A characterization of certain 3-dimensional affino-projective spaces (Q762766)

A linear space S is a set of elements called points together with a family of distinguished subsets called lines. A planar space is a linear space provided with a family of distinguished linear subspaces called planes. An affino-projective plane of order k is a linear space \(\pi\) obtained from a projective plane of order k by deleting n collinear points \((0\leq n\leq k+1)\). In particular \(\pi\) is a projective plane (respectively a punctured projective plane, an affine plane with one point at infinity, an affine plane) if \(n=0\) (resp. \(n=1,k,k+1).\) The problem to classify the planar spaces satisfying the conditions (II) and II') (described below) is studied. The following theorem describes certain classes of planar spaces: If S is a finite planar space such that (II) for every pair of planes \(\pi\) and \(\pi\) ' intersecting in exactly one point, any line intersecting \(\pi\) intersects \(\pi\) ' and (II') there are at least two planes intersecting in exactly one point, then one of the following occurs: (a) S is obtained from a 3-dimensional projective space PG(3,k) by deleting an affino- projective (but not projective) plane of order k, (b) S is obtained from a 3-dimensional projective space PG(3,k) by deleting k collinear points, (c) S is obtained by adding a new point (joined to all other points by lines of size 2) to a punctured projective plane or to an affine plane with one point at infinity or to an affine plane, (d) \(S=K_ 5\) where \(K_ 5\) denotes the planar space of five points in which all lines have two points (are of size 2) and all planes have three points.