Star Flocks

A star flock is a flock of a cone whose planes share a common point and a proper star flock is one for which this common point is unique. Linear flocks are clearly star flocks, but not proper star flocks.

We may characterize star flocks in terms of the coordinate functions in a manner analogous to that which was done for linear flocks in Theorem 6.1 [backbutton].

Theorem 7.1: A flock is a star flock if and only if the coordinate functions are linearly dependent over GF(q).

Viewing flocks in a dual setting has been an effective technique in the classical quadratic cone situation. It is especially useful in studying star flocks.

The Dual Setting

Let F be a flock of the cone C(V, S) in PG(3,q). By passing to the dual, F becomes a set of q points in PG(3,q) and the cone is the set of all planes passing through a set of lines in the plane corresponding to V, with the property that no line determined by a pair of the q points lies in any of these planes.

If F is a star flock, then the corresponding q points in the dual are coplanar.

We set up some notation for the dual setting. Let F be a flock of the cone C(V, S) with V = (0,0,0,1) and the planes of F given by f(t)X + g(t)Y + h(t)Z + W = 0. Using the standard duality, the flock becomes a set of points DF = {(f(t), g(t), h(t), 1) | t in GF(q)}. V becomes the plane with equation W' = 0. The points on a generator of C(V,S) become the set of all planes passing through a line in W' = 0. The set of lines in W' = 0, corresponding to the generators of C(V,S) will be denoted by DG, and the set of all planes passing through the lines of DG will be denoted by DC. S becomes the set of planes, denoted by DS, passing through the point (0,0,0,1) which intersect W' = 0 in a line of DG. The condition that lines formed by pairs of points of DF do not lie in the planes of DC is equivalent to the condition that these lines do not intersect the lines of DG.


Now, suppose F is a star flock of a thick cone with common point (a, b, c, 0), and coordinate functions f, g and h, where g(t) = t. Then DF is the set of points {(f(t),t,h(t),1)} lying in the plane with equation aX' + bY' + cZ' = 0. Notice that the point (0,0,0,1) is in DF. If the points of DF are collinear, then there are q other planes containing DF, each with an equation of the form a'X' + b'Y' + c'Z' = 0 (since they all contain the point (0,0,0,1)). The points (a',b',c',0) would all be common to all the planes of F, and so, F would be a linear flock. Thus, if F is a proper star flock, the points of DF are not all collinear in the plane with equation aX' + bY' + cZ' = 0.

We will return to the dual setting when we study star flocks in more detail. Star Flocks II