ABSTRACT

In this chapter, we discuss another geometry associated with flocks of quadratic cones. Some of the material is taken from Johnson [769] in abbreviated form.

There is tremendous interest in what might be called the geometry of flocks of quadratic cones in PG(3, q). This geometry includes certain translation planes whose corresponding spreads are unions of q reguli sharing a common line (see Gevaert, Johnson, and Thas [397]), certain generalized quadrangles of type (q2, q) (see Thas [1177]), and translation planes with spreads in PG(3, q) admitting Baer groups of order q (see Johnson [726] and Payne and Thas [1094]). In the last situation, there is a deficiency-one partial flock of a quadratic cone due to the work of Johnson [726]. Furthermore, partial flocks of deficiency one may be extended uniquely to flocks by Payne and Thas [1094]. There are also connections to sets of ovals, called ‘herds’ (see, e.g., [1094]), the existence of which provides a more general extension theory when q is even (see Storme and Thas [1165]). The reader interested in these and other connections is referred to the survey article by Johnson and Payne [792].