ABSTRACT
This chapter discusses generalizations of subgeometry partitions of projective
is called ‘spread retraction’ and geometric lifting. We recall that in 1976, A.A. Bruen and J.A. Thas [198] showed that it was possible to find partitions of the points of PG(2s− 1, q2) by projective subgeometries isomorphic to PG(s− 1, q2)’s and PG(2s− 1, q)’s. These might be called ‘mixed subgeometry partitions’. Bruen and Thas showed that there is an associated translation plane of order q2s and kernel containing GF (q). There is another construction using Segre varieties given in Hirschfeld and Thas [529] which generalizes and includes Baer subgeometry partitions of PG(2s, q2) by PG(2s, q)’s. In this latter case, there is an associated translation plane of order q2s+1 and kernel containing GF (q).
