$\DeclareMathOperator{\complex}{\mathbb{C}}$
Let $\bigvee^m(\complex^n)\subseteq (\complex^n)^{\otimes m}$ denote the space of symmetric tensors, i.e. the set of $x \in (\complex^n)^{\otimes m}$ that are invariant under permutations of the $m$ factors. The (cone over) the Veronese variety of $\bigvee^m(\complex^n)$ is the set of tensors of the form $x^{\otimes m}$ for some $x \in \complex^n$. This is one notion of the *decomposable tensors*.


Another notion is the set of tensors in $\bigvee^m(\complex^n)$ of the form $x\_1 \vee \dots \vee x\_m$, where $x\_i \in \complex^n$ for all $i \in \{1,\dots, m\}$, and
\begin{align}
x\_1 \vee \dots \vee x\_m :=\sum\_{\sigma \in S\_m} x\_{\sigma(1)}\otimes \dots \otimes x\_{\sigma(m)}.
\end{align}


Does this latter set have a name? Does it form an algebraic variety (like the Veronese does)? If so, what is its dimension, and have its secant varieties been studied?



A different notion of a decomposable symmetric tensor (besides Veronese)
Your $\vee$ is essentially multiplication of polynomials. The variety of tensors $x\_1 \vee \dotsb \vee x\_m$ corresponds to polynomials that factor as products of linear factors. Points of the (projective) variety correspond to hyperplane arrangements. Dually, they correspond to *cycles* of $m$ points (in the dual projective space).


This variety has various names including the **Chow variety**, the **variety of completely decomposable forms**, and the **split variety**. For example [Hirotachi Abo](https://www.webpages.uidaho.edu/%7Eabo/Research/Papers/split.pdf) calls it the "variety of completely decomposable $d$-forms" (his $d$ is your $m$) and denotes it $\operatorname{Split\_d}(\mathbb{P}^n)$ (also, his $n$ is your $n+1$). [Douglas A. Torrance](https://arxiv.org/pdf/1508.05546.pdf) calls it the Chow variety and uses the same notation, $\operatorname{Split\_d}$. [Yongui Guan](https://www.math.tamu.edu/%7Ejml/tamuthesis_YonghuiGuan.pdf) calls it the Chow variety and denotes it $\operatorname{Ch}\_d$. There are plenty, plenty more references which you can find by searching for those terms.


And it is indeed a variety. If you buy the correspondence with zero-dimensional cycles, that's pretty clearly closed–the points might move around and collide, but it's still a zero-cycle. If you want the equations, [Guan](https://www.math.tamu.edu/%7Ejml/tamuthesis_YonghuiGuan.pdf) seems to have some information; that's just what I found online. Better references would be [Gelfand-Kapranov-Zelevinsky](https://mathscinet.ams.org/mathscinet-getitem?mr=1264417) and [Landsberg's book](https://mathscinet.ams.org/mathscinet-getitem?mr=2865915) if you have access to them.


Rank with respect to this variety is called Chow rank, split rank, decomposable rank, etc. [Catalisano, et al](https://arxiv.org/abs/1511.07789) say that for a monomial $M$, an $M$-decomposition of a form $F$ (where $\deg(F)=\deg(M)$) is an expression for $F$ as a sum of terms given by evaluating $M$ at linear forms. Waring rank is $x\_1^d$-rank, and Chow (or split, or decomposable) rank is $(x\_1 \dotsm x\_d)$-rank.


If I may also respond to one of the comments: Indeed, Comon's conjecture was that the rank and symmetric rank of symmetric tensors would be equal. Symmetric rank is the same as Waring rank. [Shitov](https://arxiv.org/abs/1705.08740) gave a counterexample to that conjecture. The analogous conjecture for border ranks, that the border rank and border symmetric rank (border Waring rank) of a symmetric tensor are equal, seems to be still open (as far as I know, as of September 2020).


