For q = 32h*',h 2 0, we investigate the intersections of Hermitian and Ree ovoids of the generalized hexagon H ( q ) . 0 1996 John Wiley & Sons, h e .
1. Introduction
A finite generalized hexagon of order (s, t ) , s, t 2 1 is a 1 -(v, s + 1, t + 1) design S = (F', B,Z) whose incidence graph has girth 12 and diameter 6, also denoted by S(s, t ) . If s = t , S is said to have order s. Generalized hexagons (and more generally, generalized polygons) were introduced by Tits [12]. The only known (up to duality) finite generalized hexagons of order s > 1 arise from the Chevalley groups G2(q) and have order q, q power of a prime. They are due to Tits [12]. We denote the Ca(q)-hexagon An ovoid of a generalized hexagon of order s is a set of s3 + 1 points mutually at distance 6. A spread of a generalized hexagon of order s is defined dually. For example, consider the split-Cayley Moufang generalized hexagon H ( q ) embedded in the nonsingular quadric Q(6, q ) (see Tits [ 121). Let PG(5, q ) be a hyperplane of PG(6, q) such that PG(5, q ) n Q is an elliptic quadric Q-. Then the lines of H ( q ) on Q-constitute a spread of the generalized hexagon H ( q ) ([9]). Further 0 is an ovoid of H ( q ) if and only if 6 is an ovoid of the polar space Q(6, q ) ([lo]). So H ( q ) always has a spread. by H h ) .