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

Abstract

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≤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($\sqrt q$+1) for an integer s≥2 and the set of points of PG(3,q) not covered byS is the disjoint union of s Baer subgeometriesPG(3,$\sqrt q$). We also discuss maximal partial spreads in PG(3,p ³), p=p ^h₀ , p ₀ prime, p ₀ ≥ 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 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 subgeometryPG(5,p) in PG(3,p ³). In PG(3,p ³),p square, for maximal partial spreads of deficiency δ ≤ 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 δ, the set of holes is a disjoint union of subgeometries PG(2t+1,$\sqrt q$), which implies that δ ≡ 0 (mod$\sqrt q$+1) and, when (2t+1)($\sqrt q$-1) <q-1, that δ ≥ 2($\sqrt q$+1). In the latter case, the set of holes is the disjoint union of projected subgeometries PG(3t+2,$\sqrt[3]{q}$) and this implies δ ≡ 0 (mod q ^2/3+q ^1/3+1). A more general result is also presented.