On a-semiaffine planes with invisible lines (Q1106466)

The aim of the paper is to study finite weakly a-semiaffine planes. Suppose that the line set of a linear space L is partitioned into two sets, the `visible' and the `invisible' lines. For a non-incident point- line pair (P,\(\ell)\) let \(\pi\) (P,\(\ell)\) denote the number of lines through P which do not intersect \(\ell\). A weakly a-semiaffine plane is a linear space L where (among others) the following conditions hold: For \(P\not\in \ell\) we have \(\pi (P,\ell)\geq a-\ell;\) moreover \(\pi\) (P,\(\ell)\leq a\) iff \(\ell\) is visible. The main results give statements concerning the degrees of points and visible lines. For example, if L is of order n then every point has degree \(n+1\) or n and every visible line has degree \(n+2-a\) or \(n+1-a\). Remark: If the set of invisible lines is empty then L is an a-semiaffine plane [see the first author: Einführung in die endliche Geometrie. II: Projektive Räume. (1983; Zbl 0502.51001)].