Slices of the unitary spread

Abstract

We prove that slices of the unitary spread of Q + (7, q), q ≡ 2 (mod 3), can be partitioned into five disjoint classes. Slices belonging to different classes are non-equivalent under the action of the subgroup of P ΓO + (8, q) fixing the unitary spread. When q is even, there is a connection between spreads of Q + (7, q) and symplectic 2-spreads of P G(5, q) (see [7] and [8]). We determine all possible non-equivalent symplectic 2-spreads arising from the unitary spread of Q + (7, q), q = 2 2h+1 . Some of these already appeared in [14]. When q = 3 h we classify, up to the action of the stabilizer in P ΓO(7, q) of the unitary spread of Q(6, q), those among its slices producing spreads of the elliptic quadric Q − (5, q).