## 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.

**Sections**