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arXiv · 1403.5754

Subgeometries and linear sets on a projective line

Michel Lavrauw·Corrado Zanella
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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.

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Michel Lavrauw, Corrado Zanella. 2014-09-29. Subgeometries and linear sets on a projective line. https://arxiv.org/abs/1403.5754

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