Scattered linear sets of $PG(3,q^n)$ generated by collineations between stars of lines

Let and be two points of and let be a collineation between the stars of lines with vertices and that fixes the line . In this paper we prove that the set of points of intersection of corresponding lines under , under the hypothesis that generates , is either the union of two degenerate -sets in two planes through the line (see Donati and Durante, 2014) or the union of a scattered -linear set of rank with the line We also determine the intersection configurations of two scattered -linear sets of rank of both meeting the line in a -linear set of pseudoregulus type with transversal points and .