$\DeclareMathOperator\GL{GL}$Let $X$, $Y$, $Z$ be $\mathbb{C}$-vector spaces, and let $A\subseteq X$ and $B\subseteq Y$ and $C\subseteq Z$ be linear subspaces. Let $V=X \otimes Y \otimes Z$, acted on algebraically by $G\mathrel{:=}\GL(X)\times \GL(Y)\times \GL(Z)$ in the standard way. Let $W=A \otimes B \otimes C \subseteq V$.

For each $l \in \mathbb{N}$, define \begin{align}\tag{1}\label{1} S_l=\{v \in V : \dim(\overline{G \cdot v} \cap W)\leq l\}. \end{align}

Is it true that $S_l \subseteq V$ is Zariski closed for all $l \in \mathbb{N}$? If this statement is true, I would prefer a proof that uses classical algebraic group theory/algebraic geometry, as I am not very familiar with sheaves and schemes.

Note that if $W=V$, then this becomes a very standard result in algebraic group theory (see e.g. Lemma 1.4 in Brion - Introduction to actions of algebraic groups).



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