Most secant varieties of tangential varieties to Veronese varieties are nondefective

TL;DR

This paper proves that most secant varieties of tangential varieties to Veronese embeddings have the expected dimension, resolving a long-standing conjecture by combining induction, specialization, and computer-assisted proofs for challenging cases.

Contribution

The main novelty is the proof of the conjecture for the cubic case ($d=3$), including computational verification of large base cases.

Findings

Most secant varieties are nondefective, matching expected dimensions.

The proof reduces to verifying challenging base cases via computer calculations.

Successfully handled the largest base case involving a tangential variety in high-dimensional projective space.

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 Id\'a 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 $\mathbb{P}^{88559}$.