Generalized quadrangles and flocks of cones
A flock of the quadratic cone K of PG(3,q) is a partition of K but its vertex into disjoint conics. It is called linear if the planes of the q conics of such a flock all contain a common line. A flock is linear if and only if there corresponds a Desarguesian translation plane to it. W. M. Kantor has proved that with a set of q upper triangular \(2\times 2\) matrices over GF(q) of a certain type, there corresponds a generalized quadrangle of order \((q^ 2,q)\). In this paper, the author proves that with such a set of q matrices, there corresponds a flock of the quadratic cone of PG(3,q), and conversely with each flock of the quadratic cone there corresponds such a set of matrices. Using this relationship, new flocks and new generalized quadrangles are obtained. Some interesting open problems are also noted.
scientific article; zbMATH DE number 4054454
scientific article