From now on we let $X$ be a given smooth connected projective $K$-analytic space in the sense of Berkovich. By a model of $X$ we will mean a normal and projective $S$-variety $\mathcal{X}$ together with the data of an isomorphism $\mathcal{X}^{\mathrm{an}}_{K}\simeq X$. The set $\mathcal{M}_{X}$ of models of $X$ is non-empty thanks to the non-Archimedean GAGA principle. Given $\mathcal{X}^{\prime},\mathcal{X}$ in $\mathcal{M}_{X}$ we write $\mathcal{X}^{\prime}\geq\mathcal{X}$ if there exists a vertical blow-up $\mathcal{X}^{\prime}\to\mathcal{X}$. This turns $\mathcal{M}_{X}$ (modulo isomorphism) into a directed set.

For any model $\mathcal{X}$ of $X$ and any irreducible component $E$ of the $\mathcal{X}_{0}$ there exists a unique point $x_{E}\in X\simeq\mathcal{X}_{K}^{\mathrm{an}}$ whose center of $\mathcal{X}$ is the generic point of $E$. Such points will be called divisorial points.¹¹ 1 Divisorial points are called Shilov boundaries in [YZ09]. The set $X^{\mathrm{div}}$ of divisorial points is dense in $X$, see Corollary 2.4 and also [Poi11].