Abstract. A projective variety X is 'k-weakly defective' when its intersection with a general (k + 1)-tangent hyperplane has no isolated singularities at the k + 1 points of tangency. If X is k-defective, i.e. if the k-secant variety of X has dimension smaller than expected, then X is also k-weakly defective. The converse does not hold in general. A classification of weakly defective varieties seems to be a basic step in the study of defective varieties of higher dimension. We start this classification here, describing all weakly defective irreducible surfaces. Our method also provides a new proof of the classical Terracini's classification of k-defective surfaces.

Abstract. 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 ≤ b ≤ c, our method proves that the decomposition is unique (i.e. k-identifiability holds) for general tensors of rank k, as soon as k ≤ (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 × 4 × 4 tensors of rank 6, an interesting case because of its connection with the study of DNA strings.

We propose a new sufficient condition for verifying whether general rank-r complex tensors of arbitrary order admit a unique decomposition as a linear combination of rank-1 tensors. A practical algorithm is proposed for verifying this condition, with which it was established that in all spaces of dimension less than 15000, with a few known exceptions, listed in the paper, generic identifiability holds for ranks up to one less than the generic rank of the space. This is the largest possible rank value for which generic identifiability can hold, except for spaces with a perfect shape. The algorithm can also verify the identifiability of a given specific rank-r decomposition, provided that it can be shown to correspond to a nonsingular point of the rth order secant variety. For sufficiently small rank, which nevertheless improves upon the known bounds for specific identifiability, some local equations of this variety are known, allowing us to verify this property. As a particular example of our approach, we prove the identifiability of a specific 5 × 5 × 5 tensor of rank 7, which cannot be handled by the conditions recently provided in [I. Domanov and L. De Lathauwer, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 876-903]. Finally, we also present a surprising new class of weakly defective Segre varieties that nevertheless turns out to admit a generically unique decomposition.

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We prove that the general tensor of size 2^n and rank k has a unique decomposition as the sum of decomposable tensors if k< 0.9997*2^n/(n+1) (the constant 1 being the optimal value). Similarly, the general tensor of size 3^n and rank k has a unique decomposition as the sum of decomposable tensors if k< 0.998*3^n/(2n+1) (the constant 1 being the optimal value)