The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

Alessandra Bernardi, Enrico Carlini, Maria Virginia Catalisano, Alessandro Gimigliano and Alessandro Oneto
Additional contact information
Alessandra Bernardi: Dipartimento di Matematica, Università di Trento, 38123 Trento, Italy
Enrico Carlini: Dipartimento di Scienze Matematiche, Politecnico di Torino, 10129 Turin, Italy
Maria Virginia Catalisano: Dipartimento di Ingegneria Meccanica, Energetica, Gestionale e dei Trasporti, Università degli studi di Genova, 16145 Genoa, Italy
Alessandro Gimigliano: Dipartimento di Matematica, Università di Bologna, 40126 Bologna, Italy
Alessandro Oneto: Barcelona Graduate School of Mathematics, and Universitat Politècnica de Catalunya, 08034 Barcelona, Spain

Mathematics, 2018, vol. 6, issue 12, 1-86

Abstract: 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.

Keywords: additive decompositions; secant varieties; Veronese varieties; Segre varieties; Segre-Veronese varieties; Grassmannians; tensor rank; Waring rank; algorithm (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2018
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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