Decompositions and Terracini loci of cubic forms of low rank

We study Waring rank decompositions for cubic forms of rank n + 2 in n + 1 variables. In this setting, we prove that if a concise form has more than one non-redundant decomposition of length n+2, then all such decompositions share at least n−3 elements, and the remaining elements lie in a special configuration. Following this result, we give a detailed description of the (n+2)-th Terracini locus of the third Veronese embedding of n-dimensional projective space.