Lifting Subregular Spreads

Abstract

Let S k denote a set of k reguli in a Desarguesian affine plane Σ 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 Σ 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 Σ q 2 'lift' to sets of mutually disjoint reguli in Σ q 2s . Thus, the existence of a subregular spread in P G(3, q) produces an infinite class of subregular spreads in spaces P G(3, q s ).

A New Class of Translation Planes Constructed by Hyper-regulus Replacement

“…When n = 2, there are a variety of known planes that may be obtained from a Desarguesian affine plane by the replacement of a set of mutually disjoint reguli; the so-called 'subregular planes'. The interested reader is directed to Johnson [5] for a survey of the known subregular planes.…”

Translation planes constructed by multiple hyper-regulus replacement

“…There are a number of important translation planes that may be constructed in this manner, most of which are discussed in Johnson [12].…”