2.1.1. According to [Reference KatoKat94, Definition 9.1], a monoidal space $(T,{\mathcal{M}}_{T})$ is a topological space $T$ endowed with a sharp sheaf of monoids ${\mathcal{M}}_{T}$ , where sharp means that ${\mathcal{M}}_{T,t}^{\times }=\{1\}$ for every $t\in T$ . We often denote the monoidal space simply by $T$ .

A morphism of monoidal spaces is a pair $(f,\unicode[STIX]{x1D711}):(T,{\mathcal{M}}_{T})\rightarrow (T^{\prime },{\mathcal{M}}_{T^{\prime }})$ such that $f:T\rightarrow T^{\prime }$ is a continuous function of topological spaces and $\unicode[STIX]{x1D711}:f^{-1}({\mathcal{M}}_{T})\rightarrow {\mathcal{M}}_{T^{\prime }}$ is a sheaf homomorphism such that $\unicode[STIX]{x1D711}_{t}^{-1}(\{1\})=\{1\}$ for every $t\in T$ .

2.1.4. A monoidal space isomorphic to the monoidal space $\text{Spec}\,P$ for some monoid $P$ is called an affine Kato fan. A monoidal space is called a Kato fan if it has an open covering consisting of affine Kato fans. In particular, we call a Kato fan integral, saturated, of finite type or fs if it admits a cover by the spectra of monoids with the respective properties.

2.1.5. A morphism of fs Kato fans $F^{\prime }\rightarrow F$ is called a subdivision if it has finite fibers and the morphism

$$\begin{eqnarray}\text{Hom}(\text{Spec}\,\mathbb{N},F^{\prime })\rightarrow \text{Hom}(\text{Spec}\,\mathbb{N},F)\end{eqnarray}$$

is a bijection. By allowing subdivisions, a Kato fan might take the following shape.

The strategy of the proof of Proposition 2.1.6 goes back to [Reference Kempf, Knudsen, Mumford and Saint-DonatKKMS73] and relies on a sequence of particular subdivisions of the Kato fan, the so-called star and barycentric subdivisions [Reference Abramovich, Chen, Marcus, Ulirsch, Wise, Baker and PayneACMUW16, Example 4.10].

Proof. First, we show that every such generic point is a point of $F$ . Let $E_{1},\ldots ,E_{r}$ be irreducible components of $D_{\mathscr{X}}$ and let $x$ be a generic point of the intersection $E_{1}\cap \cdots \cap E_{r}$ . We set $d=\dim {\mathcal{O}}_{\mathscr{X},x}$ . Since $\mathscr{X}^{+}$ is log-regular, we know that ${\mathcal{O}}_{\mathscr{X},x}/{\mathcal{I}}_{\mathscr{X},x}$ is regular and that

(2.2.4) $$\begin{eqnarray}d=\dim {\mathcal{O}}_{\mathscr{X},x}/{\mathcal{I}}_{\mathscr{X},x}+\text{rank}\,{\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}.\end{eqnarray}$$

We denote by $V({\mathcal{I}}_{\mathscr{X},x})$ the vanishing locus of the ideal ${\mathcal{I}}_{\mathscr{X},x}$ in $\text{Spec}\,({\mathcal{O}}_{\mathscr{X},x})$ . We want to prove that ${\mathcal{I}}_{\mathscr{X},x}=\mathfrak{m}_{x}$ . We assume the contrary, hence that ${\mathcal{I}}_{\mathscr{X},x}\subsetneq \mathfrak{m}_{x}$ . This assumption implies that there exists $j$ such that $V({\mathcal{I}}_{\mathscr{X},x})\nsubseteq E_{j}$ : indeed, if the vanishing locus is contained in each irreducible component $E_{i}$ , that is,

$$\begin{eqnarray}V({\mathcal{I}}_{\mathscr{X},x})\subseteq E_{1}\cap \cdots \cap E_{r}\subseteq \overline{\{x\}},\end{eqnarray}$$

then ${\mathcal{I}}_{\mathscr{X},x}\supseteq \mathfrak{m}_{x}$ . From the assumption of log-regularity it follows that the vanishing locus $V({\mathcal{I}}_{\mathscr{X},x})$ is a regular subscheme, and, moreover, that $\mathscr{X}^{+}$ is Cohen–Macaulay by [Reference KatoKat94, Theorem 4.1]. Thus, there exists a regular sequence $(f_{1},\ldots ,f_{l})$ in ${\mathcal{I}}_{\mathscr{X},x}$ , where $l$ is the codimension of $V({\mathcal{I}}_{\mathscr{X},x})$ , that is,

$$\begin{eqnarray}\dim {\mathcal{O}}_{\mathscr{X},x}/{\mathcal{I}}_{\mathscr{X},x}=d-l.\end{eqnarray}$$

Moreover, by the equality (2.2.4), $\text{rank}\,{\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}=l$ .

We claim that the residue classes of these elements $f_{i}$ in ${\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}$ are linearly independent. Assume the contrary. If $l=1$ the proposition holds because $f_{1}$ is not a unit. Otherwise, $l\geqslant 2$ and, up to renumbering the $f_{i}$ , there exist an integer $e$ with $1<e\leqslant l$ , non-negative integers $a_{1},\ldots ,a_{l}$ , not all zero, and a unit $u$ in ${\mathcal{O}}_{\mathscr{X},x}$ such that

$$\begin{eqnarray}f_{1}^{a_{1}}\cdot \ldots \cdot f_{e-1}^{a_{e-1}}=u\cdot f_{e}^{a_{e}}\cdot \ldots \cdot f_{l}^{a_{l}}.\end{eqnarray}$$

This contradicts the fact that $(f_{1},\ldots ,f_{l})$ is a regular sequence in ${\mathcal{I}}_{\mathscr{X},x}$ . Thus, the classes $\overline{f_{1}},\ldots ,\overline{f_{l}}$ are independent in ${\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}$ . As we also have the equality $\text{rank}\,{\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}=l$ , it follows that these classes generate ${\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}\otimes _{\mathbb{Z}}\mathbb{Q}$ .

