Decompositions and Terracini loci of cubic forms of low rank

TL;DR

This paper investigates the structure of minimal decompositions of certain cubic forms, revealing conditions under which multiple decompositions share common elements and describing the associated Terracini loci.

Contribution

It provides new insights into the uniqueness and structure of Waring rank decompositions for cubic forms of rank n+2 in n+1 variables, including the description of Terracini loci.

Findings

Multiple decompositions share at least n-3 elements

Remaining elements are in a special configuration

Detailed description of the Terracini locus for the third Veronese embedding

Abstract

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.