Lifting Subregular Spreads

Abstract.

Let S _k denote a set of k reguli in a Desarguesian affine plane $\sum_{q^2}$ of order q ². It is shown that, for every odd integer s > 1, there is a corresponding set S ^s _k of k reguli in any Desarguesian plane $\sum_{q^{2s}}$ of order q ^2s such the line intersection properties of the reguli of S ^s _k are inherited from those of S _k . Hence, sets of mutually disjoint reguli in $\sum_{q^2}$ ‘lift’ to sets of mutually disjoint reguli in $\sum_{q^{2s}}$. Thus, the existence of a subregular spread in PG(3, q) produces an infinite class of subregular spreads in spaces PG(3, q ^s).