NESTS OF SIZE q-\ AND ANOTHER FAMILY OF TRANSLATION PLANES

We define a nest of reguli to be a collection P of reguli in a regular spread S of PG(3, q) such that every line of S is contained in exactly zero or two reguli of P. This generalizes the notion of a chain of reguli first introduced by Bruen. If U denotes the lines of S contained in the reguli of P and V is a partial spread of PG(3, q) covering the same points as U, then (S\U) U Vis a spread of PG(3, q) yielding a (potentially new) translation plane which is 2-dimensional over its kernel. In this paper we construct replaceable nests of size q — 1 for each odd prime power, q, thereby generating an (apparently new) infinite family of translation planes of order q. The geometric properties as well as collineation groups of the corresponding spreads are studied in some detail. In addition a very interesting relationship between these nests and the known chains of reguli is discovered. This relationship could lead to the construction of more chains and hence more translation planes.