Decompositions and Terracini loci of cubic forms of low rank

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.

arXiv comment15 pages. Final version

References in corpus (10)

• Computing symmetric rank for symmetric tensors
• On the rank of a binary form
• The Hitchhiker guide to: Secant Varieties and Tensor Decomposition
• On generic identifiability of symmetric tensors of subgeneric rank
• Stratification of the fourth secant variety of Veronese variety via the symmetric rank
• Identifiability of homogeneous polynomials and Cremona Transformations
• Identifiability beyond Kruskal's bound for symmetric tensors of degree 4
• On the partially symmetric rank of tensor products of W-states and other symmetric tensors
• On the identifiability of ternary forms
• A generalization of Kruskal's theorem on tensor decomposition

Cited by in corpus (2)

• Ample bodies and Terracini loci of projective varieties
• Geometry of first nonempty Terracini loci