Definitions: Flock, Linear Flock, Star Flock, Flat Cone, Critical Cone


Proposition 9.1: The flocks of all non-flat cones in PG(3,3) that have flocks are linear. The flocks of any cone are star flocks.

Pf: There are only two planes other than the carrier plane in a flock of PG(3,3). Thus, there are only two primary baselines. These may coincide, in which case the flock is linear, or not, in which case their point of intersection is in all three planes and the flock is a proper star flock. Thus, any flock of any cone admitting a flock is a star flock.

Now, suppose that the two primary baselines are distinct (i.e., the flock is a proper star flock). By Proposition 5.2 there is a secondary baseline through the point of intersection. These three baselines form the entire baseline configuration. The complement of the union of the baselines thus consists of the points on the remaining line through this point of intersection (not including that point). Thus, the critical cone is flat. So, a non-flat cone can not admit a proper star flock.


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