On the secant varieties of tangential varieties

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 $\mathrm{\mid }{\mathcal{O}}_{({\mathbb{P}}^1{)}^n}(d_1,\dots ,d_n)\mathrm{\mid }$ 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 $\mathrm{\mid }{L}^{\otimes t}\mathrm{\mid }$ is not defective.