Kantor-Payne Type Flocks

These flocks were originally defined algebraically for quadratic cones by Kantor ([Ka86]) in odd characteristic and by Payne ([Pa85]) in even characteristic. The algebraic description can be given uniformly by F = F(t,t3,t5)(in the original description in odd characteristic one would have to use a different quadratic cone to obtain the flock in this form). We will attempt in this section to provide a purely geometric description of the flocks and ultimately use that as the definition of this family.

We shall work exclusively in the dual setting and our final results will be framed in this setting. Notationally, we shall drop the use of primes that were used in earlier sections dealing with the dual setting. We will start with a careful examination of the flock F = F(t,t3,t5) and abstract from it the geometric properties needed to insure that F is a flock of a thick cone.

We first observe that the points of DF = {(t,t3,t5,1)| t in GF(q)} lie on the quadratic cone C: {(x,y,z,w)| y2 = xz} with vertex (0,0,0,1). The second observation to make is that these points also lie on the cubic surface K: {(x,y,z,w)| x2y = zw2}. The complete intersection of C and K consists of the points of DF, the line of intersection, m, of the planes Y = 0 and Z = 0 (which is a generator of C), and the point A = (0,0,1,0) in W = 0. Note that the vertex of the cone, V = (0,0,0,1) is in K and lies on the line m.

There are several lines that are contained in the cubic surface K, besides m. First observe that the intersection of K with the plane W=0 consists of the lines X=0 and Y=0 in that plane (this is actually three lines since X=0 is counted twice). Let n denote first (X=0) of these lines and r the second. Every other line contained in K must intersect this plane at a point on one of these lines. The lines m and n play special roles.

Proposition 11.5.1 : Each point of K not on lines m or n is on a unique line of K which joins a point of m with a point of n. There are q+1 such cross lines. Each point of m is on a unique cross line. In odd characteristic, each point of n, other than (0,1,0,0) and (0,0,1,0), is on 0 or 2 cross lines and those two points are on unique cross lines. In even characteristic, each point of n is on a unique cross line and the cross lines are mutually skew.

The cubic surface K

Since K has different properties depending on the characteristic of the underlying field, we will consider each case separately.

Odd Characteristic

The mapping (x,y,z,w)mapsto(-x,-y,-z,w) is an involutory homology of PG(3,q) which fixes each point of W=0 and the point V. Each line through V is stabilized, and so the cone C is stabilized. This map also stabilizes K and therefore it stabilizes the intersection of C and K. Since m is stabilized and A and V are fixed by this mapping, we see that the points of DF are mapped to themselves by this involution.

Consider a line joining V to another point, P, of DF. Under the mapping, P is sent to another point P' of DF on this line. If the line VPP' contained another point of DF it would have 4 points of K on it and therefore be a line of K since K is a cubic surface. However, besides m there is only one line of K through V, and this is easily seen to be the line of intersection of the planes X=0 and Z=0, which contains no point of DF other than V. Therefore, lines joining V to a point of DF contain exactly 3 points of DF. Any line joining two points of DF which does not pass through V can have no further point of DF since the points of DF lie on a cone which is a quadratic set.

Consider a plane determined by 3 non-collinear points of DF. If V is one of the points, then the plane contains the two lines through V determined by the other two points. This plane intersects C only in these two lines, and so, contains exactly 5 points of DF on two intersecting lines. If V is not in the plane, then the plane intersects C in a conic and all the DF points must lie on this conic. The plane intersects K in a cubic curve since K is a cubic surface. By Bezout's theorem, the maximum size of the intersection of a quadratic and a cubic curve in a plane, which have no common component, is 6 (counted with multiplicities). Since a conic has a single component, in order to have a common component, the cubic curve would have to contain the entire component which would require that at least q-1 points of DF are coplanar. But then, since q is at least 3, a point P and its image P' under the mapping would be in this plane and thus V, a contradiction. [Note, it is easy to show in the current setting that this is impossible algebraically, however we are looking ahead and trying to avoid algebraic arguments when possible] Thus, since there is no common component, the maximum intersection is 6. However, this plane must meet the line m in a single point (if it had two points of m it would also contain V), and this point is in the intersection of C and K, but not a point of DF. Therefore, any plane can meet DF in at most 5 points (which lie on a conic if V is not in the plane).

Even Characteristic

In even characteristic, the mapping used for the odd characteristic case is just the identity, and no homology with center V exists which stabilizes K other than the identity.