On the equivalence of linear sets

Let $L$ be a linear set of pseudoregulus type in a line $\ell$ in $Σ^*=\mathrm{PG}(t-1,q^t)$, $t=5$ or $t>6$. We provide examples of $q$-order canonical subgeometries $Σ_1,\, Σ_2 \subset Σ^*$ such that there is a $(t-3)$-space $Γ\subset Σ^*\setminus (Σ_1 \cup Σ_2 \cup \ell)$ with the property that for $i=1,2$, $L$ is the projection of $Σ_i$ from center $Γ$ and there exists no collineation $ϕ$ of $Σ^*$ such that $Γ^ϕ=Γ$ and $Σ_1^ϕ=Σ_2$. Condition (ii) given in Theorem 3 in Lavrauw and Van de Voorde (Des. Codes Cryptogr. 56:89-104, 2010) states the existence of a collineation between the projecting configurations (each of them consisting of a center and a subgeometry), which give rise by means of projections to two linear sets. It follows from our examples that this condition is not necessary for the equivalence of two linear sets as stated there. We characterize the linear sets for which the condition above is actually necessary.

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