Spreads Covered by Pseudo-Reguli

In this chapter, we consider spreads in PG(3; K) that are covered by pseudo-reguli that share a given line, which we call ‘conical spreads’in PG(3; K); and we consider spreads which are covered by pseudo-reguli that share two given lines, which we call ‘ruled spreads’and formulate the corresponding theory. In the following chapter, we consider conical and hyperbolic ‡ocks that correspond to spreads that are unions of reguli, sharing either one or two common components. However, there are translation planes whose spreads are unions of derivable nets that are not given by reguli over the …eld or projective space in question. This is true both for the …nite and in…nite cases. Thus, we include here a study of a more general situation than encountered in spreads corresponding to ‡ocks. Although it seems natural enough to consider this study in the con-

text of derivable nets, this material originated not with derivable nets but with the consideration of ‡ocks of quadric sets. In the next chapter, we sketch part of the theory interconnecting coverings of quadrics by planes to the analysis of spreads covered by reguli. However, we choose to work from the general to the speci…c in this instance. This and the following chapters on ‡ocks of hyperbolic and quadratic cones are also in the author’s Subplanes text with a few changes in form. We include these chapters here for convenience and completeness.