Computing germs of a projective curve Suppose $X$ is a projective curve given by $$X:=Z(x_1^n+x_2^n-x_3^n)\subseteq\mathbb{P}_\mathbb{C}^2$$
Then, I want to compute the germs $\mathcal{O}_{X,P}$ and the maximal ideal $M_P:=\{\varphi\in\mathcal{O}_{X,P} : \varphi (P)=0\}$ of $\mathcal{O}_{X,P}$, for all $P=(a_1,a_2,a_3)\in X$.
We have,
\begin{align}
\mathcal{O}_{X,P}& =\{f/g : f,g\in k[x_1,x_2,x_3]/\langle x_1^n+x_2^n-x_3^n\rangle \mbox{ are homogeneous }\, \deg (f)=\deg (g), g(P)\ne 0\}\\
& =\{f/g : f,g\mbox{ are homogenous},\deg (f)=\deg (g),\; g\not\in \langle x_1-a_1,x_2-a_2,x_3-a_3\rangle\}
\end{align}
I am unable to proceed any further. Is it isomorphic to some nice ring ? Any hints or ideas would be helpful. Thanks.
 A: The general situation is very strange!    
Given a smooth projective curve $X$,  each of its local rings $\mathcal O_{X,x}$ is the personal property of $X$: no other (= not isomorphic) curve in the universe has a single local ring isomorphic to $\mathcal O_{X,x}$!
And why is that?
Because the fraction field $\operatorname {Rat}(X)=\operatorname {Frac }(\mathcal O_{X,x})$ charactrizes $X$: this is the amazing equivalence of categories between function fields of dimension one and smooth projective curves over an algebraically closed field (cf.  Hartshorne, Chapter I, Corollary 6.12, page45).
So different curves cannot have isomorphic local rings.
Since there are continuously many such curves (they are organized into a so called moduli space), there are continuously many local rings and there is no hope to explicitly describe them all.  
To sum up, such local rings are not isomorphic to any otherwise defined "nice rings".   
Edit
Yuchen judicious comment reminds me  I should have said that completing the local rings leads to a drastic simplification:
Every completion $\widehat{ \mathcal O_{X,x}}$ is isomorphic to the ring of formal power series $\mathbb C[[z]]$    
Also by analytification, a less brutal completion process , every  local ring of a smooth curve becomes a local ring $\mathcal O_{X,x}^{an}$ isomorphic to the ring of convergent power series $\mathbb C\{z\}$ (=power series with nonzero radius of convergence).
Summing up we have the hierarchy $$ \mathcal O_{X,x}\subsetneq  \mathcal O_{X,x}^{an}=  \mathbb C\{z\}\subsetneq  \widehat{ \mathcal O_{X,x}}= \mathbb C[[z]] $$
Finally, over an arbitrary field there is another type of completion, called henselization and essential for étale cohomology, which has some pleasant  properties.