Let $g_{j}$ be a non-zero element of the ideal ${\mathcal{I}}_{\mathscr{X},x}$ that vanishes along $E_{j}$ : it necessarily exists as otherwise $E_{j}$ is not a component of the divisor $D_{\mathscr{X}}$ . Then $g_{j}$ satisfies

$$\begin{eqnarray}g_{j}^{N}=v\cdot f_{1}^{b_{1}}\cdot \ldots \cdot f_{l}^{b_{l}}\end{eqnarray}$$

with $b_{i}\in \mathbb{Z}$ , $v$ is a unit in ${\mathcal{O}}_{\mathscr{X},x}$ and $N$ is a positive integer. As $g_{j}$ vanishes along the irreducible component $E_{j}$ , at least one of the functions $f_{1},\ldots ,f_{l}$ has to vanish along $E_{j}$ : assume that it is  $f_{1}$ .

On the one hand, as $f_{1}$ is identically zero on $E_{j}$ , the trace of $E_{j}$ on $V({\mathcal{I}}_{\mathscr{X},x})$ has codimension at most $l-1$ in $E_{j}$ at the point $x$ . On the other hand, we assumed that $V({\mathcal{I}}_{\mathscr{X},x})$ is not contained in $E_{j}$ and it has codimension $l$ in ${\mathcal{O}}_{\mathscr{X},x}$ . Then the trace of $E_{j}$ on $V({\mathcal{I}}_{\mathscr{X},x})$ has codimension $l$ in $E_{j}$ at $x$ . This is a contradiction. We conclude that the ideal ${\mathcal{I}}_{\mathscr{X},x}$ is equal to the maximal ideal $\mathfrak{m}_{x}$ , therefore $x$ is a point of $F$ .

It remains to prove the converse implication: every point $x$ of the fan $F$ is a generic point of an intersection of irreducible components of $D_{\mathscr{X}}$ . Let $x$ be a point of $F$ : by construction of Kato fan $F$ , the maximal ideal of ${\mathcal{O}}_{\mathscr{X},x}$ is equal to ${\mathcal{I}}_{\mathscr{X},x}$ , thus it is generated by elements in ${\mathcal{M}}_{\mathscr{X},x}$ . The zero locus of such an element is contained in $D_{\mathscr{X}}$ by definition of the logarithmic structure on $\mathscr{X}^{+}$ . Therefore, the zero locus of a generator of $\mathfrak{m}_{x}$ in ${\mathcal{M}}_{\mathscr{X},x}$ is a union of irreducible components of the trace of $D_{\mathscr{X}}$ on $\text{Spec}\,{\mathcal{O}}_{\mathscr{X},x}$ and $x$ is a generic point of the intersection of all such irreducible components.◻

2.2.7. The construction of the Kato fan of a log scheme defines a functor from the category of log-regular log schemes to the category of Kato fans. Indeed, given a morphism of log schemes $\mathscr{X}^{+}\rightarrow \mathscr{Y}^{+}$ , we consider the embedding of the associated Kato fan $F_{\mathscr{X}}$ in $\mathscr{X}^{+}$ and the canonical morphism $\mathscr{Y}^{+}\rightarrow F_{\mathscr{Y}}$ : the composition

$$\begin{eqnarray}F_{\mathscr{X}}{\hookrightarrow}\mathscr{X}^{+}\rightarrow \mathscr{Y}^{+}\rightarrow F_{\mathscr{ Y}}\end{eqnarray}$$

functorially induces a map between associated Kato fans. Moreover, this association preserves strict morphisms [Reference UlirschUli17, Lemma 4.9].

2.3.2. It follows that given any subdivision $F^{\prime }\rightarrow F$ of the Kato fan F associated with a log-regular log scheme $\mathscr{X}^{+}$ , we can construct a log scheme over $\mathscr{X}^{+}$ with prescribed associated Kato fan $F^{\prime }$ . Combining this fact with Propositions 2.1.6 and 2.2.6 yields a construction of resolutions of log schemes in the following sense: for any log-regular log scheme over $S^{+}$ we can find a birational modification by a regular log scheme with snc special fiber. Moreover, the morphism of log schemes ${\mathscr{X}^{\prime }}^{+}\rightarrow \mathscr{X}^{+}$ is obtained by a log blow-up [Reference NiziołNiz06, Theorem 5.8].

