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

Let \(X\subset \mathbb {P}^r\) be an integral and non-degenerate variety. For any \(q\in \mathbb {P}^r\) its X-rank \(r_X(q)\) is the minimal cardinality of a finite subset of X whose linear span contains q. The solution set \(\mathcal {S}(X,q)\) of \(q\in \mathbb {P}^r\) is the set of all \(S\subset X\) such that \(\#S=r_X(q)\) and S spans q. We prove that if \(X\ne \mathbb {P}^r\) there is at least one q with \(\#\mathcal {S}(X,q)>1\) and that for almost all pairs (X, q) we have \(\dim \mathcal {S}(X,q)>0\).