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.

Keywords

• additive decompositions
• secant varieties
• Veronese varieties
• Segre varieties
• Segre-Veronese varieties
• Grassmannians
• tensor rank
• Waring rank
• algorithm