On the secant varieties of tangential varieties

TL;DR

This paper computes the dimensions of secant varieties of tangential varieties for Veronese embeddings, establishing non-defectiveness results and providing formulas for specific cases using classical and recent theorems.

Contribution

It extends known results on secant varieties by computing their dimensions for Veronese embeddings and proves non-defectiveness under certain conditions.

Findings

Dimension formulas for secant varieties of tangential varieties of Veronese embeddings.

Non-defectiveness of tangential varieties for certain Segre-Veronese embeddings.

Existence of a threshold for line bundle tensor powers ensuring non-defectiveness.

Abstract

Let $X\subset \mathbb{P}^r$ be an integral and non-degenerate variety. Let $\sigma _{a,b}(X)\subseteq \mathbb{P}^r$, $(a,b)\in \mathbb{N}^2$, be the join of $a$ copies of $X$ and $b$ copies of the tangential variety of $X$. Using the classical Alexander-Hirschowitz theorem (case $b=0$) and a recent paper by H. Abo and N. Vannieuwenhoven (case $a=0$) we compute $\dim \sigma _{a,b}(X)$ in many cases when $X$ is the $d$-Veronese embedding of $\mathbb{P}^n$. This is related to certain additive decompositions of homogeneous polynomials. We give a general theorem proving that $\dim \sigma _{0,b}(X)$ is the expected one when $X=Y\times \mathbb{P}^1$ has a suitable Segre-Veronese style embedding in $\mathbb{P}^r$. As a corollary we prove that if $d_i\ge 3$, $1\le i \le n$, and $(d_1+1)(d_2+1)\ge 38$ the tangential variety of $(\mathbb{P}^1)^n$ embedded by $|\mathcal{O} _{(\mathbb{P} ^1)^n}(d_1,\dots ,d_n)|$ is not defective and a similar statement for $\mathbb{P}^n\times \mathbb{P}^1$. For an arbitrary $X$ and an ample line bundle $L$ on $X$ we prove the existence of an integer $k_0$ such that for all $t\ge k_0$ the tangential variety of $X$ with respect to $|L^{\otimes t}|$ is not defective.