Generalised quadrangles IV: flock quadrangles

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 {\mathcal{C}} is the set of zeros of a nondegenerate quadratic form on {\mathrm{PG}(2,q)}. Embed {\mathrm{PG}(2,q)} as a hyperplane {\pi} in {\mathrm{PG}(3,q)} and take a point {P} not on {\pi}. For each of the {q+1} points of {\mathcal{C}} there is a unique line through such a point and {P}. Let {\mathcal{K}} be the set of all {q^2+q+1} points on these {q+1} lines. The set {\mathcal{K}} is called a quadratic cone with vertex {P}. Now {\mathsf{P}\Gamma\mathsf{L}(4,q)} acts transitively on the set of pairs {(P,\pi)} of points {P} and hyperplanes {\pi} of {\mathrm{PG}(3,q)} where {\pi} does not contain {P}, and the stabiliser of such a pair induces {\mathrm{GL}(3,q)} on {\pi} and so acts transitively on the set of conics contained in {\pi} . Thus all quadratic cones of {\mathrm{PG}(3,q)} are equivalent.

An easy way to construct a quadratic cone is to take the zeros of the degenerate quadratic form {Q(x)=x_2^2-x_1x_3}, where {x=(x_1,x_2,x_3,x_4)}. Here we take {P} to be {\langle (0,0,0,1)\rangle} and {\pi=\langle (1,0,0,0),(0,1,0,0),(0,0,1,0)\rangle}. The zeros of {Q} on {\pi} form the conic {\{\langle (1,t,t^2,0)\rangle\mid t\in\mathsf{GF}(q)\}\cup \{\langle (0,0,1,0) \rangle\}}. Note that for any plane {\pi'} of {\mathsf{PG}(3,q)} the set of zeros of {Q} on {\pi'} forms a conic. This is all reminiscent of the classical case of a cone in {\mathbb{R}^3}, 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 {\mathcal{K}} with vertex {P} is a partition of {\mathcal{K}\backslash\{P\}} into {q} disjoint conics. Each conic is the intersection of {\mathcal{K}} with a plane. Let {\ell} be a line of {\mathrm{PG}(3,q)} which intersects {\mathcal{K}} trivially. Then {\ell} is contained in {q+1} planes, one of which contains {P}. Each of the remaining {q} planes containing {\ell} meets each of the lines which make up {\mathcal{K}} and hence meets {\mathcal{K}} in {q+1} points. Such a set of {q+1} points is a conic. Morever, since the intersection of two planes through {\ell} is {\ell}, the {q} conics we obtain are all disjoint and so we get a flock. Such a flock is called a linear flock.

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