If $s\in\Delta_{\mathcal{X}}$ is a rational point then Theorem 3.11 yields a vertical blow-up $\mathcal{X}^{\prime}$ such that $\emb_{\mathcal{X}^{\prime}}(s)=x_{E^{\prime}}$ for some irreducible component $E^{\prime}$ of $\mathcal{X}^{\prime}$. Conversely, it $\emb_{\mathcal{X}}(s)$ is a divisorial point then the corresponding valuation takes rational values on the local equations of the components of $\mathcal{X}_{0}$, which shows that $s$ is a rational point of $\Delta_{\mathcal{X}}$. ∎