Let \(f\in S^d \mathbb C^{n+1}\) be a homogeneous polynomial of degree \(d\) in \(n+1\) variables. The Chow rank of \(f\) is the minimal integer \(s\) such that \(f\) may be written as \[ f = \ell_{1,1}\cdots \ell_{1,d} + \cdots + \ell_{s,1}\cdots \ell_{s,d}, \] where the \(\ell_{i,j}\) are linear forms. This is an important instance of an additive decomposition for a tensor. Tensor decompositions are by now a large field with deep geometric and algebraic roots and yet possess a vast number of applications in many contexts such as complexity, information theory, and machine learning among others.
One geometric feature of the subject arises when one asks what is the Chow rank of a generic \(f\in S^d \mathbb C^{n+1}\). Let \(\mathcal{C}_{d,n}\subset \mathbb P^{\binom{n+d}{d}-1}\) be the projective variety parameterizing products of linear forms in \(S^d \mathbb C^{n+1}\). The variety \(\mathcal{C}_{d,n}\) is called the Chow variety. Computing the Chow rank of a generic \(f\in S^d \mathbb C^{n+1}\) is equivalent to finding the smallest \(s\) such that \(\sigma_s(\mathcal{C}_{d,n}) = \mathbb P^{\binom{n+d}{d}-1}\), where \(\sigma_s(\mathcal{C}_{d,n})\) is the \(s\)-th secant variety of the Chow variety. The topic of secant varieties is a delightful chapter of classical algebraic geometry that has attracted more attention in the last decades, partly because of its natural role in additive decompositions and applications thereof.
This nice paper is a contribution to determining dimensions of secants of Chow varieties. The main result is that all secant varieties \(\sigma_s(\mathcal{C}_{d,n})\) have expected dimensions for:

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any \(n\) and \(d=3\),

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\(n=3\) and any \(d\).

The methods are very combinatorial and rely on a lattice construction generalising a method due to Brambilla and Ottaviani. The base cases of the inductions are treated with a computer-assisted proof.