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 2r less than or equal to q+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 q(2)+1-r lines, then r=s(root q+1) for an integer s greater than or equal to 2 and the set of points of PG(3,q) not covered by S is the disjoint union of s Baer subgeometries PG(3,root q). We also discuss maximal partial spreads in PG(3,p(3)), p=p(0)(h), p(0) prime, p(0)greater than or equal to 5, h greater than or equal to 1, p not equal 5. We show that if p is non-square, then the minimal possible deficiency of such a spread is equal to p(2)+p+1, and that if such a maximal partial spread exists, then the set of points of PG(3,p(3)) not covered by the lines of the spread is a projected subgeometry PG(5,p) in PG(3,p(3)). In PG(3,p(3)), p square, for maximal partial spreads of deficiency delta less than or equal to p(2)+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 delta, the set of holes is a disjoint union of subgeometries PG(2t+1,root q), which implies that delta=0 (mod root q+1) and, when (2t+1)(root q-1)<q-1, that delta greater than or equal to 2(root q+1). In the latter case, the set of holes is the disjoint union of projected subgeometries PG(3t+2,(3)root q) and this implies delta=0(mod q(2/3)+q(1/3)+1). A more general result is also presented.
spreads partial spreads projective spaces