Vertex properties of maximum scattered linear sets of PG(1,q^n)

In this paper we investigate the geometric properties of the configuration consisting of a subspace Γ and a canonical subgeometry Σ in PG(n−1,qn), with Γ∩Σ=0̸. The idea motivating is that such properties are reflected in the algebraic structure of the linear set which is projection of Σ from the vertex Γ. In particular we deal with the maximum scattered linear sets of the line PG(1,qn) found by Lunardon and Polverino (2001) and recently generalized by Sheekey (2016). Our aim is to characterize this family by means of the properties of the vertex of the projection as done by Csajbók and the first author of this paper for linear sets of pseudoregulus type. With reference to such properties, we construct new examples of scattered linear sets in PG(1,q6), yielding also to new examples of MRD-codes in Fq6×6 with left idealizer isomorphic to Fqjavax.xml.bind.JAXBElement@2fd8e0d8.