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.

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

References in corpus (8)

• Identifiability of parameters in latent structure models with many observed variables
• Geometric lower bounds for generalized ranks
• Symplectic bundles on the plane, secant varieties and Lüroth quartics revisited
• Osculating Varieties of Veronesean and their higher secant varieties
• Special effect varieties in higher dimension
• A counterexample to Strassen's direct sum conjecture
• Every Ternary Quintic is a Sum of Ten Fifth Powers
• Projective and birational geometry of Grassmannians and other special varieties