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!



A problem on Jets in algebraic geometry
Let $\mathfrak{m}$ (resp. $\mathfrak{m}\_0$) be the maximal ideal sheaf at $x$ (resp. $x\_0$). Put $L:=\mathcal{O}(d\_1,d\_2)$.


If I understand correctly, you are asking whether the restriction map 
$$H^0(X,L)\rightarrow (L/\mathfrak{m}^rL)\times (L/\mathfrak{m}\_0^rL)$$ (or maybe $\mathfrak{m}^{r+1}$, this depends on your conventions) is surjective. Because of the exact sequence
$$0\rightarrow \mathfrak{m}^r\mathfrak{m}\_0^rL \rightarrow L \rightarrow (L/\mathfrak{m}^rL)\times (L/\mathfrak{m\_0}^rL)\rightarrow 0\ ,$$
this is equivalent to the vanishing of $H^1(X, \mathfrak{m}^r\mathfrak{m}\_0^rL)$; this vanishing is guaranteed by Serre's theorem for $L$ ample enough, that is, $d\_1,d\_2\gg 0$.


