Grassman Secants, Identifiability, and Linear Systems of Tensors

For any irreducible non-degenerate variety X in P^r, we give a criterion for the (k,s)-identifiability of X. If k<s<r$, then the (k,s)-identifiability holds for X if and only if the s-identifiability holds for the Segre product P^k x X. Moreover, if the s-th secant variety of X is not defective and it does not fill the ambient space, then we can produce a family of pairs (k,s) for which the (k,s)-identifiability holds for X.