A Geometric Relationship Between Equivalent Spreads

By Andrè theory, it is well known how to algebraically convert a spread in a projective space to an equivalent spread (representing the same translation plane) in a projective space of different dimension and of different order (corresponding to a subfield of the kernel). The goal of this paper is to establish a geometric connection between any two such equivalent spreads by embedding them in a subspace and a subgeometry of an ambient projective space. The connection can be viewed as a generalization of a construction due to Hirschfeld and Thas.