2.4.1. Given morphisms of fs log schemes $f_{1}:\mathscr{X}_{1}^{+}\rightarrow \mathscr{Y}^{+}$ and $f_{2}:\mathscr{X}_{2}^{+}\rightarrow \mathscr{Y}^{+}$ , their fibred product exists in the category of log schemes. It is obtained by endowing the usual fibred product of schemes

2.4.3. In general, the fibred product is not fs, but the category of fs log schemes also admits fibred products. Keeping the same notation, the following is a local description of the fibred product in the category of fine and saturated log schemes

$$\begin{eqnarray}\mathscr{X}_{1}^{+}\times _{\mathscr{ Y}^{+}}^{\mathit{ fs}}\mathscr{X}_{2}^{+}=(\mathscr{X}_{1}^{+}\times _{\mathscr{ Y}^{+}}\mathscr{X}_{2}^{+})\times _{\mathbb{ Z}[{Q_{1}\oplus }_{P}Q_{2}]}\mathbb{Z}[({Q_{1}\oplus }_{P}Q_{2})^{\text{sat}}]\end{eqnarray}$$

in terms of the charts for $f_{1}$ and $f_{2}$ [Reference BultotBul15, 3.6.16]. We remark that the two fibred products above may not only have different log structures, but also the underlying schemes may differ. Nevertheless, this obviously does not occur when the monoid ${Q_{1}\oplus }_{P}Q_{2}$ is saturated.

2.4.4. Log-smoothness is preserved under fs base change and composition [Reference Gabber and RameroGR04, Proposition 12.3.24]. In particular, if $f_{1}:\mathscr{X}_{1}^{+}\rightarrow \mathscr{Y}^{+}$ is log-smooth and $\mathscr{X}_{2}^{+}$ is log-regular, then $\mathscr{X}_{1}^{+}\times _{\mathscr{Y}^{+}}^{\mathit{fs}}\mathscr{X}_{2}^{+}$ is log-regular, by [Reference KatoKat94, Theorem 8.2].

Consider log-smooth morphisms of fs log schemes $\mathscr{X}_{1}^{+}\rightarrow \mathscr{Y}^{+}$ and $\mathscr{X}_{2}^{+}\rightarrow \mathscr{Y}^{+}$ . The sheaves of logarithmic differentials are related by the isomorphism

(2.4.5) $$\begin{eqnarray}p_{1}^{\ast }\unicode[STIX]{x1D6FA}_{\mathscr{X}_{1}^{+}/\mathscr{Y}^{+}}^{\log }\oplus p_{2}^{\ast }\unicode[STIX]{x1D6FA}_{\mathscr{X}_{2}^{+}/\mathscr{Y}^{+}}^{\log }\simeq \unicode[STIX]{x1D6FA}_{\mathscr{X}_{1}^{+}\times _{\mathcal{Y}^{+}}^{\mathit{fs}}\mathscr{X}_{2}^{+}/\mathscr{Y}^{+}}^{\log }\end{eqnarray}$$

by [Reference Gabber and RameroGR04, Proposition 12.3.13]. Furthermore, by assumption of log-smoothness over $S^{+}$ the logarithmic differential sheaves are locally free of finite rank [Reference KatoKat94, Proposition 3.10] and we can consider their determinants; they are called log canonical bundles and denoted by $\unicode[STIX]{x1D714}^{\log }$ . The following isomorphism is a direct consequence of (2.4.5):

(2.4.6) $$\begin{eqnarray}p_{1}^{\ast }\unicode[STIX]{x1D714}_{\mathscr{X}_{1}^{+}/\mathscr{Y}^{+}}^{\log }\otimes p_{2}^{\ast }\unicode[STIX]{x1D714}_{\mathscr{X}_{2}^{+}/\mathscr{Y}^{+}}^{\log }\simeq \unicode[STIX]{x1D714}_{\mathscr{X}_{1}^{+}\times _{\mathcal{Y}^{+}}^{\mathit{fs}}\mathscr{X}_{2}^{+}/\mathscr{Y}^{+}}^{\log }.\end{eqnarray}$$

2.4.7. Similarly to the construction of fibred products of fs log schemes, the category of fs Kato fans admits fibred products: on affine Kato fans $F=\text{Spec}\,P$ and $G=\text{Spec}\,Q$ over $H=\text{Spec}\,T$ , $F\times _{H}G$ is the spectrum of the amalgamated sum $(P\oplus _{T}Q)^{\text{sat}}$ in the category of fs monoids [Reference UlirschUli19, Proposition 2.4] and on the underlying topological spaces this coincides with the usual fibred product.

2.4.10. For any pair of points $(x,y)$ in $F_{\mathscr{X}}\times _{F_{\mathscr{T}}}F_{\mathscr{Y}}$ , we denote by $n(x,y)$ the number of preimages of $(x,y)$ in the Kato fan of $\mathscr{Z}^{+}$ under the local isomorphism (2.4.9).

Proof. Let $z^{\prime }$ be a preimage of the pair $(x^{\prime },y^{\prime })$ . By Proposition 2.4.8, there exists an open neighborhood $U_{z^{\prime }}$ of $z^{\prime }$ such that the restriction of $F_{\mathscr{Z}}\rightarrow F_{\mathscr{X}}\times _{F_{\mathscr{T}}}F_{\mathscr{Y}}$ to $U_{z^{\prime }}$ is an isomorphism onto its image. In particular, $(x,y)$ lies in this image. Thus, there exists a unique preimage of $(x,y)$ that is contained in $U_{z^{\prime }}$ . It follows that $n(x^{\prime },y^{\prime })\geqslant n(x,y)$ .◻

2.5.1. We recall that a log-regular log scheme $\mathscr{X}^{+}$ is said to be semistable if the special fiber is reduced. We will see that semistability is a sufficient condition to establish injectivitiy of the local isomorphism (2.4.9).

