What is the ring of germ of regular functions $\mathcal{O}_{X,p}$ when $X=\mathbb{V}(xy)\subset\mathbb{A}^2$ and for $p=(0,0)$ and $p=(0,1)$?
I am not clear about the definition of $\mathcal{O}_{X,p}$, so it will be greatly appreciated if you can show me in detail how to calculate the above two rings so I may get a better understanding of the concept
In general, for a variety $V \subset \mathbb A^2$, the ring $\mathcal O_{V,p}$ is just the localization of $k[x,y]/I_V$ at the prime ideal corresponding to $p$.
So if $p=(0,0)$, you localize at $\mathfrak m=(x,y)$. This means you are allowed to divide by everything outside $\mathfrak m$. But in this case, the ideal is contained in $\mathfrak m$, so "nothing changes", in the sense that the generators of the ideal is not affected. The elements of the local ring are now fractions $\frac{f}{g}$ where $g \not\in (x,y)$.
However, if you localize at $(x,y-1)$ (corresponding to $p=(0,1)$), then $y$ becomes a unit, so the local ring is $k[x,y]/(x) \approx k[y]$ (localized at $(y-1)$). This reflects the fact that near $p$, the variety just looks like a line, whereas near $(0,0)$, the variety looks like a cross.