Lifts of nuclei in finite projective spaces

Summary

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 B_n 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 B_n is called a nucleus of B_n if every line through it meets B_n (exactly once, of course). The set of all nuclei of B_n is denoted by N(B_n). The following two fundamental results are well known.

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

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