Let $k$ be a perfect field, let $n$ and $m$ be two positive integers.
Consider $X=\mathbb P_k^n\times \mathbb P_k^m$. Let $x_0=(1,0,\cdots,0;1,0,\cdots,0)\in X$ be fixed.
For any pair of integers $(d_1,d_2)$, let $\mathcal O(d_1,d_2)=pr_1^\ast \mathcal O(d_1)\otimes pr_2^\ast \mathcal O(d_2)$.
Let $A$ be a affine space(vector space) over $k$ which contains all sections of $\mathcal O(d_1,d_2)$.
For any $x\in X$ and any integer $r\geq 0$, let $B(x,r)$ be the affine space whose points are $r$-jets of sections of $\mathcal O(d_1,d_2)$.
For any $x\neq x_0$ and any two positive numbers $N$ and $M$, if $d_1$ and $d_2$ are large enough, I believe that the canonical restriction map is surjective $A\longrightarrow B(x_0,N)\times B(x,M)$.
But I don't know how to show it. Any comment is very welcome!