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Sets of Disjoint Lines in PG(3, q)

Published online by Cambridge University Press:  20 November 2018

Dale M. Mesner
Dale M. Mesner*
Affiliation:
Purdue University and University of North Carolina
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Let ∑ be a projective space PG(3, q) of dimension 3 and finite order q. Then ∑ contains (q + 1)(q2 + 1) points and an equal number of planes, and (q2 + 1) (q2 + q + 1) lines. It will be convenient to consider lines and planes as sets of points and to treat the incidence relation as set inclusion. Each plane contains q2 + q + 1 points and an equal number of lines. Each line contains q + 1 points and is contained in an equal number of planes. Each point is contained in q2 + q + 1 planes and an equal number of lines.

A spread of lines of ∑ is a set of q2 + 1 lines of ∑ which are pairwise disjoint, or skew; it can also be defined as a set of lines such that each point (or each plane) is incident with exactly one of the lines.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 273 - 280
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Bruck, R. H. and Bose, R. C., The construction of translation planes from projective spaces, J. Algebra, 1 (1964), 85–102.Google Scholar
2. Bruck, R. H. and Bose, R. C., Linear representations of projective planes in projective spaces, J. Algebra, 4 (1966), 117–172.Google Scholar
3. Veblen, O. and Young, J. W., Projective Geometry, Vol. 1 (Boston, 1910).Google Scholar