On the dimension of contact loci and the identifiability of tensors

Let X in P^r be an integral and non-degenerate variety. Set n:= dim (X). We prove that if the (k+n-1)-secant variety of X has (the expected) dimension (k+n-1)(n+1)-1<r and X is not uniruled by lines, then X is not k-weakly defective and hence the k-secant variety satisfies identifiability, i.e. a general element of it is in the linear span of a unique S in X with card(S) =k. We apply this result to many Segre-Veronese varieties and to the identifiability of Gaussian mixtures G{1,d}. If X is the Segre embedding of a multiprojective space we prove identifiability for the k-secant variety (assuming that the (k+n-1)-secant variety has dimension (k+n-1)(n+1)-1<r, this is a known result in many cases), beating several bounds on the identifiability of tensors.