ISSN:
1573-7586
Keywords:
spreads
;
partial spreads
;
projective spaces
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
Notes:
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 0 h , 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1008305824113
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