2.5.2. Given a log-regular log scheme $\mathscr{X}^{+}$ over $S^{+}$ , the morphism $f:\mathscr{X}^{+}\rightarrow S^{+}$ is called saturated if for any $x\in \mathscr{X}$ the morphism ${\mathcal{C}}_{S,f(x)}\rightarrow {\mathcal{C}}_{\mathscr{X},x}$ on the stalks of the characteristic sheaves is a saturated morphism of monoids; that is, if for any $x$ and any morphism $u:{\mathcal{C}}_{S,f(x)}\rightarrow P$ of fs monoids, the amalgamated sum ${{\mathcal{C}}_{\mathscr{X},x}\oplus }_{{\mathcal{C}}_{S,f(x)}}P$ is still a saturated monoid.

Following the work by T. Tsuji in an unpublished 1997 preprint, Vidal in [Reference Vidal, Adolphson, Baldassarri, Berthelot, Katz and LoeserVid04] defines the notion of saturation indexes for morphisms of log schemes. In the case of a log-regular log scheme $\mathscr{X}^{+}$ over $S^{+}$ , the saturation index at a point $x\in \mathscr{X}_{k}$ is the least common multiple of the multiplicities in $\mathscr{X}_{k}$ of the prime components of $\mathscr{X}_{k}$ passing through $x$ . The following criterion holds.

Proof. Since $\mathscr{X}^{+}$ is a semistable log-regular log scheme over $S^{+}$ , the saturation index of $\mathscr{X}^{+}\rightarrow S^{+}$ at any point $x$ in $\mathscr{X}_{k}$ is $1$ . Thus, by Lemma 2.5.3 the morphism $\mathscr{X}^{+}\rightarrow S^{+}$ is saturated, and it follows that the fibred product in the category of log schemes coincides with the fibred product in the category of fs log schemes. This fact is crucial in the sequel of the proof to describe explicitly the divisor $D_{\mathscr{Z}}$ , and to characterize the points of $\mathscr{Z}_{k}$ .

We write $D_{\mathscr{X}}=\unicode[STIX]{x1D6E5}_{\mathscr{X}}+\mathscr{X}_{k}$ , where $\unicode[STIX]{x1D6E5}_{\mathscr{X}}=\sum _{i}\unicode[STIX]{x1D6E5}_{\mathscr{X},i}$ is the horizontal part and $\mathscr{X}_{k}=\sum _{i}E_{\mathscr{X},i}$ ; and $D_{\mathscr{Y}}=\unicode[STIX]{x1D6E5}_{\mathscr{Y}}+\mathscr{Y}_{k,\text{red}}$ , with $\unicode[STIX]{x1D6E5}_{\mathscr{Y}}=\sum _{j}\unicode[STIX]{x1D6E5}_{\mathscr{Y},j}$ and $\mathscr{Y}_{k,\text{red}}=\sum _{j}E_{\mathscr{Y},j}$ . The divisor $D_{\mathscr{Z}}$ , associated to the log structure of the fs fibred product, is given by $D_{\mathscr{Z}}=\unicode[STIX]{x1D6E5}_{\mathscr{Z}}+\mathscr{Z}_{k,\text{red}}$ , where $\unicode[STIX]{x1D6E5}_{\mathscr{Z}}=\unicode[STIX]{x1D6E5}_{\mathscr{X}}\times \mathscr{Y}+\mathscr{X}\times \unicode[STIX]{x1D6E5}_{\mathscr{Y}}=\sum _{l}\unicode[STIX]{x1D6E5}_{\mathscr{Z},l}$ and $\mathscr{Z}_{k,\text{red}}=\sum _{h}E_{\mathscr{Z},h}.$ As $k$ is algebraically closed, the irreducible components of $\mathscr{Z}_{k}=\mathscr{X}_{k}\times _{k}\mathscr{Y}_{k}$ are given by the products of irreducible components of $\mathscr{X}_{k}$ and $\mathscr{Y}_{k}$ ; that is, for any $h$ we have $E_{\mathscr{Z},h}=E_{\mathscr{X},i_{h}}\times _{k}E_{\mathscr{Y},j_{h}}$ .

The points in the special fiber $\mathscr{Z}_{k}$ are characterized as follows:

$$\begin{eqnarray}z=(x,y,s,\mathfrak{p})\quad \text{and}\quad {\mathcal{O}}_{\mathscr{Z},z}=({\mathcal{O}}_{\mathscr{X},x}\otimes _{R}{\mathcal{O}}_{\mathscr{Y},y})_{\mathfrak{p}},\end{eqnarray}$$

where $x$ and $y$ are points of $\mathscr{X}^{+}$ and $\mathscr{Y}^{+}$ both mapped to $s$ , while $\mathfrak{p}$ is a prime ideal of the tensor product of residue fields $\unicode[STIX]{x1D705}(x)\otimes _{k}\unicode[STIX]{x1D705}(y)$ . We assume now that $z$ is a point in $F_{\mathscr{Z}}\cap \mathscr{Z}_{k}$ , and see the restrictions on $x$ , $y$ , and $\mathfrak{p}$ that follow from this assumption.

1. (i) We claim that the projections $x$ and $y$ are points of the respective Kato fans. By Lemma 2.2.3, $z$ is a generic point of an intersection of irreducible components of $D_{\mathscr{Z}}$ .

