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Open AccessArticle

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

1
Dipartimento di Matematica, Università di Trento, 38123 Trento, Italy
2
Dipartimento di Scienze Matematiche, Politecnico di Torino, 10129 Turin, Italy
3
Dipartimento di Ingegneria Meccanica, Energetica, Gestionale e dei Trasporti, Università degli studi di Genova, 16145 Genoa, Italy
4
Dipartimento di Matematica, Università di Bologna, 40126 Bologna, Italy
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Barcelona Graduate School of Mathematics, and Universitat Politècnica de Catalunya, 08034 Barcelona, Spain
*
Author to whom correspondence should be addressed.
The initial elaboration of this work had Anthony V. Geramita as one of the authors. The paper is dedicated to him.
Mathematics 2018, 6(12), 314; https://doi.org/10.3390/math6120314
Received: 9 October 2018 / Revised: 13 November 2018 / Accepted: 14 November 2018 / Published: 8 December 2018
(This article belongs to the Special Issue Decomposability of Tensors)
We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject. View Full-Text
Keywords: additive decompositions; secant varieties; Veronese varieties; Segre varieties; Segre-Veronese varieties; Grassmannians; tensor rank; Waring rank; algorithm additive decompositions; secant varieties; Veronese varieties; Segre varieties; Segre-Veronese varieties; Grassmannians; tensor rank; Waring rank; algorithm
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Bernardi, A.; Carlini, E.; Catalisano, M.V.; Gimigliano, A.; Oneto, A. The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition. Mathematics 2018, 6, 314.

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