On the complement of the set of \(n\)-collinear points in \(PG(3,n)\). (English) Zbl 1202.51009
A linear space is planar if there exists a family of at least two subspaces (planes) with the property that any three points not on a common line lie in exactly one plane. The author proves the following two main results about planar spaces. If \(S\) is a planar space with lines of length \(n+1-s\) and \(n+1\) and of no other length such that for every point \(p\) the quotient geometry \(S/p\) is either a punctured projective plane of order \(n\) or a projective plane of order \(n \geq 2s+1\), with \(s\geq 1\), then \(S\) is either the complement of a point, the complement of a line or the complement of \(n\) collinear points in \(PG(3,n).\) If \(S\) is a finite planar space with lines of length \(n+1-s\) and \(n+1\), with \(s\geq 1\), such that for every point \(p\) the quotient geometry \(S/p\) is a projective plane of order \(n\), then \(S\) is the complement of either a point or a line in \(PG(3,n)\).
Reviewer: Steven T. Dougherty (Scranton)
MSC:
| 51E26 | Other finite linear geometries |
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