   If every irreducible component of $D_{\mathscr{Z}}$ containing $z$ is in the special fiber, then the intersection is given by

   $$\begin{eqnarray}\mathop{\bigcap }_{h=1}^{r}E_{\mathscr{ Z},h}=\mathop{\bigcap }_{h=1}^{r}E_{\mathscr{ X},i_{h}}\times E_{\mathscr{Y},j_{h}}=\bigg(\mathop{\bigcap }_{h=1}^{r}E_{\mathscr{ X},i_{h}}\bigg)\times \bigg(\mathop{\bigcap }_{h=1}^{r}E_{\mathscr{ Y},j_{h}}\bigg).\end{eqnarray}$$
   The projections $x$ and $y$ of $z$ are generic points of strata in $\mathscr{X}_{k}$ and $\mathscr{Y}_{k}$ , hence lie in the respective Kato fans.

   Otherwise, let $\unicode[STIX]{x1D6E5}_{\mathscr{Z},1},\ldots ,\unicode[STIX]{x1D6E5}_{\mathscr{Z},r^{\prime }}$ be the horizontal components of $D_{\mathscr{Z}}$ passing through $z$ . We consider the subscheme $\mathscr{D}=\bigcap _{l=1}^{r^{\prime }}\unicode[STIX]{x1D6E5}_{\mathscr{Z},l}$ of $\mathscr{Z}$ , endowed with the divisorial log structure $D_{\mathscr{D}}=\mathscr{D}_{k,\text{red}}$ . Then, $\mathscr{D}^{+}=(\mathscr{D},D_{\mathscr{D}})$ is a log-regular log scheme and $z$ is a point in the Kato fan of $\mathscr{D}^{+}$ that corresponds to the generic point of an intersection of irreducible components of $D_{\mathscr{D}}$ , that are all contained in the special fiber. Since

   $$\begin{eqnarray}\mathscr{D}=\mathop{\bigcap }_{l=1}^{r^{\prime }}\unicode[STIX]{x1D6E5}_{\mathscr{Z},l}=\bigg(\mathop{\bigcap }_{l=1}^{r^{\prime \prime }}\unicode[STIX]{x1D6E5}_{\mathscr{X},l}\times \mathscr{Y}\bigg)\cap \bigg(\mathop{\bigcap }_{l=r^{\prime \prime }+1}^{r^{\prime }}\mathscr{X}\times \unicode[STIX]{x1D6E5}_{\mathscr{Y},l}\bigg)=\bigg(\mathop{\bigcap }_{l=1}^{r^{\prime }}\unicode[STIX]{x1D6E5}_{\mathscr{X},l}\bigg)\times \bigg(\mathop{\bigcap }_{l=r^{\prime \prime }+1}^{r^{\prime }}\unicode[STIX]{x1D6E5}_{\mathscr{Y},l}\bigg)\end{eqnarray}$$
   is itself a fibred product of log-smooth log schemes, we conclude by the previous case.

2. (ii) We claim that the ideal $\mathfrak{p}$ is minimal. Since $z$ lies in $F_{\mathscr{Z}}$ , we have that $\dim {\mathcal{O}}_{\mathscr{Z},z}=\text{rank}{\mathcal{C}}_{\mathscr{Z},z}^{\text{gp}}$ . At the level of characteristic sheaves

   $$\begin{eqnarray}\text{rank}{\mathcal{C}}_{\mathscr{Z},z}^{\text{gp}}=\text{rank}{\mathcal{C}}_{\mathscr{ X},x}^{\text{gp}}+\text{rank}{\mathcal{C}}_{\mathscr{ Y},y}^{\text{gp}}-1\end{eqnarray}$$
   holds, since the stalk at $z$ of the characteristic sheaf of $\mathscr{Z}^{+}$ is equal to ${\mathcal{C}}_{\mathscr{Z},z}={{\mathcal{C}}_{\mathscr{X},x}\oplus }_{\mathbb{N}}{\mathcal{C}}_{\mathscr{Y},y}$ and the morphism $\mathbb{N}\rightarrow {\mathcal{C}}_{\mathscr{X},x}$ is saturated. As $x$ and $y$ are both points in the associated Kato fans, the equality between the dimension of the local ring and the rank of the groupification of the characteristic sheaves holds for $x$ and $y$ as well, and we obtain
   $$\begin{eqnarray}\displaystyle \dim {\mathcal{O}}_{\mathscr{Z},z}=\text{rank}{\mathcal{C}}_{\mathscr{Z},z}^{\text{gp}}=\text{rank}{\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}+\text{rank}{\mathcal{C}}_{\mathscr{Y},y}^{\text{gp}}-1=\dim {\mathcal{O}}_{\mathscr{X},x}+\dim {\mathcal{O}}_{\mathscr{Y},y}-1. & & \displaystyle \nonumber\end{eqnarray}$$
   By log-regularity of $\mathscr{Z}^{+}$ we have the inequality $\dim {\mathcal{O}}_{\mathscr{Z},z^{\prime }}\geqslant \text{rank}{\mathcal{C}}_{\mathscr{Z},z^{\prime }}^{\text{gp}}$ at any point $z^{\prime }$ , hence the equality that holds for $z$ necessarily implies that $\mathfrak{p}$ is a minimal prime ideal of $\unicode[STIX]{x1D705}(x)\otimes _{k}\unicode[STIX]{x1D705}(y)$ .

Let $(x,y)$ be a pair of points of $F_{\mathscr{X}}\times _{F_{S}}F_{\mathscr{Y}}$ that lie in the respective special fibers. In order to determine the number $n(x,y)$ of preimages of $(x,y)$ in $F_{\mathscr{Z}}$ , we need to study the number of minimal prime ideals of the tensor product $\unicode[STIX]{x1D705}(x)\otimes _{k}\unicode[STIX]{x1D705}(y)$ . Since the residue field $k$ is an algebraically closed field, the tensor product $\unicode[STIX]{x1D705}(x)\otimes _{k}\unicode[STIX]{x1D705}(y)$ is a domain; in particular, it has a unique minimal prime ideal, namely $0$ . We obtain that $n(x,y)=1$ .

