Ovoidal packings of P G ( 3 , q ) for even q

Bhaskar Bagchi, N.S. Narasimha Sastry
2013 Discrete Mathematics  
We show that any set of n pairwise disjoint ovals in a finite projective plane of even order 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
more » ... onsist of the transversals to the pair xy, 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.
doi:10.1016/j.disc.2013.05.021 fatcat:nl7lqgllpneyxmr4g3wzwzotmu