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Coordinates and Normalizations

Let C = C(V, S) be a cone in PG(3,q) with a flock F. We shall assume that one of the planes of F, designated , is chosen as the carrier plane of the cone (i.e., S is contained in ). We can introduce projective coordinates, (x, y, z, w), in PG(3,q) so that V has coordinates (0, 0, 0, 1) and the plane has equation W = 0. Furthermore, if C is a non-flat cone, we can, without loss of generality, assume that S contains the points A = (1, 0, 0, 0), B = (0, 1, 0, 0) and C = (0, 0, 1, 0). [Should the need arise, if C is the very degenerate cone over a single point, we may assume that that point is B, and, for other flat cones we will assume that S contains A and B.]

To each plane of the flock F, we associate a unique element of the field GF(q), subject only to the restriction that 0 is associated to the plane . With this indexing of the planes of F, and since the planes do not pass through V, there exist functions f, g and h : GF(q)mapstoGF(q), so that the planes with t running through GF(q), have equations:

: f(t)X + g(t)Y + h(t)Z + W = 0.
These three functions will be called the coordinate functions of the flock.

The only condition that we have, so far, imposed on the coordinate functions is that f(0) = g(0) = h(0) = 0. We shall use the notation F = F(f,g,h) to denote the flock F with coordinate functions f, g and h.

In several references, the coordinate functions are used to define a set of q 2 × 2 upper triangular matrices

.

This set of matrices is called the clan associated to the flock F. [In the literature you will find references to q-clans, 2-clans, -clans and -clans. These modifiers reflect the type of cone involved.] Under certain restrictions, in particular, in the quadratic cone setting, the clan has algebraic properties which have been exploited in the study of their flocks. In the general setting, however, the clan is merely a notational device, and we shall, in general, avoid its use until we have specialized the discussion.

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


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