Decompositions and Terracini loci of cubic forms of low rank (Q6630012)

Let \(F\) be a degree \(d\) complex homogeneous polynomial in \(n+1\) variables. Its rank (or additive rank) is the minimal number of \(d\)-powers of linear forms whose sum is \(F\). The closure of the set of all \(d\)-forms of rank \(r\) is a variety, the \(r\)-th secant variety of the \(d\)-Veronese embedding of \(\mathbb{P}^n\). For applications it is important to know if the additive decomposition with \(r\) addenda is unique, up to permutations. For general forms of prescribed rank this is the case with a few exceptions [\textit{L. Chiantini} et al., Trans. Am. Math. Soc. 369, No. 6, 4021--4042 (2017; Zbl 1360.14021)]. For non-generic forms a classical criterion for the uniqueness of the additive decomposition is the classical Kruskal's criterion for tensors adapted to the symmetric case. Just outside the range of Kruskal's criterion non-uniquess appears and it is important to describe all possible additive decompositions. We say that \(F\) is concise if there no linear change of coordinates for which \(F\) depends on at most \(n\) variables. A concise form has rank at least \(n+1\). Additive decompositions with (potentially) more than \(r\) addenda, all of them needed, are called irredudant. In this paper we have \(d=3\) and a complete descricription of non-uniqueness for irredundant decompositions with \(n+2\) terms (all decompositions have at least \(n-3\) common addenda, while the other \(5\) addenda are in a special configuration. As an application the authors give a complete description of the \((n+2)\)-th Terracini locus of the third Veronese embedding of \(\mathbb{P}^n\).\N\NFor the entire collection see [Zbl 1545.13003].