Scattered Linear Sets and Pseudoreguli

Abstract

In this paper, we show that one can associate a pseudoregulus with every scattered linear set of rank $3n$ in $\mathrm{PG}(2n-1,q^3)$. We construct a scattered linear set having a given pseudoregulus as associated pseudoregulus and prove that there are $q-1$ different scattered linear sets that have the same associated pseudoregulus. Finally, we give a characterisation of reguli and pseudoreguli in $\mathrm{PG}(3,q^3)$.