Linear spaces with many projective planes

Abstract

We assume that in a linear space (P , L) there is a non-empty set M of points with the property that every plane containing a point of M is a projective plane. In section 3 an example is given that in general (P , L) is not a projective space. But if M can be completed by two points to a generating set of P , then (P , L) is a projective space. (1991): 51A05, 51D99, 51E15. Mathematics Subject Classification

Twisted Fano spaces and their classification, linear completions of systems of triangle perspectives

“…By the type of ξ we mean the triple ξ(1, 2), ξ(2, 3), ξ (1,3). Then, up to an isomorphism, there are exactly 8 systems K which can be completed to a Steiner triple system.…”

“…The point is to prove that the structures in T are pairwise nonisomorphic. If K has type (ix) then T = PG (3,2) and if K has type (x) then K =: B 0 has the (unique) centre (cf. 2.8); clearly, PG(3, 2) ∼ = B 0 , and in view of 3.9, K ∼ = PG(3, 2), B 0 for K of type (i)-(v).…”