On scattered linear sets of pseudoregulus type in PG(1,qt)

Scattered linear sets of pseudoregulus type in PG(1,qt)PG(1,qt) have been defined and investigated in [19]. The aim of this paper is to continue such an investigation. Properties of a scattered linear set of pseudoregulus type, say LL, are proved by means of three different ways to obtain LL: (i) as projection of a q-order canonical subgeometry [20], (ii) as a point set whose image under the field reduction map is the hypersurface of degree t   in PG(2t−1,q)PG(2t−1,q) studied in [10], (iii) as exterior splash, by the correspondence described in [15]. In particular, given a canonical subgeometry Σ of PG(t−1,qt)PG(t−1,qt), necessary and sufficient conditions are given for the projection of Σ with center a (t−3)(t−3)-subspace to be a linear set of pseudoregulus type. Furthermore, the q-order sublines are counted and geometrically described.