# The Hitchhiker guide to: Secant Varieties and Tensor Decomposition

## TL;DR

This paper surveys the classical and recent results on secant varieties of Veronese, Grassmannian, and Segre varieties, which are fundamental in understanding tensor decompositions and ranks in algebraic geometry.

## Contribution

It compiles and discusses known results and open problems related to secant varieties of key algebraic varieties used in tensor analysis.

## Key findings

- Secant varieties stratify tensors by rank.
- Veronese, Grassmannian, Segre varieties are central in tensor decomposition.
- Many open problems remain in the study of these secant varieties.

## 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.