A Characterization of Lines in Finite Lie Incidence Geometries of Classical Type

TL;DR

This paper classifies special point sets in finite classical Grassmannian geometries, extending classical results, and relates these sets to geometric lines and blocking sets in finite geometries.

Contribution

It provides a new classification of point sets in finite classical geometries, extending Bose & Burton's classical result to broader contexts.

Findings

Sets are either lines, hyperbolic lines, or ovoids.

Classifies geometric lines in finite classical geometries.

Connects these sets to blocking sets.

Abstract

We consider any classical Grassmannian geometry $\Gamma$; that is, any projective or polar Grassmann space. Suppose every line in $\Gamma$ contains $s+1$ points. Then we classify all sets of points in $\Gamma$ of cardinality $s+1$, with the property, that no object of opposite type in the corresponding building, is opposite every point of the set. It turns out that such sets are either lines, or hyperbolic lines in symplectic residues, or ovoids in large symplectic subquadrangles of rank 2 residues in characteristic 2. This is a far-reaching extension of a famous and fundamental result of Bose & Burton from the 1960s. We describe a new way to classify geometric lines in finite classical geometries and how our results correspond to blocking sets.