Definitions: Flock, Projectively Equivalent, Herd Equivalent, C-Equivalent,Critical Cone, Strongly Equivalent

Proposition 10.1: Let F be a flock, whose planes have the equations f(t)X + g(t)Y + h(t)Z + W = 0, of the cone C = C(V,S). Then, for any fixed non-zero d in GF(q), the set of q planes, F', whose equations are df(t)X + dg(t)Y + dh(t)Z + W = 0, is a flock of the same cone, strongly, properly C-equivalent to F.

Pf: Consider the element of PGL(4,q) whose action on the planes of PG(3,q) is given by the diagonal matrix, diag[d,d,d,1]. Clearly, this element maps the q planes of F to those of F'. The action of this element on the points of PG(3,q) is given by the diagonal matrix, diag[d-1, d-1, d-1, 1], and so, it fixes the vertex V = (0,0,0,1). These flocks are therefore projectively equivalent. Furthermore, this element stabilizes the plane W=0 by fixing every point in the plane, in particular the points of S. The cone, C(V,S) is therefore invariant under the mapping, and the two flocks are strongly C-equivalent. Finally, we need to show that the critical cone is preserved. At a point (a,b,c,0) of the critical cone of F, we have that af(t) + bg(t) + ch(t) is a permutation. The function associated to this point in the herd space of F' is adf(t) + bdg(t) + cdh(t) = d(af(t) + bg(t) + ch(t)) and is a permutation iff d0. Since our mapping fixes the points (a,b,c,0) we see that it stabilizes the critical cone, and so, F and F' are herd equivalent.