Stratification of the fourth secant variety of Veronese varieties via the symmetric rank (Q2841521)

The \(r\)-th secant variety \(\sigma_r(X)\) of the \(d\)-Veronese embedding \(X\) of \(\mathbb P^n\) corresponds to the set of symmetric tensors \(n\times n\times\cdots\times n\) (\(d\) times) whose \textit{(symmetric) border rank} is \(\leq r\), i.e. tensors which are limit of tensors of symmetric rank \(\leq r\). Equivalently, \(\sigma_r(X)\) corresponds to the set of forms of degree \(d\) in \(n+1\) variables, which are limit of sums of \(r\) powers of linear forms.NEWLINENEWLINEThe general \(T\in\sigma_r(X)\) has symmetric rank \(r\), but this is false for special tensors. First of all, \(\sigma_r(X)\) contains all tensors of symmetric rank \(\leq r\), as it contains \(\sigma_{r-1}(X)\). Moreover, it may also contain some tensors of symmetric rank strictly bigger than \(r\).NEWLINENEWLINEThe stratification, by the symmetric rank, of \(\sigma_r(X)\setminus \sigma_{r-1}(X)\) was previously known up to \(r= 3\). The authors extend our knowledge by providing a complete description of the stratification of \(\sigma_4(X)\setminus \sigma_3(X)\), for any value of \(n\) and \(d\).