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
5
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
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Revised: 13 November 2018
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Accepted: 14 November 2018
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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.
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Keywords:
additive decompositions; secant varieties; Veronese varieties; Segre varieties; Segre-Veronese varieties; Grassmannians; tensor rank; Waring rank; algorithm
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MDPI and ACS Style
Bernardi, A.; Carlini, E.; Catalisano, M.V.; Gimigliano, A.; Oneto, A. The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition. Mathematics 2018, 6, 314. https://doi.org/10.3390/math6120314
AMA Style
Bernardi A, Carlini E, Catalisano MV, Gimigliano A, Oneto A. The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition. Mathematics. 2018; 6(12):314. https://doi.org/10.3390/math6120314
Chicago/Turabian StyleBernardi, Alessandra, Enrico Carlini, Maria Virginia Catalisano, Alessandro Gimigliano, and Alessandro Oneto. 2018. "The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition" Mathematics 6, no. 12: 314. https://doi.org/10.3390/math6120314
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