The Hitchhiker guide to: Secant Varieties and Tensor Decomposition

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.

arXiv commentThis paper belongs to the special issue "Decomposability of Tensors", ed. Prof. Luca Chiantini

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Cited by in corpus (17)

• Secant non-defectivity via collisions of fat points
• Waring, tangential and cactus decompositions
• Entangled subspaces and generic local state discrimination with pre-shared entanglement
• Identifiability and singular locus of secant varieties to Grassmannians
• Non-defectivity of Segre-Veronese varieties
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