Subgeometries and linear sets on a projective line

• Published in: Finite fields and their applications Vol. 34; pp. 95 - 106
• Main Authors: Lavrauw, Michel, Zanella, Corrado
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.07.2015
• Subjects: Finite projective line; Linear set; Subgeometry; Tangent splash; Subgeometry; 51E20; Linear set; Tangent splash; Finite projective line
• ISSN: 1071-5797; 1090-2465
• DOI: 10.1016/j.ffa.2015.01.006

We define the splash of a subgeometry on a projective line, extending the definition of [1] 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 [4], and show that clubs correspond to tangent splashes. We obtain a 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.