Elenco seminari del ciclo di seminari
“GEOMETRIA ALGEBRICA E TENSORI”

Abstract. A tensor T of rank k is identifiable when it has a unique decomposition in terms of rank-1 tensors. There are cases in which the identifiability fails over C, for general tensors of fixed rank. The failure, often, is due to the existence of an elliptic normal curve through general points of the corresponding variety of rank-1 tensors. After a brief introduction to the subject, we prove the existence of non-empty euclidean open subsets of some varieties of real k-rank tensors, whose elements have 2 complex decompositions, but are identifiable over R. Moreover we provide examples of non-trivial euclidean open subsets in certain spaces of symmetric tensors and of almost unbalanced tensors, whose elements have real rank equal to the complex rank and are identifiable over R but not over C. On the contrary, there are examples of tensors of given real rank, for which identifiability over R can't hold in non-trivial open subsets. These results have been obtained in collaboration with Cristiano Bocci and Luca Chiantini.

Binary real forms of degree d admit as typical ranks all the integers between floor(d/2)+1 and d. We investigate the boundary between the open subset of rank r forms and the open subset of rank r+1. These boundaries are known only in the extreme cases, by Lee-Sturmfels (between rank floor(d/2)+1 and floor(d/2)+2) and Comon-Ottaviani (between rank d-1 and d). We investigate the intermediate boundaries. In the talk I will present our new results, focusing on the case of degree 7 forms. This is work in progress with G.Stagliano'.

The strength of a homogeneous polynomial is the smallest length of an additive decomposition as sum of reducible forms. It is called slice rank if we additionally require that the reducible forms have a linear factor. Geometrically, the slice rank corresponds to the smallest codimension of a linear space contained in the hypersurface defined by the form. Due to this relation, it is well-known and easy to compute the value of the general slice rank and also to show that the set of forms with bounded slice rank is Zariski-closed. In this talk, I will present the following results from recent joint works with A. Bik, E. Ballico and E. Ventura: (1) the set of forms with bounded strength is not always Zariski-closed: this is an asymptotic result in the number of variables proved by using the theory of polynomial functors; (2) for general forms, strength and slice rank are equal: this is proved by showing that the largest component of the secant variety of the variety of reducible forms is the secant variety of the variety of forms with a linear factor.

Secant defectivity of projective varieties is classically approached via dimensions of linear systems with multiple base points in general position. The latter can be studied via degenerations. We exploit a technique that allows some of the base points to collapse together. We deduce a general result which we apply to prove a conjecture by Abo and Brambilla: for c≥3 and d≥3, the Segre-Veronese embedding of Pm×Pn in bidegree (c,d) is non-defective.

The study of the decompositions of the powers of a quadratic form, also called representations, is a very classical problem. Many examples appear several times even in old literature, especially for the real case. Our purpose is to deal with this problem by a modern point of view, with the final aim of determining its rank and its border rank. The main instrument we used is the apolarity theory, by which it is possible to determine suitable decompositions of a given form, just analyzing its apolar ideal, that in the case of the powers of quadrics results to be generated by harmonic polynomials. This approach also allows us to determine the border rank in the case of three variables.

Discuterò su come alcuni problemi di geometria proiettiva elementare possono essere discussi e risolti con il metodo di Veronese.

The classical Waring problem for homogeneous polynomials can be translated into geometric terms, using the notion of defectivity and identifiability for secant varieties. The defectivity problem was completely solved by Alexander-Hirschowitz using classical degeneration techniques. On the other hand identifiability has recently been addressed by Mella and Galuppi. In this talk I will briefly explain the relationship between defectivity and identifiability in a more general setting and give bounds for a generalized Waring problem, introduced by Fröberg, Ottaviani and Shapiro. In particular we will see how the union of classical degeneration techniques combine with techniques borrowed from toric geometry, allowing us to give very sharp bounds on identifiability and defectivity in a much more general context. In the last part of the talk I will show how to generalize the previous approach to singular toric varieties. This is a joint work (in progress) with Elisa Postinghel.