Let $V,W$ be vector spaces and $X\subset V\times W.$ If $X$ is the zero set of a collection of bilinear maps then it satisfies the following properties:
- $(0,w),(v,0)\in X$ for all $v,w.$
- If $(v,w)\in X$ then $(\alpha v,w),(v, \alpha w)\in X$ for any scalar $\alpha.$
- If $(v_1,w),(v_2,w)\in X$ then $(v_1+v_2,w)\in X$ and the same for the second coordinate.
Now suppose a set $X\subsetneq V\times W$ satisfies conditions 1-3. Then is there always a non-zero bilinear map vanishing on $X?$ If so, is $X$ the zero set of a collection of bilinear maps?
These questions can be phrased in terms of the subspace spanned by $X$ when viewed in $V\otimes W,$ but that's as far as I've been able to get.