Partitioning the complement of a simplex in PG(e,qd+1) into copies of PG(d,q)

In some projective spaces the complement of a simplex can be partitioned into disjoint copies of a higher or lower dimensional projective space of a different order. More precisely, let d and e be positive integers with e 2. We exhibit an embedding of PG(d,q) in PG(e,q^d+1) and show that the complement of a simplex in PG(e,q^d+1) may be partitioned into disjoint copies of embedded PG(d,q)'s, Each embedded PG(d,q) spans PG(e,q^d+1) whenever d e. These results are also true for PG(d,F) and PG(e,K) for infinite fields for which deg_F K=d+1 and the field extension is normal, separable, and cyclic.This research was supported in part by Grant Number A8027, Natural Sciences and Engineering Research Council of Canada.