On generic identifiability of 3-tensors of small rank

We introduce an inductive method for the study of the uniqueness of decompositions of tensors, by means of tensors of rank 1. The method is based on the geometric notion of weak defectivity. For three-dimensional tensors of type (a, b, c), a\le b\le c, our method proves that the decomposition is unique (i.e. k-identifiability holds) for general tensors of rank k, as soon as k\le (a+1)(b+1)/16. This improves considerably the known range for identifiability. The method applies also to tensor of higher dimension. For tensors of small size, we give a complete list of situations where identifiability does not hold. Among them, there are 4\times4\times4 tensors of rank 6, an interesting case because of its connection with the study of DNA strings.