The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

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 cla...

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Bibliographic Details
Published in:Mathematics (Basel) 2018-12, Vol.6 (12), p.314
Main Authors: Bernardi, Alessandra, Carlini, Enrico, Catalisano, Maria, Gimigliano, Alessandro, Oneto, Alessandro
Format: Article
Language:English
Subjects:
additive decompositions
Algebra
algorithm
Data analysis
Decomposition
Geometry
Grassmannians
Mathematical analysis
Mathematics
secant varieties
Segre varieties
Segre-Veronese varieties
Signal processing
tensor rank
Tensors
Veronese varieties
Waring rank
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math6120314