Segre varieties contained in hyperplanes

Question

Recall, the Segre embedding is a map $\sigma: \mathbb{P}^m \times \mathbb{P}^n \to \mathbb{P}^{(m+1)(n+1)-1}$ given by \begin{equation} ([x_0:\cdots:x_m],[y_0:\cdots:y_n]) \mapsto [x_0y_0:x_0y_1: \cdots: x_iy_j:\cdots: x_my_n] \end{equation}

Call the image of $\sigma$ the Segre variety $\Sigma_{m,n}$. I vaguely remember reading somewhere that no Segre variety is contained in a hyperplane in $\mathbb{P}^{(m+1)(n+1)-1}$. This certainly feels true enough (since the defining equations for $\Sigma_{m,n}$ are quadratic, not linear), but writing down a proof in the general case seems difficult.

In fact, we may consider the following generalization. Let $X \subseteq \mathbb{P}^m$ and $Y \subseteq \mathbb{P}^n$ be varieties. The Segre product of $X$ and $Y$ is the image $\sigma(X \times Y)$ in $\mathbb{P}^{(m+1)(n+1)-1}$. Can someone supply a proof or reference that if $X$ and $Y$ are nonempty, then $\sigma(X \times Y)$ is never contained in a hyperplane? If this turns out to be false, what conditions on $X$ and $Y$ are needed to guarantee this is true?

Edit: It seems that this is false if either $X$ or $Y$ consists of points only. So perhaps one needs to require that the varieties be at least one-dimensional.

Answer 1

a) It is indeed true that $\Sigma_{m,n}$ is not contained in a hyperplane and, contrary to your fears, writing down a proof is not difficult. Actually it is child's play:

A hyperplane in $\mathbb{P}^{(m+1)(n+1)-1}$ has equation $\sum a_{ij}z_{ij}=0$ and to say that it contains $\Sigma_{m,n}$ means that $\sum a_{ij}x_iy_j=0 \:$ for all $(x,y)\in \mathbb{P}^m \times \mathbb{P}^n$.
But this forces every $a_{ij}$ to be zero [do you see which pair $ (x_i,y_j) \in \mathbb{P}^m \times \mathbb{P}^n$ shows that?], which is impossible for the equation of a hyperplane.

b) However it is quite possible that $\sigma(X \times Y)$ is contained in a hyperplane for subvarieties $X \subseteq \mathbb{P}^m$ and $Y \subseteq \mathbb{P}^n$:

Just take for $X$ the hyperplane $x_0=0$ of $\mathbb{P}^m$and for $Y$ the whole of $\mathbb{P}^n$.
The image $\sigma(X \times Y)$ is then contained in each of the hyperplanes $z_{00}=0,z_{01}=0,\cdots,z_{0n}=0 $ .