![Using codes with sms plus gx](https://loka.nahovitsyn.com/162.jpg)
Īn example of a BLT-set is the linear one and is constructed as follows. Hence meets two lines of, that is, every line of meets 0 or 2 lines of. Thus we have obtained lines incident with, contradicting the fact that each point of lies on exactly lines. Since no line of meets more than 2 lines of, it follows that this gives lines incident with. Considering each of the lines of we obtain points collinear with. Then for each line of, the GQ property implies that for each point on, is collinear with a unique point of and in particular there is a unique point of collinear with. Let be the point of intersection and suppose that does not meet any of the remaining lines of. Now let be a line of which meets some line of. BLT-sets are named after Laura Bader, Guglielmo Lunardon and Jef Thas who first studied them in 1990. Such a flock is called a linear flock.Ī BLT-set of lines of is a set of disjoint lines such that no line of meets more than two lines of. Morever, since the intersection of two planes through is, the conics we obtain are all disjoint and so we get a flock. Each of the remaining planes containing meets each of the lines which make up and hence meets in points. Then is contained in planes, one of which contains. Let be a line of which intersects trivially. Each conic is the intersection of with a plane.
![four flocks four flocks](https://i.ytimg.com/vi/79_n94dgUnw/maxresdefault.jpg)
![four flocks four flocks](https://img.yumpu.com/9139273/1/716x1122/shepherd-of-peace-inspires-flock-canton-public-library.jpg)
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.Ī flock of a quadratic cone with vertex is a partition of into disjoint conics. Note that for any plane of the set of zeros of on forms a conic. Thus all quadratic cones of are equivalent.Īn easy way to construct a quadratic cone is to take the zeros of the degenerate quadratic form, where. 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. The set is called a quadratic cone with vertex. Let be the set of all points on these lines. For each of the points of there is a unique line through such a point and. Embed as a hyperplane in and take a point not on. Recall that a conic is the set of zeros of a nondegenerate quadratic form on. I have relied heavily on Maska Law’s PhD thesis which is available from Ghent’s PhD theses in finite geometry page. 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.
![four flocks four flocks](https://i.ytimg.com/vi/BzIu3O7ziDk/maxresdefault.jpg)
In this post I wish to discuss flock generalised quadrangles.
![Using codes with sms plus gx](https://loka.nahovitsyn.com/162.jpg)