A Characterization of the Natural Embedding of the Split Cayley Hexagon in PG(6,q) by Intersection Numbers in Finite Projective Spaces of Arbitrary Dimension

TL;DR

This paper characterizes the natural embedding of the split Cayley hexagon in PG(6,q) by analyzing intersection numbers of a specific line set in finite projective spaces, establishing conditions under which it corresponds to the hexagon.

Contribution

It provides a characterization of the split Cayley hexagon embedding in PG(6,q) based on intersection properties of line sets in finite projective spaces.

Findings

The set L corresponds to the lines of a split Cayley hexagon H(q).

The intersection properties uniquely determine the embedding in PG(6,q).

The characterization applies to line sets with specified incidence properties.

Abstract

We prove that a non-empty set L of at most q^5+q^4+q^3+q^2+q+1 lines of PG(n, q) with the properties that (1) every point of PG(n,q) is incident with either 0 or q+1 elements of L, (2) every plane plane of PG(n, q) is incident with either 0, 1 or q+1 elements of L, (3) every solid of PG(n, q) is incident with either 0, 1, q+1 or 2q+1 elements of L, and (4) every 4-dimensional subspace of PG(n, q) is incident with at most q^3-q^2+4q elements of L, is necessarily the set of lines of a split Cayley hexagon H(q) naturally embedded in PG(6, q).