We show how methods of algebraic geometry can produce criteria for the identifiability of specific tensors that reach beyond the range of applicability of the celebrated Kruskal criterion. More specifically, we deal with the symmetric identifiability of symmetric tensors in Sym 4 (C n+1 ), i.e., quartic hypersurfaces in a projective space P n , that have a decomposition in 2n + 1 summands of rank 1. This is the first case where the reshaped Kruskal criterion no longer applies. We present an effective algorithm, based on efficient linear algebra computations, that checks if the given decomposition is minimal and unique. The criterion is based on the application of advanced geometric tools, like Castelnuovo's lemma for the existence of rational normal curves passing through a finite set of points, and the Cayley-Bacharach condition on the postulation of finite sets. In order to apply these tools to our situation, we prove a reformulation of these results, hereby extending classical results such as Castelnuovo's lemma and the analysis of Geramita, Kreuzer, and Robbiano, Cayley-Bacharach schemes and their canonical modules, Trans. Amer. Math.

Let D = {D1, . . . , D } be a multi-degree arrangement with normal crossings on the complex projective space P n , with degrees d1, . . . , d ; let Ω 1 P n (log D) be the logarithmic bundle attached to it. First we prove a Torelli type theorem when D has a sufficiently large number of components by recovering them as unstable smooth irreducible degree-di hypersurfaces of Ω 1 P n (log D). Then, when n = 2, by describing the moduli spaces containing Ω 1 P 2 (log D), we show that arrangements of a line and a conic, or of two lines and a conic, are not Torelli. Moreover we prove that the logarithmic bundle of three lines and a conic is related with the one of a cubic. Finally we analyze the conic-case.

We report about the state of the art on complex and real generic identifiability of tensors, we describe some of our recent results obtained in [6] and we present perspectives on the subject.

Abstract. We prove that a general polynomial vector (f 1 , f 2 , f 3 ) in three homogeneous variables of degrees (3, 3, 4) has a unique Waring decomposition of rank 7. This is the first new case we are aware, and likely the last one, after five examples known since 19th century and the binary case. We prove that there are no identifiable cases among pairs (f 1 , f 2 ) in three homogeneous variables of degree (a, a + 1), unless a = 2, and we give a lower bound on the number of decompositions. The new example was discovered with Numerical Algebraic Geometry, while its proof needs Nonabelian Apolarity.

We describe a new method to determine the minimality and identifiability of a Waring decomposition A of a specific form (symmetric tensor) T in three variables. Our method, which is based on the Hilbert function of A, can distinguish between forms in the span of the Veronese image of A, which in general contains both identifiable and not identifiable points, depending on the choice of coefficients in the decomposition. This makes our method applicable for all values of the length r of the decomposition, from 2 up to the generic rank, a range which was not achievable before. Though the method in principle can handle all cases of specific ternary forms, we introduce and describe it in details for forms of degree 8.