A geometric characterisation of Desarguesian spreads

TL;DR

This paper offers a geometric characterization of Desarguesian spreads in projective spaces, focusing on spreads with normal elements in general position, advancing understanding of their structure and properties.

Contribution

It introduces a new geometric characterization of Desarguesian spreads based on the configuration of normal elements in projective spaces.

Findings

Characterization of (n-1)-spreads with normal elements in general position

Geometric criteria for identifying Desarguesian spreads

Extension of characterization to higher-dimensional projective spaces

Abstract

We provide a characterisation of $(n-1)$-spreads in $\mathrm{PG}(rn-1,q)$ that have $r$ normal elements in general position. In the same way, we obtain a geometric characterisation of Desarguesian $(n-1)$-spreads in $\mathrm{PG}(rn-1,q)$, $r>2$.