My question may be absolutely elementary and is probably answered in 19th century. A reference or a short clear argument would be highly appreciated.

Let $V_1, \ldots V_n$ be finite dimensional vector spaces over the same field (may assume complex numbers). What are $GL(V_1)\times \ldots \times GL(V_n)$-orbits on $V_1 \otimes \ldots \otimes V_n$?

The only invariant of an orbit I can see is "a multirank" $(k_1, \ldots k_n)$ where $k_i$ is the dimension of support of an element in $V_i$. The multirank satisfies inequalities $k_i \leq \prod_{j\neq i} k_j$. Would it be too naive to suggest that orbits are in 1-1 correspondence with legal multiranks?