Embedded varieties, X-ranks and uniqueness or finiteness of the solutions

Let X ⊂ P r be an integral and non-degenerate variety. For any q ∈ P rits X-rank rX(q) is the minimal cardinality of a finite subset of X whose linear span contains q. The solution set S(X, q) of q ∈ Pr is the set of all S ⊂ X such that #S = rX(q) and S spans q. We prove that if X 6= Pr there is at least one q with #S(X, q) > 1 and that for almost all pairs (X, q) we have dim S(X, q) > 0.