Set $S:=\operatorname{Spec}k[\![{t}]\!]$. Following the convention of [MN15], we define a model of $X$ to be a normal scheme ${\mathcal{X}}$, flat and of finite type (but possibly non-proper) over $S$, together with an identification of the generic fiber of the structure morphism $\pi\colon{\mathcal{X}}\to S$ with $X$.

For any two models ${\mathcal{X}}$, ${\mathcal{X}}^{\prime}$, the identifications of the generic fibers with $X$ induces a unique birational map ${\mathcal{X}}^{\prime}\dashrightarrow{\mathcal{X}}$. We say that ${\mathcal{X}}^{\prime}$ dominates ${\mathcal{X}}$ if this map is a morphism. Any two models can be dominated by a third.

For any model ${\mathcal{X}}$ and every irreducible component $E$ of ${\mathcal{X}}_{0}$, we set $b_{E}:=\operatorname{ord}_{E}({t})$, and view the divisorial valuation

$$v_{E}:=b_{E}^{-1}\operatorname{ord}_{E}$$

as an element of $X^{\operatorname{val}}\subset X^{\mathrm{an}}$. The set of such points is a dense subset $X^{\mathrm{div}}\subset X^{\mathrm{an}}$.

We usually denote by ${\mathcal{X}}_{0}=\sum_{i\in I}b_{i}E_{i}$ the irreducible decomposition of the central fiber, and write $E_{J}:=\bigcap_{i\in J}E_{i}$ for $J\subset I$. We say that ${\mathcal{X}}$ is snc if (${\mathcal{X}}$ is regular and) ${\mathcal{X}}_{0,\mathrm{red}}$ has simple normal crossing support. Since $k$ has characteristic $0$, this means that each non-empty $E_{J}$ is smooth over $k$, of codimension $|J|$ in ${\mathcal{X}}$.

More generally, a model ${\mathcal{X}}$ is toroidal if ${\mathcal{X}}\setminus{\mathcal{X}}_{0}\subset{\mathcal{X}}$ is a strict toroidal embedding in the sense of [KKMS], i.e. is formally isomorphic, at each closed point of ${\mathcal{X}}_{0}$, to the inclusion of ${\mathbb{G}}_{m,k}^{n+1}$ in a toric $k$-variety, and such that each $E_{i}$ is normal (which then implies that each non-empty $E_{J}$ is normal).

Every model ${\mathcal{X}}$ contains a largest snc Zariski open subset ${\mathcal{X}}_{\mathrm{snc}}\subset{\mathcal{X}}$. By Temkin’s version of Hironaka’s theorem [Tem12], ${\mathcal{X}}$ is dominated by an snc model ${\mathcal{X}}^{\prime}$ such that the induced birational morphism $\mu\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ is projective, and an isomorphism over ${\mathcal{X}}_{\mathrm{snc}}$.

If ${\mathcal{X}}$ is a model of $X$, the set ${\mathcal{X}}^{\mathrm{an}}\subset X^{\mathrm{an}}$ of semivaluations that admit a center (or reduction) on ${\mathcal{X}}_{0}$ is a closed subset; it can be viewed as the generic fiber of a suitable formal scheme [MN15, 2.2.2]. By the valuative criterion of properness, we have ${\mathcal{X}}^{\mathrm{an}}={\mathcal{X}}^{\prime\mathrm{an}}$ for each proper morphism of models ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$, and ${\mathcal{X}}^{\mathrm{an}}=X^{\mathrm{an}}$ if ${\mathcal{X}}$ is proper (over $S$, that is). The reduction map $c_{\mathcal{X}}\colon{\mathcal{X}}^{\mathrm{an}}\to{\mathcal{X}}_{0}$, taking a semivaluation to its center, is anticontinuous.⁷⁷ 7 Anticontinuity means that the inverse image of an open set is closed.

The set ${\mathcal{X}}^{\mathrm{div}}:={\mathcal{X}}^{\mathrm{an}}\cap X^{\mathrm{div}}$ consists of all divisorial valuations $v$ on $F({\mathcal{X}})=F(X)$ that are centered on ${\mathcal{X}}_{0}$, trivial on $k$ and such that $v({t})=1$.