On the equivalence of linear sets

Let L be a linear set of pseudoregulus type in a line $$\ell $$ℓ in $$\varSigma ^*={\mathrm {PG}}(t-1,q^t)$$Σ∗=PG(t-1,qt), $$t=5$$t=5 or $$t>6$$t>6. We provide examples of q-order canonical subgeometries $$\varSigma _1,\, \varSigma _2 \subset \varSigma ^*$$Σ1,Σ2⊂Σ∗ such that there is a $$(t-3)$$(t-3)-subspace $$\varGamma \subset \varSigma ^*\setminus (\varSigma _1 \cup \varSigma _2 \cup \ell )$$Γ⊂Σ∗\(Σ1∪Σ2∪ℓ) with the property that for $$i=1,2$$i=1,2, L is the projection of $$\varSigma _i$$Σi from center $$\varGamma $$Γ and there exists no collineation $$\phi $$ϕ of $$\varSigma ^*$$Σ∗ such that $$\varGamma ^{\phi }=\varGamma $$Γϕ=Γ and $$\varSigma _1^{\phi }=\varSigma _2$$Σ1ϕ=Σ2. 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.