Let $\mathcal{X}$ be a proper $S$-variety. Its generic fiber $\mathcal{X}_{K}$ is in particular a proper $K$-scheme. As a topological space, its $K$-analytification $\mathcal{X}_{K}^{\mathrm{an}}$ in the sense of Berkovich is compact and can be described as follows (cf. [Ber90, Theorem 3.4.1]). Choose a finite cover of $\mathcal{X}_{K}$ by Zariski open subsets of the form $U=\spec A$ where $A$ is a $K$-algebra of finite type. The Berkovich space $U^{\mathrm{an}}$ is defined as the set of all multiplicative seminorms $|\cdot|_{x}:A\to\mathbf{R}_{+}$ extending the given absolute value of $K$, endowed with the topology of pointwise convergence. It is common usage to write $|f(x)|:=|f|_{x}$. The space $\mathcal{X}_{K}^{\mathrm{an}}$ is then obtained by gluing together the open sets $U^{\mathrm{an}}$. There is a canonical continuous map $s_{\mathcal{X}}:\mathcal{X}_{K}^{\mathrm{an}}\to\mathcal{X}_{K}$, locally defined on $U^{\mathrm{an}}$ by setting

$$s_{\mathcal{X}}(x)=\left\{f\in A,\,|f(x)|=0\right\}.$$

The seminorm $|\cdot|_{x}$ defines a norm on the residue field $\kappa(s_{\mathcal{X}}(x))$, extending the given absolute value on $K$. The completion of $\kappa(s_{\mathcal{X}}(x))$ with respect to this norm is denoted $\mathcal{H}(x)$. It is the residue field at $x$ of the natural structure sheaf of $\mathcal{X}_{K}^{\mathrm{an}}$, that we will however not explicitely use.

Given $x\in\mathcal{X}_{K}^{\mathrm{an}}$ denote by $R_{x}$ the corresponding valuation ring in $\kappa(s_{\mathcal{X}}(x))$. By the valuative criterion of properness, the map $T_{x}:=\spec R_{x}\to S$ admits a unique lift $T_{x}\to\mathcal{X}$ mapping the generic point to $s_{\mathcal{X}}(x)$. In line with valuative terminology [Va00], we call the image of the closed point of $T_{x}$ in $\mathcal{X}$ the center of $x$ on $\mathcal{X},$ and denote it by $c_{\mathcal{X}}(x)$. It is a specialization of $s_{\mathcal{X}}(x)$ in $\mathcal{X}$. It also belongs to $\mathcal{X}_{0}$ since it maps to the closed point of $S$ by construction. The map $c_{\mathcal{X}}:\mathcal{X}_{K}^{\mathrm{an}}\to\mathcal{X}_{0}$ so defined is anti-continuous. It is referred to as the reduction map in rigid geometry.

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].

Let $\mathcal{X}$ be a model of $X$. Each vertical fractional ideal sheaf $\mathfrak{a}$ on $\mathcal{X}$ defines a continuous function $\log|\mathfrak{a}|\in C^{0}(X)$ by setting

(2.1)

$$\log|\mathfrak{a}|(x):=\max\left\{\log|f(x)|,\,f\in\mathfrak{a}_{c_{\mathcal{X}}(x)}\right\}.$$

In particular, each vertical Cartier divisor $D\in\Div_{0}(\mathcal{X})$ defines a vertical fractional ideal sheaf $\mathcal{O}_{\mathcal{X}}(D)$, hence a continuous function

$$\varphi_{D}:=\log|\mathcal{O}_{\mathcal{X}}(D)|.$$

Note that $\varphi_{\mathcal{X}_{0}}$ is the constant function $1$ since $\log|\varpi|^{-1}=1$. Since models are assumed to be normal, a vertical divisor $D$ is uniquely determined by the values $\varphi_{D}(x_{E})$ at divisorial points $x_{E}$, and we have in particular $\varphi_{D}\geq 0$ iff $D$ is effective. The map $D\mapsto\varphi_{D}$ extends by linearity to $\Div_{0}(\mathcal{X})_{\mathbf{R}}\to C^{0}(X)$.

Following [Yua08] we introduce the following terminology.

We shall also occasionally consider the similarly defined spaces $\mathcal{D}(X)_{\mathbf{Z}}$ and $\mathcal{D}(X)_{\mathbf{R}}$.

As a matter of terminology, we say that a model $\mathcal{X}$ is a determination of a model function $\varphi$ if $\varphi=\varphi_{D}$ for some $D\in\Div_{0}(\mathcal{X})_{\mathbf{Q}}$. By the above remarks we have a natural isomorphism

$$\varinjlim_{\mathcal{X}\in\mathcal{M}_{X}}\Div_{0}(\mathcal{X})_{\mathbf{Q}}\simeq\mathcal{D}(X)\subset C^{0}(X).$$

The next result summarizes the key properties of model functions. Since our setting does not require any machinery from rigid geometry we provide direct arguments for the convenience of the reader.

Thanks to the “Boolean ring version” of the Stone-Weierstrass theorem, we get as a consequence the following crucial result, which is equivalent to [Gub98, Theorem 7.12] (compare [Yua08, Lemma 3.5] and the remark following it).

Corollary 2.3 in turn implies the following result, which corresponds to [YZ09, Lemma 2.4]. We reproduce the short proof for completeness.

The collection of finite dimensional spaces $\Div_{0}(\mathcal{X})^{*}_{\mathbf{R}}\simeq\mathcal{D}(\mathcal{X})^{*}_{\mathbf{R}}$ endowed with the transpose of pull-back morphisms on divisors and the topology of the pointwise convergence forms an inductive system, and we have:

The image of this map will be described in Corollary 3.2.