On the complement of the set of n-collinear points in PG(3, n)

Abstract

Let \({S = (\mathcal{P}, \mathcal{L}, \mathcal{H})}\) be the finite planar space obtained from the 3-dimensional projective space PG(3, n) of order n by deleting a set of n-collinear points. Then, for every point \({p\in S}\), the quotient geometry S/p is either a projective plane or a punctured projective plane, and every line of S has size n or n + 1. In this paper, we prove that a finite planar space with lines of size n + 1 − s and n + 1, (s ≥ 1), and such that for every point \({p\in S}\), the quotient geometry S/p is either a projective plane of order n or a punctured projective plane of order n, is obtained from PG(3, n) by deleting either a point, or a line or a set of n-collinear points.