Subgeometries and linear sets on a projective line
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 , 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.
Keywords
Cite
@article{arxiv.1403.5754,
title = {Subgeometries and linear sets on a projective line},
author = {Michel Lavrauw and Corrado Zanella},
journal= {arXiv preprint arXiv:1403.5754},
year = {2014}
}