Linear sets from projection of Desarguesian spreads

Every linear set in a projective space is the projection of a subgeometry, and most known characterizations of linear sets are given under this point of view. For instance, scattered linear sets of pseudoregulus type are obtained by considering a Desarguesian spread of a subgeometry and projecting from a vertex which is spanned by all but two director spaces. In this paper we introduce the concept of linear sets of h-pseudoregulus type, which turns out to be projected from the span of an arbitrary number of director spaces of a Desarguesian spread of a subgeometry. Among these linear sets, we characterize those which are h-scattered and solve the equivalence problem between them; a key role is played by an algebraic tool recently introduced in the literature and known as Moore exponent set. As a byproduct, we classify asymptotically h-scattered linear sets of h-pseudoregulus type.