{1,2}-semi-affine planar spaces.

Let (S,L) be a finite linear space, that is, a finite set S whose elements we call points, and L a family of parts in S, whose elements we call lines, such that any line has at least two points, two distinct points are contained in just one line and |L|≥2. A subspace in (S,L) is a subset S′ in S such that for any X,Y∈S′, X≠Y, the line joining them belongs to S′. Suppose a family P of subspaces in (S,L) exists such that |P|≥2, every π∈P contains three noncollinear points and through three noncollinear points there is only one element of P. The triple (S,L,P) is called a planar space; the elements of P are called planes. Let (X,l) be a pair consisting of a point X∈S and a line l∈L with X∉l; let π(X,l) be the number of lines on X not meeting l and let H:={π(X,l):X∈S, l∈L, X∉l}. (S,L) is also called an H-semiaffine plane. Let n+1 be the maximum number of lines on a point; then the integer n is called the order of (S,L,P). In this paper the following result is proved: Let (S,L,P) be a finite planar space such that every plane of P is a {1,2}-semiaffine plane of order ≥5 and n+1 is the number of planes through every line of L. Then (S,L,P) is one of the following examples:(a) PG(3,n)∖π; (b) PG(3,n)∖{π∪X}, with X∉π; (c) PG(3,n)∖{π∪l}, with l⊄π; (d) PG(3,n)∖{π∪π′}, where X, l, π are a point, a line, and a plane of PG(3,n), respectively, and π′ is a plane of PG(3,n) different from π.