Let $(x,y)$ be a pair of points in $F_{\mathscr{X}}\times _{F_{S}}F_{\mathscr{Y}}$ whose closures intersect the special fibers, namely there exist $x^{\prime }\in F_{\mathscr{X}}\cap \mathscr{X}_{k}$ and $y^{\prime }\in F_{\mathscr{Y}}\cap \mathscr{Y}_{k}$ such that $x^{\prime }$ is in the closure of $x$ and $y^{\prime }$ in the closure of $y$ . Then, by the previous part of the proof and by Lemma 2.4.11, we have $n(x,y)\leqslant n(x^{\prime },y^{\prime })=1$ .◻

3.1.1. Let $\mathscr{X}^{+}$ be a log-regular log scheme over $S^{+}$ . Let $x$ be a point of the associated Kato fan $F$ . Denote by $F(x)$ the set of points $y$ of $F$ such that $x$ lies in the closure of $\{y\}$ , and by ${\mathcal{C}}_{F(x)}$ the restriction of ${\mathcal{C}}_{F}$ to $F(x)$ . Denote by $\text{Spec}\,{\mathcal{C}}_{\mathscr{X},x}$ the spectrum of the monoid ${\mathcal{C}}_{\mathscr{X},x}={\mathcal{C}}_{F,x}$ . Then there exists a canonical isomorphism of monoid spaces

$$\begin{eqnarray}(F(x),{\mathcal{C}}_{F(x)})\rightarrow \text{Spec}\,{\mathcal{C}}_{\mathscr{X},x}:y\mapsto \{s\in {\mathcal{C}}_{\mathscr{X},x}\,|\,s(y)=0\},\end{eqnarray}$$

where the expression $s(y)=0$ means that $s^{\prime }(y)=0$ for any representative $s^{\prime }$ of $s$ in ${\mathcal{M}}_{\mathscr{X},x}$ . In particular, we obtain a bijective correspondence between the faces of the monoid ${\mathcal{C}}_{\mathscr{X},x}$ and the points of $F(x)$ , and for every point $y$ of $F(x)$ , a surjective cospecialization morphism of monoids

$$\begin{eqnarray}\unicode[STIX]{x1D70F}_{x,y}:{\mathcal{C}}_{\mathscr{X},x}\rightarrow {\mathcal{C}}_{\mathscr{X},y}\end{eqnarray}$$

which induces an isomorphism of monoids

$$\begin{eqnarray}S^{-1}{\mathcal{C}}_{\mathscr{X},x}/(S^{-1}{\mathcal{C}}_{\mathscr{X},x})^{\times }\cong {\mathcal{C}}_{\mathscr{ X},x}/S\xrightarrow[{}]{{\sim}}{\mathcal{C}}_{\mathscr{X},y},\end{eqnarray}$$

where $S$ denotes the monoid of elements $s$ in ${\mathcal{C}}_{\mathscr{X},x}$ such that $s(y)\neq 0$ .

3.1.2. For each point $x$ in $F$ , we denote by $\unicode[STIX]{x1D70E}_{x}$ the set of morphisms of monoids

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}:{\mathcal{C}}_{\mathscr{X},x}\rightarrow (\mathbb{R}_{{\geqslant}0},+)\end{eqnarray}$$

such that $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70B})=1$ for every uniformizer $\unicode[STIX]{x1D70B}$ in $R$ . We endow $\unicode[STIX]{x1D70E}_{x}$ with the topology of pointwise convergence, where $\mathbb{R}_{{\geqslant}0}$ carries the usual Euclidean topology. Note that $\unicode[STIX]{x1D70E}_{x}$ is a polyhedron, possibly unbounded, in the real affine space

$$\begin{eqnarray}\{\unicode[STIX]{x1D6FC}:{\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}\rightarrow (\mathbb{R},+)\,|\,\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70B})=1\text{ for every uniformizer }\unicode[STIX]{x1D70B}\text{ in }R\}.\end{eqnarray}$$

If $y$ is a point of $F(x)$ , then the surjective cospecialization morphism $\unicode[STIX]{x1D70F}_{x,y}$ induces a topological embedding $\unicode[STIX]{x1D70E}_{y}\rightarrow \unicode[STIX]{x1D70E}_{x}$ that identifies $\unicode[STIX]{x1D70E}_{y}$ with a face of $\unicode[STIX]{x1D70E}_{x}$ .

3.1.3. We denote by $T$ the disjoint union of the topological spaces $\unicode[STIX]{x1D70E}_{x}$ with $x$ in $F$ . On the topological space $T$ , we consider the equivalence relation ${\sim}$ generated by couples of the form $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FC}\circ \unicode[STIX]{x1D70F}_{x,y})$ where $x$ and $y$ are points in $F$ such that $x$ lies in the closure of $\{y\}$ and $\unicode[STIX]{x1D6FC}$ is a point of $\unicode[STIX]{x1D70E}_{y}$ .

