Finite planar spaces with projective points (Q2655545)

If \(({\mathcal S},{\mathcal L},{\mathcal P})\) is a planar space then a point is projective if every line incident with it meets every plane in the space and every pencil of lines with this point as its center has constant size \(k+1\). It is shown that any planar space with a projective point, where \({\mathcal S}\) is not the union of three concurrent lines incident with the projective point, has size either \(s+1\) or \(k+1\), with \(k \leq s\). The authors prove that if \(k=s\) a planar space with a projective point is the 3-dimensional finite projective space \(PG(3,s)\) and if \(k=s-1\), \(s \neq 3\), that it is the 3-dimensional finite affine space \(AG(3,2)\) with a point at infinity.