (the split Cayley hexagon H(3)). We exhibit some unexpected and nice geometric properties. As an application, we provide an elementary and geometric construction of a G2(3)-GAB of type G˜2. We also exhibit new ovoid-spread pairings.