Bilinear flocks

A flock in PG(3, q) is a set of q planes which do not contain the vertex of a cone and have the property that the intersections of the planes of the flock with the cone partition the points of the cone except for the vertex. In this paper, we examine flocks, called bilinear flocks, where the planes of the flock pass through at least one of two distinct lines, called supporting lines in PG(3, q). We classify and provide examples of cones that admit bilinear flocks whose supporting lines intersect in PG(3, q). We also examine bilinear flocks whose supporting lines are skew, providing an example and also showing that this situation can not occur under certain conditions.