Hyper-reguli and non-André quasi-subgeometry partitions of projective spaces Hyper-reguli and non-André quasi-subgeometry partitions of projective spaces.

A question raised by Ostrom on the existence of hyper-reguli that are not André hyper-reguli is completely determined for hyper-reguli of order $q^{t}$ where t is composite. Various new types of ‘generalized André replacements’ are constructed that produce many new classes of generalized André planes. In general, for t=ds, new non-André quasi-subgeometry partitions are constructed of ${\it PG}(s-1,q^{d})$ by quasi-subgeometries isomorphic to PG$(ds/e-1,q^{e})$, for various divisors e of d. When $d=2$, this produces new non-André sub-geometry partitions of PG $(2s-1,q^{2})$ by subgeometries isomorphic to PG $(2s-1,q)$ and PG$(s-1,q^{2})$. [ABSTRACT FROM AUTHOR]