Classes and equivalence of linear sets in PG(1,qn)

The equivalence problem of Fq-linear sets of rank n of PG(1,qn) is investigated, also in terms of the associated variety, projecting configurations, Fq-linear blocking sets of Rédei type and MRD-codes. We call an Fq-linear set LU of rank n in PG(W,Fqn)=PG(1,qn)simple if for any n-dimensional Fq-subspace V of W, LV is PΓL(2,qn)-equivalent to LU only when U and V lie on the same orbit of ΓL(2,qn). We prove that U={(x,Trqn/q(x)):x∈Fqn} defines a simple Fq-linear set for each n. We provide examples of non-simple linear sets not of pseudoregulus type for n>4 and we prove that all Fq-linear sets of rank 4 are simple in PG(1,q4).