Definitions: Coordinate Functions, Flock, Linear Flock


Theorem 6.1: A flock is a linear flock if and only if the three coordinate functions are scalar multiples of each other.

Pf: Let F be a flock of the cone C(V, S) whose coordinate functions are scalar multiples of each other. There exist constants a, b and c (not all zero), and a function k, such that f(t) = ak(t), g(t) = bk(t) and h(t) = ck(t). Notice that k(t) = 0 if and only if t = 0, otherwise F would not consist of q distinct planes. Any plane, of F, with t not 0, meets in the line ak(t)X + bk(t)Y + ck(t)Z = 0, i.e., aX + bY + cZ = 0. Thus, the flock is linear.

On the other hand, if F is a linear flock, the common line of intersection lies in and has an equation of the form aX + bY + cZ = 0 with not all of the coefficients being 0. Given any permutation p(t) of GF(q), with p(0) = 0, the planes which contain this line and do not pass through V = (0,0,0,1), have equations of the form ap(t)X + bp(t)Y + cp(t)Z + W = 0. The coordinate functions of this flock are therefore given by f(t) = ap(t), g(t) = bp(t) and h(t) = cp(t).


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