Scattered linear sets generated by collineations between pencils of lines
Let \(A\) and \(B\) be two points of \(\mathrm{{PG}}(2,q^n)\), and let \(\Phi \) be a collineation between the pencils of lines with vertices \(A\) and \(B\). In this paper, we prove that the set of points of intersection of corresponding lines under \(\Phi \) is either the union of a scattered \(\mathrm{{GF}}(q)\)-linear set of rank \(n+1\) with the line \(AB\) or the union of \(q-1\) scattered \(\mathrm{{GF}}(q)\)-linear sets of rank \(n\) with \(A\) and \(B\). We also determine the intersection configurations of two scattered \(\mathrm{{GF}}(q)\)-linear sets of rank \(n+1\) of \(\mathrm{{PG}}(2,q^n)\) both meeting the line \(AB\) in a \(\mathrm{{GF}}(q)\)-linear set of pseudoregulus type with transversal points \(A\) and \(B\).
linear sets collineations subgeometries 51e21 05b25