Subgeometries and linear sets on a projective line

TL;DR

This paper explores the relationship between subgeometries and linear sets on projective lines, establishing that splashes of subgeometries are linear sets and vice versa, and introduces new concepts like clubs and tangent splashes.

Contribution

It generalizes the concept of splashes to arbitrary dimensions, proves their equivalence to linear sets, and introduces the notion of clubs, providing a comprehensive framework for tangent splashes.

Findings

A splash of a subgeometry on a projective line is always a linear set.

Every linear set on a projective line can be realized as a splash of some subgeometry.

Conditions for a splash to be a scattered linear set are characterized.

Abstract

We define the splash of a subgeometry on a projective line, extending the definition of \cite{BaJa13} to general dimension and prove that a splash is always a linear set. We also prove the converse: each linear set on a projective line is the splash of some subgeometry. Therefore an alternative description of linear sets on a projective line is obtained. We introduce the notion of a club of rank $r$, generalizing the definition from \cite{FaSz2006}, and show that clubs correspond to tangent splashes. We determine the condition for a splash to be a scattered linear set and give a characterization of clubs, or equivalently of tangent splashes. We also investigate the equivalence problem for tangent splashes and determine a necessary and sufficient condition for two tangent splashes to be (projectively) equivalent.