Equivalence of Flocks

We will find it useful to introduce various notions of equivalence in discussing general flocks.

Let F1 and F2 be two flocks, each containing the plane W = 0, we will say that F1 and F2 are projectively equivalent if there exists a collineation of PG(3,q) which fixes the point V and maps the planes of F1 to those of F2. If F1 and F2 are both flocks of the same cone, C = C(V,S), we will say that they are C-equivalent if they are projectively equivalent under a collineation which stabilizes the cone C. In the special case that F1 and F2 have the same critical cone C, when they are C-equivalent we shall say that they are herd equivalent. Alternatively, two flocks are herd equivalent if and only if they have the same set of baselines (although the sets of primary and secondary baselines need not be preserved). Note that if C' is a proper subcone of a critical cone C, then two flocks can be herd equivalent without being C'-equivalent and vice versa. If flocks F1 and F2 are C'-equivalent and also herd equivalent we say that they are properly C'-equivalent.

If, given two projectively equivalent flocks F1 and F2, there is a collineation mapping F1 to F2 which stabilizes the plane W = 0, we will say that F1 and F2 are strongly equivalent, otherwise they are weakly equivalent.

In the classical quadratic cone case, "equivalence" has meant what we are now calling C- equivalence, where C is a quadratic cone.

To solidify these ideas, consider the proof of:

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.

Since the herd space of a flock F is completely determined by F, we may refer to two herd spaces as being projectively equivalent iff their defining flocks are projectively equivalent. Similarly, we may refer to two herd spaces as being herd equivalent if their corresponding flocks are herd equivalent. Herd equivalence is a very strict form of equivalence. For instance, while it is clear that any two linear flocks have projectively equivalent herd spaces, these two herd spaces are herd equivalent if and only if the common line of the two flocks (and hence the entire flock) is the same.