Let $R$ be a complete discrete valuation ring with fraction field $K$ and residue field $k$. We shall assume that $k$ has characteristic zero. We let $t\in R$ be a uniformizing parameter and normalize the corresponding absolute value on $K$ by $\log|t|^{-1}=1$. Note that $R\simeq k[\![t]\!]$ and $K\simeq k(\!(t)\!)$, see for instance [Ser68]. Write $S:=\spec R$.

Let $X$ be a smooth projective $K$-variety, i.e. an integral (but not necessarily geometrically integral) smooth projective $K$-scheme. A model of $X$ is a normal, flat and projective $S$-scheme $\mathcal{X}$ with $X$ as its generic fiber. We denote by $\mathcal{X}_{0}$ its special fiber, and by $\Div_{0}(\mathcal{X})$ the group of vertical Cartier divisors, i.e. those supported in $\mathcal{X}_{0}$. We write $\Div_{0}(\mathcal{X})_{\mathbf{R}}$ accordingly.

Let $\mathcal{M}_{X}$ be the set of all isomorphism classes of models of $X$. Given $\mathcal{X}^{\prime},\mathcal{X}$ in $\mathcal{M}_{X}$ we write $\mathcal{X}^{\prime}\geq\mathcal{X}$ if there exists a morphism $\mathcal{X}^{\prime}\to\mathcal{X}$ obtained by blowing up an ideal sheaf co-supported on the special fiber of $\mathcal{X}$. This turns $\mathcal{M}_{X}$ into a directed set.

Given a model $\mathcal{X}$, let $(E_{i})_{i\in I}$ be the set of irreducible components of the special fiber. For each subset $J\subset I$ set $E_{J}:=\bigcap_{j\in J}E_{j}$. A regular model $\mathcal{X}$ is an SNC model if the special fiber has simple normal crossing support and $E_{J}$ is irreducible (or empty) for each $J\subset I$.

As a topological space, the Berkovich space $X^{\mathrm{an}}$ attached to the given smooth projective $K$-variety $X$ is compact and can be described as follows (cf. [Ber90, Theorem 3.4.1]). Choose a finite cover of $X$ by affine 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 $|\cdot|:A\to\mathbf{R}_{+}$ extending the given absolute value of $K$, endowed with the topology of pointwise convergence. The space $X^{\mathrm{an}}$ is obtained by gluing the open sets $U^{\mathrm{an}}$.

There is a natural equivalence of categories between projective $K$-analytic spaces and projective $K$-schemes, see [Ber90, §3.4]. In the sequel we shall therefore always identify a projective $K$-scheme with its associated Berkovich space and write $X^{\mathrm{an}}=X$.

Let $\mathcal{X}$ be a model of $X$. To each irreducible component $E$ of the special fiber is associated a divisorial valuation $\ord_{E}$ of the function field of $X$. After rescaling and exponentiating, this gives rise to an element $x_{E}\in X$ called a divisorial point. The set $X^{\mathrm{div}}$ of divisorial points is dense in $X$.

When $\mathcal{X}$ is an SNC model, we can refine this construction. Write the special fiber as $\mathcal{X}_{0}=\sum_{i\in I}b_{i}E_{i}$. The dual complex $\Delta_{\mathcal{X}}$ of $\mathcal{X}$ is the simplicial complex whose vertices correspond to the irreducible components $E_{i}$ and whose simplices correspond to nonempty intersections $E_{J}$. We can equip $\Delta_{\mathcal{X}}$ with an (integral) affine structure and embed it in the Berkovich space $X$ as follows.

Consider a subset $J\subset I$ with $E_{J}\neq\emptyset$ and pick $w=(w_{j})_{j\in J}$ with $w_{j}\geq 0$ and $\sum_{j\in J}b_{j}w_{j}=1$. Let $\xi_{J}$ be the generic point of $E_{J}$ and pick a system $(z_{j})_{j\in J}$ of regular parameters for $\mathcal{O}_{\mathcal{X},\xi_{J}}$ with $z_{j}$ defining $E_{j}$. By Cohen’s structure theorem, $\widehat{\mathcal{O}}_{\mathcal{X},\xi_{J}}\simeq\kappa(\xi_{J})[[z_{j},j\in J]]$. Let $v_{J,w}$ be the restriction to $\mathcal{O}_{\mathcal{X},\xi}$ of the monomial valuation on this power series ring, taking value $w_{j}$ on $z_{j}$, i.e. $v_{J,w}\left(\sum_{\alpha\in\mathbf{N}^{J}}c_{\alpha}z^{\alpha}\right)=\min\left\{\sum_{j\in J}w_{j}\alpha_{j}\mid c_{\alpha}\neq 0\right\}$. Then $e^{-v_{J,w}}\in X$. This defines an embedding $\emb_{\mathcal{X}}:\Delta_{\mathcal{X}}\to X$, and the parameters $w$ equip $\Delta_{\mathcal{X}}$ with an affine structure.

There is also a retraction $p_{\mathcal{X}}:X\to\Delta_{\mathcal{X}}$, defined as follows. Any point $x\in X$ admits a center on $\mathcal{X}$. This is the unique point $\xi=c_{\mathcal{X}}(x)\in\mathcal{X}_{0}$ such that $|\varphi|_{x}\leq 1$ for $\varphi\in\mathcal{O}_{\mathcal{X},\xi}$ and $|\varphi|_{x}<1$ for $\varphi\in\mathfrak{m}_{\mathcal{X},\xi}$. Let $J\subset I$ be the maximal subset such that $\xi\in E_{J}$. Then $p_{\mathcal{X}}(x)\in\Delta_{\mathcal{X}}$ corresponds to the monomial valuation with weight $-\log|z_{j}|_{x}$, $j\in J$.

We have $p_{\mathcal{X}}=\id$ on $\Delta_{\mathcal{X}}$. If $\mathcal{Y}$ dominates $\mathcal{X}$, then $\Delta_{\mathcal{X}}\subset\Delta_{\mathcal{Y}}$ and $p_{\mathcal{X}}\circ p_{\mathcal{Y}}=p_{\mathcal{X}}$. The retractions induce a homeomorphism of $X$ onto the inverse limit $\varprojlim\Delta_{\mathcal{X}}$.

In order to keep notation light, we shall identify $\Delta_{\mathcal{X}}$ with its image in $X$ under $\emb_{\mathcal{X}}$. Note that this convention differs from the one adopted in [BFJ11]. A point in $X$ lying in some dual complex $\Delta_{\mathcal{X}}$ is called quasi-monomial, and the set of such points is denoted by $X^{\mathrm{qm}}$.