ABSTRACT
In this chapter, we consider the possibility of covering the conics of a quadric set. This material is loosely based on the article by Cherowitzo and the author [32] and modi…ed for this text. Normally, we would consider a quadric set in PG(3; K), for K a
…eld, as either an elliptic quadric, a hyperbolic quadric, or a quadratic cone. In this chapter, we consider also -cones as well and could further consider oval-cones or ovoids and ask in what sense one could consider a parallelism of such geometric objects. Naturally, there are some problems when trying to consider a gen-
eral de…nition of a parallelism. If G is an elliptic quadric, hyperbolic quadric, or quadratic cone, we would require the plane intersections to be conics. But, if G is an ovoid, which is not an elliptic quadric, the plane intersections would only be required to be ovals. If G is an oval-cone, the plane intersections would be also be ovals, but if G is an -cone, we would merely require the plane intersections not to contain a generating line (this would also work as a requirement for an oval-cone). There are some additional complications in the in…nite case for
ovoids, since a ‡ock could conceivably miss either 0, 1; or 2 points of G. If a parallelism is a set of ‡ocks, should we distinguish between ‡ocks that miss 0 points and those that miss 2 points within the set of ‡ocks of a parallelism? Initially, at least, we shall choose to ignore this potential problem. Also, we might wish to consider ‘maximal partial parallelisms’, and again we delay such discussions. We shall also be considering ‡ocks of Minkowski planes in a later chapter.
