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

Abstract

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 $\dim \mathcal {S}(X,q)>0$.