Subgeometry and Quasi-subgeometry Partitions
Norman Johnson · 2010
A subgeometry partition of a nite projective space is simply a partition of the projective space by subgeometries isomorphic to nite projective spaces of various dimensions over various elds. The rst examples of subgeometry partitions were partitions of PG(2; q2) by subgeometries isomorphic to PG(2; q)s, the so-called Baer subgeometry partitions. These partitions originated in the text [73] by Hirschfeld and Thas. There is a so-called lifting process, which we term geometric lifting that produces from a Baer subgeometry partition a planar-spread that corresponds to a translation plane of order q3 with projective 2-spread in PG(5; q). The associated translation planes of order q3 and kernel containing GF (q) will often admit a collineation group that is ag-transitive. Regarding the group action on the line at in nity, a ag-transitive group is simply a group that is transitive on the line at in nity. In this case, and in the cases of most interest, this collineation group is cyclic of order q3 + 1. The subgroup of order q + 1 together with the kernel homology group of order q 1 together with the zero mapping generate a eld isomorphic to GF (q), which contains a cyclic subgroup of order q2 1 that acts on the translation plane with orbits of length q+ 1. The existence of a group of order q2 1 containing the kernel group of order q 1, such that by adjoining the zero map one obtains a eld R, over which the point set as a vector space is important to the theory. This retraction group is the key to the understanding of how to construct subgeometry partitions (see Johnson [113]). Basically, the orbits of length q + 1 retract to subgeometries isomorphic to PG(2; q) that cover the associated projective space (the 6-dimensional vector space over GF (q) becomes 3-dimensional over R and produces the subgeometries isomorphic to PG(2; q2)). The geometric lifting process generalizes to partitions of PG(3; q2) by PG(1; q2)s and PG(3; q)s, the so-called mixed partitions and, in this case, produces translation planes of order q4 with kernel containing GF (q) and again there is an associated group of order q2 1 containing the kernel homology group of order q 1, and there is an associated eld R, whose multiplicative group now has component orbits of length 1 or q+1. The orbits of length q+1 retract to subgeometries isomorphic to PG(3; q), and the orbits of length 1 retract to subgeometries isomorphic to PG(1; q2). The existence of the retraction group with eld isomorphic to GF (q2) was shown in Johnson [113]. Furthermore, in Jha and Johnson [84], it is shown that it is pos- sible to use algebraic liftingto produce from a translation plane of order q2 with spread in PG(3; q), a corresponding spread of order q4 in PG(3; q2), wherein lies a retraction group. Considered over the subkernel isomorphic to GF (q), it turns out that there is a retraction group of order q2 1 with associated eld isomorphic to GF (q2) that constructs an associated subgeometry partition of PG(3; q2) by PG(3; q)s and PG(1; q2)s: We shall use a variation of this process to construct new subgeometry partitions from focal-spreads.