Subgeometries and linear sets on a projective line

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.