A remark on symplectic spreads

We first note that each element of a symplectic spread of PG(2n-1,2^r) either intersects a suitable nonsingular quadric in a subspace of dimension n-2 or is contained in it, then we prove that this property characterises symplectic spreads of PG(2n-1,2^r). As an application, we show that a translation plane of order q^n , q even, with kernel containing GF(q), is defined by a symplectic spread if and only if it contains a maximal arc of the type constructed by J.A. Thas.