Let $X$ be a smooth proper scheme over the non-archimedean field $K$. We assume that $K$ carries a discrete valuation $val$ such that $val(K^{\times})={{\bf Z}}$.

A model has no nontrivial automorphisms. Thus, the stack of equivalence classes of models is in fact a set, which we denote by $Mod_{X}$. It carries a natural partial order. Namely, we say that ${\cal X}_{1}\geq{\cal X}_{2}$ if there exists a map ${\cal X}_{1}\to{\cal X}_{2}$ over $Spec({{\cal O}}_{K})$. Such a map is automatically unique.

By the resolution of singularities, in the case $char\,k=0$ we know that every model is dominated by a model with normal crossings.

The set of equivalence classes of snc models will be denoted by $Mod_{X}^{snc}$. It is a filtered partially ordered set. The order is given by dominating maps of models which give the identity automorphism on the generic fiber.

It is easy to show that starting with any model with normal crossings and applying blow-ups centered at certain self-intersection loci of the special fiber we can get a snc model. In what follows we use snc models only. This choice is dictated by convenience and not by necessity. Working with snc models has the advantage that all definitions and calculations can be made very transparent. The reader can consult [Be2] for the approach in the general case, without the use of the resolution of singularities.

Let ${\cal X}$ be an snc model and $I=I_{{\cal X}}$ the set of irreducible components of ${\cal X}^{0}_{red}$. Denote by $D_{i}\subset{\cal X}$ the divisor corresponding to $i\in I$. For any finite non-empty subset $J\subset I$ put

$$D_{J}:=\bigcap_{j\in J}D_{j}\,\,.$$

By the snc property the set $D_{J}$ is either empty or is a smooth connected proper variety over $k$ of dimension $\dim(D_{J})=(n-|J|+1)$. For a divisor $D_{i}\subset{\cal X}^{0}$ we denote by $d_{i}\in{\bf Z}_{>0}$ the order of vanishing of $u$ at $D_{i}$, where $u\in K$ is an uniformizing element, $val_{K}(u)=1$. Equivalently, $d_{i}$ is the multiplicity of $D_{i}$ in ${\cal X}^{0}$.

Clearly, $S_{\cal X}$ is a nonempty connected CW-complex. We will also consider the cone over $S_{\cal X}$:

$$C_{\cal X}({\bf R}):=\left\{\sum_{i\in I}a_{i}\langle D_{i}\rangle|\,a_{i}\in{{\bf R}}_{\geq 0},\,\,\,\bigcap_{i:\,a_{i}>0}D_{i}\neq\emptyset\right\}\setminus\{0\}\subset{\bf R}^{I}\,\,.$$

Analogously, we can define $C_{\cal X}({\bf Z}),\,$.

We identify $S_{\cal X}$ with the following subset of $C_{\cal X}({\bf R})$:

$$\left\{\sum_{i\in I}a_{i}\langle D_{i}\rangle\in C_{\cal X}({\bf R})|\,\sum_{i}a_{i}d_{i}=1\right\}\,\,.$$

Obviously, we can also describe $S_{\cal X}$ as a quotient of $C_{\cal X}({\bf R})$:

$$S_{\cal X}=C_{\cal X}({\bf R})/{\bf R}^{\times}_{+}\,\,.$$