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 $X$)?