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Flocks of Arbitrary Cones

Let S be an arbitrary set of points in a projective plane embedded in PG(3,q). Let V be a point of PG(3,q)\. The cone C = C(V,S), with vertex V and carrier S is the union of the points on the lines joining V to the points of S. is called a carrier plane of the cone C.

(Note that we are permitting S to be the empty set which would give rise to the empty cone, consisting of just the point V. This is done so that we can develop a consistent terminology at very little cost.)

A flock F of cone C is a set of q planes which do not contain V such that no two planes of F meet at a point of C. The intersections of the the planes of the flock with the cone partition the points of the cone except for the vertex V.

Note that this definition of flock differs from that found in the majority of the references. The more common definition of a flock refers to the partition of the cone. However, under either definition, the term "planes of the flock" refers to the same set of q planes.

Note also that if the cone C is the empty cone, then every set of q planes not containing V is a flock of C.


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