On finite locally projective planar spaces

Abstract

Let L be a finite geometric lattice of rank 4 (i.e., a planar space) such that any two planes of L meet in a line. There is a longstanding conjecture due to W. M. Kantor which states that every such lattice can be embedded into a projective space. If L is given as above, then for every point p ϵ L, L p is a projective plane of order n (independent of p). Recently, A. Beutelspacher has shown that if L has at least n 3 points then L can be embedded into a projective space. We give an alternative proof of his result, which applies to the more general class of finite locally projective planar spaces. Furthermore, our considerations lead to some more insight into the geometrical structure of a possible counterexample to Kantor's conjecture. For example, they can be used to show that the bound on n 3 is not tight.