Some remarks on the Spin Module Representation of Sp6(2e)Sp6(2e)

We study the representation of a symplectic spread SS of PG(5,2e)PG(5,2e) on the Grassmannian of the planes of PG(5,2e)PG(5,2e). We observe that such a representation defines an ovoid of Q+(7,2e)Q+(7,2e) which, in turn, via triality defines a spread of Q+(7,2e)Q+(7,2e) one slice of which is isomorphic to SS; this allows to explicitly compute the ovoid of Q+(7,2e)Q+(7,2e) without using a triality map. Also, we apply our arguments to show that a partial symplectic ovoid of PG(3,2e)PG(3,2e) defines a particular partial spread of Q+(7,2e)Q+(7,2e).