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

An 𝔽 𝑞 -linear set 𝐿 = 𝐿 𝑈 of Λ = P G ⁡ ( 𝑉 , 𝔽 𝑞 𝑛 ) ≅ P G ⁡ ( 𝑟 − 1 , 𝑞 𝑛 ) is a set of points defined by non-zero vectors of an 𝔽 𝑞 -subspace U of V . The integer d i m 𝔽 𝑞 ⁡ 𝑈 is called the rank of L . In Lunardon and Polverino (2004) [23] , it was proven that any 𝔽 𝑞 -linear set L of Λ of rank u such that 〈 𝐿 〉 = Λ is either a canonical subgeometry of Λ or there are a ( 𝑢 − 𝑟 − 1 ) -dimensional subspace Γ of P G ⁡ ( 𝑢 − 1 , 𝑞 𝑛 ) ⊃ Λ disjoint from Λ and a canonical subgeometry Σ ≅ P G ⁡ ( 𝑢 − 1 , 𝑞 ) disjoint from Γ such that L is the projection of Σ from Γ onto Λ. The subspace Γ is called the vertex of the projection. In this article, we will show a method to reconstruct the vertex Γ for a particular class of linear sets of rank 𝑢 = 𝑛 ⁢ ( 𝑟 − 1 ) in P G ⁡ ( 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 P G ⁡ ( 1 , 𝑞 𝑛 ) introduced from 2018 onwards, by means of certain properties of their projection vertices, as done in Csajbók and Zanella (2016) [13] and in Zanella and Zullo (2020) [34] .