Hyper-reguli and non-André quasi-subgeometry partitions of projective spaces (Q1430538)

Suppose that \(\mathcal A\neq \mathcal B\) are partial spreads in PG\((2ds-1,q)\), each of \(N=(q^{ds}-1)/(q-1)\) mutually skew \((ds-1)\)-dimensional (projective) subspaces. If \(| A\cap B |=1\), for all \(A\in\mathcal A\), \(B\in\mathcal B\) then \(\mathcal A\) is called a hyper-regulus in PG\((2d-1,q)\) [cf. \textit{T. G. Ostrom}, J. Geom. 48, No. 1--2, 157--166 (1993; Zbl 0792.51001)]. (When \(ds=2\) then \(\mathcal A\) and \(\mathcal B\) are a pair of mutually opposite reguli.) It is well known that André planes of order \(q^{ds}\) can be obtained by replacing a certain family of hyper-reguli in a regular spread \(\mathcal S\) of PG\((2ds-1,q)\) [see \textit{J. André}, Math. Z. 60, 156--186 (1954; Zbl 0056.38503)]. One of the main results of the present paper says that `non-André hyper-reguli' exist whenever \(d,s>1\). Replacement yields then a tremendous variety of generalized André planes and, under an extra assumption, also a wealth of interesting new subgeometry partitions.