Stalk of a generic point of the special fiber

Question

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 $X$)?

Answer 1

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.