Let $S$ be a surface (so a $2$-dimensional proper $k$-scheme). My question arises from following another thread of mine.
I'm looking for conditions for $S$ which garantee that the completion of every stalk $O_{S,s}$ with respect to it's unique max ideal $m_s$ has the shape $\widehat{\mathcal{O}}_{S,s}= k[[x,y,z]]/(f_s)$.
The structure of $f_s$ might be depend on if $s$ is regular or singular. More precisely I intuitively expect that if $s$ is regular then $f_s$ is linear and therefore without loss of generality $\widehat{\mathcal{O}}= k[[x,y,z]]/(f)\cong k[[x,y]]$. Otherwise $f\in m_s ^n $ for $n\neq1$.