ABSTRACT
There is, of course, another related area of interest, partial and maximal partial spreads, of which there is a voluminous amount of constructions and theory. For example, the text by Beth, Jungnickel and Lenz [111] gives a nice presentation of many of the maximal partial spreads. We do not attempt any sort of treatment of this area other than to point out some of the nice consequences of ideas of Bruen [190]. Take any spread in PG(3, q), form the associated translation plane and choose any 2-dimensional GF (q)-subspace pio which is not a component (line). Then the net of degree q + 1 and order q2 defined by the subspace pio may or may not be a regulus net (corresponds to a regulus in PG(3, q)). If it is not a regulus, then it turns out that it is either 1 or 2 Baer subplanes of this net that can arise from 2-dimensional GF (q)-subspaces. Define a new translation net by taking the Baer subplanes of a non-regulus net constructed in the above manner together with the components of the spread not in the partial spread defining the net. This will be forced to be a maximal partial spread of either q2−q+1 or q2−q+2 subspaces. The way to see this is to note that any other 2-dimensional subspace will be forced to define a Baer subplane of the net in question. However, this will then force the net to be a regulus net. Jungnickel [815] calls such maximal partial spreads ‘of small deficiency’ and there is a corresponding theory of such structures.