The skeleton of $\mathscr{X}^{+}$ is defined as the quotient of the topological space $T$ by the equivalence relation ${\sim}$ . We denote this skeleton by $\text{Sk}(\mathscr{X}^{+})$ . It is clear that $\text{Sk}(\mathscr{X}^{+})$ has the structure of a polyhedral complex with cells $\{\unicode[STIX]{x1D70E}_{x},\,x\in F\}$ , so it comes equipped with a PL structure, and that the faces of a cell $\unicode[STIX]{x1D70E}_{x}$ are precisely the cells $\unicode[STIX]{x1D70E}_{y}$ with $y$ in $F(x)$ .

3.1.4. We note that $\unicode[STIX]{x1D70E}_{x}$ is empty for any point $x$ that does not lie in the special fiber $\mathscr{X}_{k}$ : indeed, outside the special fiber any uniformizer is an invertible element, so it is trivial in ${\mathcal{C}}_{\mathscr{X},x}$ and is mapped to $0$ by any morphism of monoids. Therefore, the construction of the skeleton associated to $\mathscr{X}^{+}$ only concerns the points in the Kato fan $F$ that lie in the special fiber. Moreover, given a generic point $x\in \mathscr{X}_{k}$ of an intersection of components of $D_{\mathscr{X}}$ , where at least one component is not in the special fiber, the corresponding face $\unicode[STIX]{x1D70E}_{x}$ is unbounded.

In other words, the skeleton associated to a log-regular scheme $\mathscr{X}^{+}$ , where $D_{\mathscr{X}}$ allows horizontal components, generalizes Berkovich’s skeletons by admitting unbounded faces in the direction of the horizontal components as well as by allowing singularities in the special fiber. It also generalizes the construction performed by Gubler, Rabinoff, and Werner in [Reference Gubler, Rabinoff and WernerGRW16] of a skeleton associated to a strictly semistable snc pair.

3.2.1. Let $\mathscr{X}^{+}$ be a log-regular log scheme over $S^{+}$ . Let $x$ be a point of the associated Kato fan $F$ . As the log structure on $\mathscr{X}^{+}$ is of finite type, the characteristic monoid ${\mathcal{C}}_{\mathscr{X},x}$ is of finite type too, and thus ${\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}$ is a free abelian group of finite rank. Hence there exists a section

$$\begin{eqnarray}\unicode[STIX]{x1D701}:{\mathcal{M}}_{\mathscr{X},x}^{\text{gp}}/{\mathcal{M}}_{\mathscr{ X},x}^{\times }\rightarrow {\mathcal{M}}_{\mathscr{X},x}^{\text{gp}}.\end{eqnarray}$$

The section $\unicode[STIX]{x1D701}$ restricts to ${\mathcal{C}}_{\mathscr{X},x}\rightarrow {\mathcal{M}}_{\mathscr{X},x}$ ; indeed, if $x\in {\mathcal{M}}_{\mathscr{X},x}$ then $\unicode[STIX]{x1D701}(\overline{x})-x\in {\mathcal{M}}_{\mathscr{X},x}^{\times }$ . Therefore we may choose a section

(3.2.2) $$\begin{eqnarray}{\mathcal{C}}_{\mathscr{X},x}\rightarrow {\mathcal{M}}_{\mathscr{X},x}\end{eqnarray}$$

of the projection homomorphism

$$\begin{eqnarray}{\mathcal{M}}_{\mathscr{X},x}\rightarrow {\mathcal{C}}_{\mathscr{X},x}\end{eqnarray}$$

and use this section to view ${\mathcal{C}}_{\mathscr{X},x}$ as a submonoid of ${\mathcal{M}}_{\mathscr{X},x}$ . Note that ${\mathcal{C}}_{\mathscr{X},x}\setminus \{1\}$ generates the ideal ${\mathcal{I}}_{\mathscr{X},x}$ of ${\mathcal{O}}_{\mathscr{X},x}$ .

We propose a generalization of [Reference Mustaţă and NicaiseMN15, Lemma 2.4.4].

Lemma 3.2.3. Let $A$ be a Noetherian ring, let $I$ be an ideal of $A$ , and let $(y_{1},\ldots ,y_{m})$ be a system of generators for $I$ . We denote by $\hat{A}$ the $I$ -adic completion of $A$ . Let $B$ be a subring of $A$ such that the elements $y_{1},\ldots ,y_{m}$ belong to $B$ and generate the ideal $B\cap I$ in $B$ . Then, in the ring $\hat{A}$ , every element $f$ of $B$ can be written as

(3.2.4) $$\begin{eqnarray}f=\mathop{\sum }_{\unicode[STIX]{x1D6FD}\in \mathbb{Z}_{{\geqslant}0}^{m}}c_{\unicode[STIX]{x1D6FD}}y^{\unicode[STIX]{x1D6FD}},\end{eqnarray}$$

where the coefficients $c_{\unicode[STIX]{x1D6FD}}$ belong to $((A\setminus I)\cap B)\cup \{0\}$ .

