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

Abstract 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 P G ( 3 , q ) , q even, has exactly q 2 + 1 common tangent lines, constituting a regular spread. Also, if q − 1 ovoids in P G ( 3 , q ) intersect pairwise exactly in two given points x ≠ y and share two tangent planes π x , π 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 x y , π x ∩ π y of skew lines. There is a similar (but more complicated) result for the common tangent lines to q ovoids in P G ( 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 P G ( 3 , q ) , q even, form a regular spread.