Definitions: Flock, Herd Space, Linear Flock

Proposition 8.1.4: The herd space of a flock F contains at least two constant functions, if and only if, F is a linear flock. In this case, the herd space contains exactly q+1 constant functions and all the other functions are scalar multiples of the same permutation. Furthermore, in the affine plane obtained by removing the line Z = 0 from the plane W = 0, all the herd space functions that are equal to each other are associated to the points which lie on a line of the parallel class containing the common line of the flock.

Pf: If the herd space of F contains the constant functions fP and fQ with PQ, then every plane of F contains both P and Q, and hence, the line joining them. Thus, F is a linear flock.

Conversely, the herd space function associated with each point of the common line of the linear flock is the constant 0 function.

There can be no other constant functions in the herd space of a linear flock besides the q+1 that are associated to the points of the common line, since otherwise there would exist three non-collinear points on all planes of the flock, and the "flock" would consist of just one plane. The carrier of the critical cone of a linear flock is the affine plane which is the complement of the common line in W = 0. Thus, the points associated to the herd of a linear flock are precisely the points of this affine plane, and so, all the herd space functions at these points are permutations. By Theorem 6.1 the planes of a linear flock have equations of the form, ak(t)X + bk(t)Y + ck(t)Z + W = 0, for some permutation k and scalars a, b and c (not all 0). Let P = (A,B,C,0) be any point of W = 0 not on the common line. Then the herd space function at P, fP(t) = aAk(t) + bBk(t) + cCk(t) = (aA + bB + cC)k(t), is a nonzero multiple of k. In the affine plane obtained by removing the line Z = 0 from the plane W = 0 (provided this line is not the common line of the flock, which is the case for all non-empty cones by our normalizations), the points (A,B,C,0) for which aA + bB + cC is a constant, lie on a line parallel to the common line of the flock.