Given a dimension \(n+1\) vector space \(V\) and a symmetric tensor \(T\in S^dV\), the symmetric rank of \(T\) is defined as the minimum integer \(r\) such that \(T\) can be written as the sum of \(r\) elements of the form \(v^{\otimes d}\) for some \(v\in V\).
Since the symmetric tensors with symmetric rank 1 can be identified, up to nonzero multiples, with the points of the Veronese variety of dimension \(n\) and degree \(d\), \(X_{n,d}\subset \mathbb{P}(S^dV)\), it follows that the \(r\)-th secant variety \(\sigma_r(X_{n,d})\) of the Veronese variety contains all the tensors of symmetric rank \(\leq r\). On the other hand, the secant variety contains also tensors which are limits of tensors of symmetric rank \(\leq r\) and which could have symmetric rank higher than \(r\).
In the paper under review the authors study the symmetric rank strata of the secant varieties to certain Veronese varieties and they give a complete description in the following cases:

–

\(\sigma_r(X_{1,d})\)

–

\(\sigma_2(X_{n,d})\)

–

\(\sigma_3(X_{n,d})\)

–

\(\sigma_r(X_{2,4})\) for \(r\leq5\).

In the first three cases they also give algorithms to compute the symmetric rank. Finally they study the rank of points on \(\sigma_2(\Gamma_{d+1})\) where \(\Gamma_{d+1}\) is an elliptic normal curve in \(\mathbb{P}^d\).
The main goal of this article is to give a purely geometric point of view on this kind of problems. In fact some of the results contained in the paper were partially known, but obtained with different approaches: see e.g. [G. Comas and M. Seiguer, “On the rank of a binary form”, arXiv:math/0112311; J. Brachat, P. Comon, B. Mourrain and E. Tsigaridas, Linear Algebra Appl. 433, No. 11–12, 1851–1872 (2010; Zbl 1206.65141); J. M. Landsberg and Z. Teitler, Found. Comput. Math. 10, No. 3, 339–366 (2010; Zbl 1196.15024)].