A geometric characterization of known maximum scattered linear sets of $\mathrm{PG}(1,q^n)$

An $\mathbb{F}_q$- linear set $L=L_U$ of $\Lambda=\mathrm{PG}(V, \mathbb{F}_{q^n}) \cong \mathrm{PG}(r-1,q^n)$ is a set of points defined by non-zero vectors of an $\mathbb{F}_q$-subspace $U$ of $V$. The integer $\dim_{\mathbb{F}_q} U$ 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 $\mathbb{F}_q$-linear set $L$ of $\Lambda$ of rank $u$ such that $\langle L \rangle=\Lambda$ is either a canonical subgeometry of $\Lambda$ or there are a $(u-r-1)$-dimensional subspace $\Gamma$ of $\mathrm{PG}(u-1,q^n) \supset \Lambda$ disjoint from $\Lambda$ and a canonical subgeometry $\Sigma \cong \mathrm{PG}(u-1,q)$ disjoint from $\Gamma$ such that $L$ is the projection of $\Sigma$ from $\Gamma$ onto $\Lambda$. The subspace $\Gamma$ is called the vertex of the projection. In this article, we will show a method to reconstruct the vertex $\Gamma$ for a peculiar class of linear sets of rank $u = n(r - 1)$ in $\mathrm{PG}(r - 1, q^n)$ called evasive linear sets. Also, we will use this result to characterize some families of linear sets of the projective line $\mathrm{PG}(1,q^n)$ introduced from 2018 onward, by means of certain properties of their projection vertices, as done in [B. Csajb\'{o}k, C. Zanella: On scattered linear sets of pseudoregulus type in $\mathrm{PG}(1, q^t)$, Finite Fields Appl. 41 (2016)] and in [C. Zanella, F. Zullo: Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1, q^n)$. Discrete Math. 343(5) (2020)].