$\newcommand{\complex}{\mathbb{C}}$ Let $Seg \subseteq \complex^M \otimes \complex^N$ be the set of elements of the form $v \otimes w$. It is well-known that a general linear subspace of dimension $(M-1)(N-1)$ intersects $Seg$ only in the point $\{0\}$, as this is the codimension of the algebraic variety $Seg$, and that every linear subspace of dimension $(M-1)(N-1)+1$ intersects $Seg$ at some non-zero point.
I would like to know if every linear subspace of dimension $(M-1)(N-1)+(M-1)$ contains in its intersection with $Seg$ a linear subspace of the form $H \otimes v$, where $H\subseteq \complex^M$ is a hyperplane and $v \in \complex^N$ is a non-zero vector.
More generally for any $k \in \{1,...,M-1\}$, does every linear subspace of dimension $(M-1)(N-1)+k$ contain in its intersection with $Seg$ a linear subspace of the form $H \otimes v$, where $H\subseteq \complex^M$ is a linear subspace of dimension $k$?
