Let $V_1,\ldots , V_k$ be vector spaces of dimensions $n_1,\ldots , n_k$ over a field of characteristic zero.

Consider the rational map $ \newcommand{\PP}{\mathbb{P}} \newcommand{\bs}{\boldsymbol} \DeclareMathOperator{\Gr}{Gr} \DeclareMathOperator{\im}{im} $ $$ \Phi : \left[\PP (V_1^*) \times \cdots \times \PP (V_k^*)\right]^j \dashrightarrow \Gr (n_1 \cdots n_k - j, \, V_1\otimes \cdots \otimes V_k )$$ defined by $$ \left([\alpha_1^i], \ldots , [\alpha_k^i]\right)_{1\le i\le j} \mapsto \{ T \mid (\alpha_1^i \otimes \cdots \otimes \alpha_k^i) (T) =0, \, \, 1\le i \le j \}. $$ Counting dimensions, we expect that the map is dominant when $$ j \, (n_1 + n_2 \cdots +n_k - k) \ge j \, \left( n_1 \cdots n_k - j \right) $$ $$ \Rightarrow \, \, \, j \ge n_1 \cdots n_k - (n_1 + n_2 \cdots +n_k - k) . $$ To see that our expectation is correct in the case where equality holds, it is enough to show generically finite fibers. Let $$ L = \Phi ( \bs{\alpha}) = \big\langle \alpha_1^1 \otimes \cdots \alpha_k^1, \, \ldots , \, \alpha_1^j \otimes \cdots \alpha_k^j \big\rangle^\perp , $$ so that $\PP (L^\perp)$ is a $(j-1)$-secant of the Segre variety $$ \Sigma_{n_1-1, \ldots, n_k-1}\subset \PP (V_1 \otimes \cdots \otimes V_k ),$$ which has dimension $(n_1 + n_2 \cdots +n_k - k).$ Since the dimensions of $\PP (L^\perp)$ and $\Sigma_{n_1-1, \ldots, n_k-1}$ are complementary, the classical "trisecant lemma" implies for generic $\bs{\alpha}$ that $$\PP (L^\perp ) \cap \Sigma _{n_1-1, \ldots, n_k-1} = \{ [\alpha_1^1 \otimes \cdots \alpha_k^1], \, \ldots , \, [\alpha_1^j \otimes \cdots \alpha_k^j] \}. $$

(See eg. Prop 1.3.3 in Russo's "Geometry of Special Varieties." The statement assumes characteristic $0,$ though I'm not sure if it's necessary for me.)

Thus, if $\Phi (\bs{\alpha} ) = \Phi (\bs{\alpha}')$ for $\bs{\alpha}$ generic, then $\bs{\alpha }$ equals $\bs{\alpha}'$ after permuting some of the $j$ factors in the domain.

I discovered the above argument for the case where $k=2$ and $(n_1, n_2) = (3,3),$ in a context apparently unrelated to tensors, and was surprised that it generalized with little effort. I am thus wondering if it is a well-known fact or if there is some simpler explanation.

A seemingly more difficult question I am also interested in: describe the constructible set $\im \Phi .$