# Stalk of a generic point of the special fiber

Im trying to understand the paper Weight functions on non-archimedean analytic spaces and the kontsevich-soibelman skeleton by Johannes Nicaise.

Let $R$ be a discrete valuation ring and $K=Frac(R)$. Let $X$ be a normal integral separated $K-$scheme of finite type and $\mathscr{X}$ a normal $R-$model.

An $R-$model $\mathscr{X}$ of $X$ is a normal flat separated $R-$scheme of finite type endowed with an isomorphism of $K-$schemes $\mathscr{X}_{K} \longrightarrow X$.

Why if we consider $E=\overline{\{\xi\}}$ an irreducible component of the special fiber $\mathscr{X}_{k}$ ( where $\xi$ is the generic point of $E$), then $\mathcal{O}_{\mathscr{X,\xi}}$ is a discrete valuation ring with fraction field $K(X)=\mathcal{O}_{X,\eta}$ (where $\eta$ is the generic point of $X$)?

• Welcome to Math.SE. Your problem presentation is tightly packed with specialized terminology, but it could be improved by using some whitespace to divide the problem setup from the question formulation. Your own thoughts about how to approach such a problem, what motivates you interest in it, or a citation to the source of the problem would all make valuable additions to the Question. – hardmath Jun 13 '18 at 20:37
• What is the "point" $\xi$ ? – random123 Jun 14 '18 at 6:08

Let $\pi : X \rightarrow Spec(R)$ is the structure morphisms. Since $X$ is normal, then irreducible and connected components of $X$ are the same.
Let $X = \cup X_i$, where $X_i$ are irreducible components. Thus each of the $X_i$ are connected components and hence open. Now, since $X$ is flat, we get each of the $X_i$ is flat over $Spec(R)$, hence $\pi(X_i)$ is also open and hence contains the generic point. It follows that the generic fiber is not irreducible, a contradiction. Thus we get $\mathscr{X}$ is irreducible.
Now since $\pi$ is flat and $X$ is irreducible and thus equi-dimensional, each of the irreducible components of all of the fibers, we get $E$ is one dimensional. Hence $\xi$ is a codimension one point in $X$ and $X$ is normal, hence the local ring $\mathcal{O}_{\mathscr{X},\xi}$ is a discrete valuation ring.