I'm looking for a reference or at least some help for the following construction.
Suppose we have $\mathbf{P}^2_k$ the projective plane over some algebraically closed field $k$ (possibly of positive characteristic, although it might not matter) and let $D$ be a divisor on it.
There is a valuation $v_D$ on the field $K:=k(s,t)$ of rational functions of $\mathbf{P}^2_k$, which is the usual divisorial valuation with centre on $D$. Now, one can complete $K$ with respect to $v$ and obtain thus the completion $\widehat{K}$.
I am wondering if there is an analogous of this operation on the "geometric side", something like completing $\mathbf{P}^2_k$ along $D$. More precisely, I would expect that this completion along $D$ had function field exactly $\widehat{L}$.
Is there something of this kind and how is it performed? Thank you.