Ovoidal packings of PG(3,q) for even q

We show that any set of n pairwise disjoint ovals in a finite projective plane of even order n has a unique common tangent. As a consequence, any set of q+1 pairwise disjoint ovoids in PG(3,q), q even, has exactly q2+1 common tangent lines, constituting a regular spread. Also, if q−1 ovoids in PG(3,q) intersect pairwise exactly in two given points x≠y and share two tangent planes πx,πy at these two points, then these ovoids share exactly (q+1)2 common tangent lines, and they consist of the transversals to the pair xy, πx∩πy of skew lines. There is a similar (but more complicated) result for the common tangent lines to q ovoids in PG(3,q) which are mutually tangent at a common point and share a common tangent plane through this point. It is also shown that the common tangent lines to any pair of disjoint ovoids of PG(3,q), q even, form a regular spread.