Complete Spans on Hermitian Varieties

Let L be a general linear complex in PG(3, q) for any prime power q. We show that when G F(q) is extended to G F(q2), the extended lines of L cover a non-singular Hermitian surface H ∼= H(3, q2) of PG(3, q2). We prove that if S is any symplectic spread PG(3, q), then the extended lines of this spread form a complete (q2 +1)-span of H . Several other examples of complete spans of H for small values of q are also discussed. Finally, we discuss extensions to higher dimensions, showing in particular that a similar construction produces complete (q3 +1)-spans of the Hermitian variety H(5, q2).