Partial t-Spreads in PG(2t+1,q)

This article first of all discusses the problem of the cardinality of maximal partial spreads in PG(3,q), q square, q>4. Let r be an integer such that 2rq+1 and such that every blocking set of PG(2,q) with at most q+r points contains a Baer subplane. If S is a maximal partial spread of PG(3,q) with q²-1-r lines, then r=s(+1) for an integer s2 and the set of points of PG(3,q) not covered byS is the disjoint union of s Baer subgeometriesPG(3,). We also discuss maximal partial spreads in PG(3,p³), p=p₀^h, p₀ prime, p₀ 5, h 1, p 5. We show that if p is non-square, then the minimal possible deficiency of such a spread is equal to p²+p+1, and that if such a maximal partial spread exists, then the set of points of PG(3,p³) not covered by the lines of the spread is a projected subgeometryPG(5,p) in PG(3,p³). In PG(3,p³),p square, for maximal partial spreads of deficiency p²+p+1, the combined results from the preceding two cases occur. In the final section, we discuss t-spreads in PG(2t+1,q), q square or q a non-square cube power. In the former case, we show that for small deficiencies , the set of holes is a disjoint union of subgeometries PG(2t+1,), which implies that 0 (mod+1) and, when (2t+1)(-1) 2(+1). In the latter case, the set of holes is the disjoint union of projected subgeometries PG(3t+2,) and this implies 0 (mod q^2/3+q^1/3+1). A more general result is also presented.