In this post I wish to discuss flock generalised quadrangles. As mentioned in the first of this series, John has already discussed these a bit in a previous post so my main aim will be to flesh that out and provide more background. I have relied heavily on Maska Law’s PhD thesis which is available from Ghent’s PhD theses in finite geometry page.
Flocks of quadratic cones
Recall that a conic is the set of zeros of a nondegenerate quadratic form on . Embed as a hyperplane in and take a point not on . For each of the points of there is a unique line through such a point and . Let be the set of all points on these lines. The set is called a quadratic cone with vertex . Now acts transitively on the set of pairs of points and hyperplanes of where does not contain , and the stabiliser of such a pair induces on and so acts transitively on the set of conics contained in . Thus all quadratic cones of are equivalent.
An easy way to construct a quadratic cone is to take the zeros of the degenerate quadratic form , where . Here we take to be and . The zeros of on form the conic . Note that for any plane of the set of zeros of on forms a conic. This is all reminiscent of the classical case of a cone in , where the intersections of a plane with the cone are the conic sections and are either a point, a circle, an ellipse, a parabola or a hyperbola.
A flock of a quadratic cone with vertex is a partition of into disjoint conics. Each conic is the intersection of with a plane. Let be a line of which intersects trivially. Then is contained in planes, one of which contains . Each of the remaining planes containing meets each of the lines which make up and hence meets in points. Such a set of points is a conic. Morever, since the intersection of two planes through is , the conics we obtain are all disjoint and so we get a flock. Such a flock is called a linear flock.