Let $V$, $W$ be varieties (affine or projective) over an algebraically closed field $K$.
Let $p \in V$ and $q \in W$. Is there a description of the local ring of $V\times W$ at $(p,q)$ in terms of the local ring of $V$ at $p$ and the local ring of $W$ at $q$? 



Local ring of product of varieties
Let $k$ be a field. The fiber product of two $k$-schemes $X,Y$ (or even locally ringed spaces, see [here](http://arxiv.org/pdf/1103.2139v1.pdf)) has as points triples $(x,y,\mathfrak{p})$, where $\mathfrak{p}$ is a prime ideal in $\kappa(x) \otimes\_k \kappa(y)$, or equivalenty a prime ideal $\mathfrak{q}$ in $\mathcal{O}\_{X,x} \otimes\_k \mathcal{O}\_{Y,y}$ which restricts to the maximal ideals in both factors. The stalk of the structure sheaf at $(x,y,\mathfrak{p})$ is $(\mathcal{O}\_{X,x} \otimes\_k \mathcal{O}\_{Y,y})\_\mathfrak{q}$. If $x,y$ are $k$-rational points, we have $\mathfrak{p}=0$ and therefore $\mathfrak{q}$ is the kernel of the natural map $\mathcal{O}\_{X,x} \otimes\_k \mathcal{O}\_{Y,y} \to k$, $f \otimes g \mapsto f(x) \otimes g(y)$. So the stalk of $X \times\_k Y$ at a rational point $(x,y)$ is a localization of the tensor product of the stalks, where those elements $\sum\_i f\_i \otimes g\_i$ with $\sum\_i f\_i(x) \cdot g\_i(y) \neq 0$ are inverted.


