Abstract

We present a synthetical construction of the lifting process introduced in [1] and apply this process to obtain a new result on the structure of sets in the plane admitting the maximal number of nuclei.

Introduction

Let Bn be a set of qn−1 + qn−2 +…+ q + 1 points, not all on a hyperplane in the n-dimensional projective space PG(n, q) over the Galois field GF(q), n ≥ 2. A point not in Bn is called a nucleus of Bn if every line through it meets Bn (exactly once, of course). The set of all nuclei of Bn is denoted by N(Bn). The following two fundamental results are well known.

Result 1.1 (Segre-Korchmáros, [7]) If a, b, c are three non-collinear nuclei of Bn, then the points of Bn on the lines ab, bc, ca are collinear.

Result 1.2 (Blokhuis-Wilbrink, [3]) If Bn is an affine set (i.e. it is contained in the complement of a hyperplane), then |N(Bn)| ≤ q − 1.

The proofs of both the previous results have been given by the authors in the two dimensional case, but it is straightforward to see that they work in arbitrary dimensions. The original proof of Result 1.2 surprisingly does not use Result 1.1.

In the plane case, an elementary derivation of Result 1.2 from Result 1.1 has been obtained in [1] by using a process called “lifting”.