Lines and opposition in Lie incidence geometries of exceptional type

TL;DR

This paper characterizes and classifies geometric lines in exceptional Lie incidence geometries, linking them to minimal blocking sets and automorphisms of these complex structures.

Contribution

It provides a new classification of geometric lines in exceptional Lie geometries and relates them to minimal blocking sets and opposition-preserving automorphisms.

Findings

Classification of geometric lines in exceptional geometries

Identification of minimal blocking sets as lines

Automorphisms characterized as opposition-preserving maps

Abstract

We characterise sets of points of exceptional Lie incidence geometries, that is, the natural geometries arising from spherical buildings of exceptional types $\mathsf{F_4}$, $\mathsf{E_6}$, $\mathsf{E_7}$, $\mathsf{E_8}$ and $\mathsf{G_2}$, that form a line using the opposition relation. With that, we obtain a classification of so-called ``geometric lines'' in many of these geometries. Furthermore, our results lead to a characterisation of geometric lines in finite exceptional Lie incidence geometries as minimal blocking sets, that is, point sets of the size of a line admitting no object opposite to all of their members, in most cases, and we classify all exceptions. As a further consequence, we obtain a characterisation of automorphisms of exceptional spherical buildings as certain opposition preserving maps.