Definitions: Cone, Flock, Blocking Set, Linear Flock


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 .

Pf: Since q is at least two, there are at least two planes in any flock of any cone. Suppose that S is a blocking set in , and that and are any two distinct planes of PG(3,q) which do not contain V. If S1 is the intersection of with C(V,S), then by lemma 3.1, S1 is a blocking set in . The intersection of and is a line l in , and so, contains a point of S1. Thus, any two planes not containing V must intersect on the cone C(V,S), and so, C(V,S) admits no flock.

Suppose now that S is not a blocking set. Then there is a line l in which contains no point of S. The set of q planes through the line l which do not contain the point V form a flock of C(V,S) (a linear flock ).


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