Cones having no Flocks

A blocking set (for lines) in a projective plane is a set of points, B, such that every line of the plane contains a point of B. A blocking set is said to be proper if it contains no line. [More reference: blocking sets]

Theorem 3.1: A cone C(V,S) in PG(3,q) has no flock if and only if S is a blocking set in its carrier plane .

The theorem follows immediately from the following lemma and the fact that if there is a line of which does not intersect S, then C(V,S) admits a linear flock based on this line.

Lemma 3.1: S is a blocking set in if and only if the intersection of any plane not containing V with the cone C(V,S) is a blocking set in that plane.

Thus, as simple examples of cones without flocks, we have cones over Baer subplanes, cones over unitals and cones over any set containing a line.