On the secant varieties of tangential varieties

Let $Xsubset mathbb{P}^r$ 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 $X=Ytimes mathbb{P}^1$ has a suitable Segre-Veronese style embedding in $mathbb{P}^r$. As a corollary we prove that if $d_ige 3$, $1le i le n$, 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 $mathbb{P}^ntimes mathbb{P}^1$. 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 $tge k_0$ the tangential variety of $X$ with respect to $|L^{otimes t}|$ is not defective.