Most secant varieties of tangential varieties to Veronese varieties are nondefective

Abstract

We prove a conjecture stated by Catalisano, Geramita, and Gimigliano in 2002, which claims that the secant varieties of tangential varieties to the $d$th Veronese embedding of the projective $n$-space ${\mathbb{P}}^n$ have the expected dimension, modulo a few well-known exceptions. As Bernardi, Catalisano, Gimigliano, and Idá demonstrated that the proof of this conjecture may be reduced to the case of cubics, i.e., $d=3$, the main contribution of this work is the resolution of this base case. The proposed proof proceeds by induction on the dimension $n$ of the projective space via a specialization argument. This reduces the proof to a large number of initial cases for the induction, which were settled using a computer-assisted proof. The individual base cases were computationally challenging problems. Indeed, the largest base case required us to deal with the tangential variety to the third Veronese embedding of ${\mathbb{P}}^{79}$ in ${\mathbb{P}}^{88559}$.

25 pages, 2 figures, extended the introduction, and added a C++ code as an ancillary file

References in corpus (4)

• An algorithm for generic and low-rank specific identifiability of complex tensors
• On generic identifiability of symmetric tensors of subgeneric rank
• Higher secant varieties of Segre-Veronese varieties
• Generic forms of low Chow rank

Cited by in corpus (7)

• The Hitchhiker guide to: Secant Varieties and Tensor Decomposition
• Waring, tangential and cactus decompositions
• Algebraic stories from one and from the other pockets
• Tangential varieties of Segre-Veronese surfaces are never defective
• Restricted Secant Varieties of Grassmannians
• Almost all subgeneric third-order Chow decompositions are identifiable
• Secant varieties of Segre-Veronese varieties ${\mathbb{P}}^m\times {\mathbb{P}}^n$ embedded by $\mathcal{O}(1,2)$ are non-defective for $n\gg m^3$, $m\ge 3$