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 q(2) + 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 not equal y and share two tangent planes pi(x), pi(y) 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, pi(x) boolean AND pi(y) of skew lines. There is a similar (but more complicated) result for the common tangent lines to g 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. (C) 2013 Elsevier B.V. All rights reserved.