History

Peter Dembowski, in his influential book Finite Geometries [1968], used the word flock to describe a partition of all but two points of an inversive plane (Möbius plane) into circles. The concept was naturally extended to other circle geometries (Minkowski planes, Laguerre planes.)

The term today is used in reference to the various quadratic sets of PG(3,q); ovoids, hyperbolic quadrics and cones. Except for the minor differences due to the different sizes of these sets, the concept can be uniformly applied.

In the case of an ovoid, O, of PG(3,q), it has been shown that all flocks are linear. This was done in the odd case by W. F. Orr [Or73] and in the even case by J. A. Thas [Th72].

For hyperbolic quadrics in the even case, J. A. Thas [Th75] has shown that all flocks are linear. For the odd case, Thas [Th75] provided a construction of non-linear flocks, and in [Th87], he conjectured that this was the only possible construction. L. Bader [Ba88] found three examples related to irregular nearfields that were not of this type (also found by H. Gevaert and N.L. Johnson, and one by R.D. Baker and G.L. Ebert). Thas [Th??] then showed that there were no others if q = 1 mod 4, and L. Bader and G. Lunardon [BL89] finished the problem by showing that no others exist in the remaining case (using translation planes).

This leaves only the classification of flocks of cones. This situation is more complex than the others in that there are many different types of flocks. Even specializing to the case where the cone is a quadratic cone, we are far from any type of classification theorem.

The interest in flocks of cones stems from the interconnections between flocks of quadratic cones and other geometrical objects. In particular certain, translation planes, generalized quadrangles, ovals and hyperovals, can be constructed from flocks of quadratic cones. The relationship between flocks of quadratic cones and translation planes (via the Klein quadric) was discovered independently by Thas [Th8?] and Walker [Wa76]. The connection between flocks of quadratic cones and elation generalized quadrangles was also discovered by Thas [Th87]. In the even characteristic case, a connection exists between flocks of a quadratic cone in PG(3,2^e) and hyperovals in PG(2,2^e). One direction of this was observed by Payne [Pa89] and the converse was expounded upon in Cherowitzo, Penttila, Pinneri and Royle [CPPR96].

Only in the case of hyperovals in planes of even characteristic has the flock connection been expanded to cones of a more general nature (Cherowitzo [Ch98a]). Analogs to these other connections have yet to be found, if they exist.


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