A geometric characterization of known maximum scattered linear sets of PG (1, qⁿ)

An Fq- linear set L=LU of =PG (V, Fₐ䂞) PG (r-1, qⁿ) is a set of points defined by non-zero vectors of an Fq-subspace U of V. The integer ₅ₐ U is called the rank of L. In G. Lunardon, O. Polverino: Translation ovoids of orthogonal polar spaces. Forum Math. 16 (2004), it was proven that any Fq-linear set L of of rank u such that L = is either a canonical subgeometry of or there are a (u-r-1) -dimensional subspace of PG (u-1, qⁿ) disjoint from and a canonical subgeometry PG (u-1, q) disjoint from such that L is the projection of from onto. The subspace is called the vertex of the projection. In this article, we will show a method to reconstruct the vertex for a peculiar class of linear sets of rank u = n (r - 1) in PG (r - 1, qⁿ) called evasive linear sets. Also, we will use this result to characterize some families of linear sets of the projective line PG (1, qⁿ) introduced from 2018 onward, by means of certain properties of their projection vertices, as done in B. Csajb\'ok, C. Zanella: On scattered linear sets of pseudoregulus type in PG (1, qᵗ), Finite Fields Appl. 41 (2016) and in C. Zanella, F. Zullo: Vertex properties of maximum scattered linear sets of PG (1, qⁿ). Discrete Math. 343 (5) (2020).