Proof. Let $f$ be an element of $B$ ; we construct an expansion for $f$ of the form (3.2.4) by induction. If $f$ belongs to the complement of $I$ , the conclusion trivially holds. Otherwise, $f$ belongs to $I$ and we can write $f$ as a linear combination of the elements $y_{1},\ldots ,y_{m}$ with coefficients in $B$ :

$$\begin{eqnarray}f=\mathop{\sum }_{j=1}^{m}b_{j}y_{j},\quad b_{j}\in B.\end{eqnarray}$$

By the induction hypothesis, we suppose that $i$ is a positive integer and that we can write every $f$ in $B$ as a sum of an element $f_{i}$ of the form (3.2.4) and a linear combination of degree- $i$ monomials in the elements $y_{1},\ldots ,y_{m}$ with coefficients in $B$ . We apply this assumption to the coefficients $b_{j}$ , hence

$$\begin{eqnarray}b_{j}=b_{j,i}+\mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FD}\in \mathbb{Z}_{{\geqslant}0}^{m} \\ |\unicode[STIX]{x1D6FD}|=i}}b_{j,\unicode[STIX]{x1D6FD}}y^{\unicode[STIX]{x1D6FD}},\quad b_{j,\unicode[STIX]{x1D6FD}}\in B.\end{eqnarray}$$

Then we can write $f$ as a sum of an element $f_{i+1}$ of the form (3.2.4) and a linear combination of degree-( $i+1$ ) monomials in the elements $y_{1},\ldots ,y_{m}$ with coefficients in $B$ :

$$\begin{eqnarray}f=\underbrace{\mathop{\sum }_{j=1}^{m}b_{j,i}y_{j}}_{f_{i+1}}+\mathop{\sum }_{j=1}^{m}\Bigg(\mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FD}\in \mathbb{Z}_{{\geqslant}0}^{m} \\ |\unicode[STIX]{x1D6FD}|=i}}b_{j,\unicode[STIX]{x1D6FD}}y^{\unicode[STIX]{x1D6FD}}\Bigg)y_{j}\end{eqnarray}$$

such that $f_{i}$ and $f_{i+1}$ have the same coefficients in degree less than or equal to $i$ . Iterating this construction, we finally find an expansion of $f$ of the required form.◻

3.2.5. Let $f$ be an element of ${\mathcal{O}}_{\mathscr{X},x}$ . Considering $A=B={\mathcal{O}}_{\mathscr{X},x}$ , $I=\mathfrak{m}_{x}$ , and a system of generators for $\mathfrak{m}_{x}$ in ${\mathcal{C}}_{\mathscr{X},x}\setminus \{1\}$ , by Lemma 3.2.3 we can write $f$ as a formal power series

(3.2.6) $$\begin{eqnarray}f=\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}}c_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FE}\end{eqnarray}$$

in $\widehat{{\mathcal{O}}}_{\mathscr{X},x}$ , where each coefficient $c_{\unicode[STIX]{x1D6FE}}$ is either zero or a unit in ${\mathcal{O}}_{\mathscr{X},x}$ . We call this formal series an admissible expansion of $f$ . We set

(3.2.7) $$\begin{eqnarray}S=\{\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}\,|\,c_{\unicode[STIX]{x1D6FE}}\neq 0\}\end{eqnarray}$$

and we denote by $\unicode[STIX]{x1D6E4}_{x}(f)$ the set of elements of $S$ that lie on a compact face of the convex hull of $S+{\mathcal{C}}_{\mathscr{X},x}$ in ${\mathcal{C}}_{\mathscr{X},x}^{\text{gp}}\otimes _{\mathbb{Z}}\mathbb{R}$ . We call $\unicode[STIX]{x1D6E4}_{x}(f)$ the initial support of $f$ at $x$ , notation which is justified by the next proposition.

Proof. If we denote by $I$ the ideal of $k(x)[{\mathcal{C}}_{\mathscr{X},x}]$ generated by ${\mathcal{C}}_{\mathscr{X},x}\setminus \{1\}$ , then it follows from [Reference KatoKat94] that there exists an isomorphism of $k(x)$ -algebras

(3.2.9) $$\begin{eqnarray}\text{gr}_{I}\,k(x)[{\mathcal{C}}_{\mathscr{X},x}]\rightarrow \text{gr}_{\mathfrak{m}_{x}}\,{\mathcal{O}}_{\mathscr{X},x}.\end{eqnarray}$$

Using this result and following the argument of [Reference Mustaţă and NicaiseMN15, Proposition 2.4.4], we show now that $f_{x}$ does not depend on the expansion of $f$ . Let

$$\begin{eqnarray}f=\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}}c_{\unicode[STIX]{x1D6FE}}^{\prime }\unicode[STIX]{x1D6FE}\end{eqnarray}$$

be another admissible expansion of $f$ with associated set $\unicode[STIX]{x1D6E4}_{x}(f)^{\prime }$ and element $f_{x}^{\prime }$ . Then

$$\begin{eqnarray}0=\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}}(c_{\unicode[STIX]{x1D6FE}}-c_{\unicode[STIX]{x1D6FE}}^{\prime })\unicode[STIX]{x1D6FE}=\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}}d_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FE}\end{eqnarray}$$

where the right-hand side is an admissible expansion obtained by choosing admissible expansions for the elements $c_{\unicode[STIX]{x1D6FE}}-c_{\unicode[STIX]{x1D6FE}}^{\prime }$ that do not lie in ${\mathcal{O}}_{\mathscr{X},x}^{\times }\cup \{0\}$ . In particular, $d_{\unicode[STIX]{x1D6FE}}(x)=c_{\unicode[STIX]{x1D6FE}}(x)-c_{\unicode[STIX]{x1D6FE}}^{\prime }(x)$ for any $\unicode[STIX]{x1D6FE}$ in $\unicode[STIX]{x1D6E4}_{x}(f)\cup \unicode[STIX]{x1D6E4}_{x}(f)^{\prime }$ . The isomorphism of graded algebras in (3.2.9) implies that the elements $d_{\unicode[STIX]{x1D6FE}}$ must all vanish, hence $\unicode[STIX]{x1D6E4}_{x}(f)=\unicode[STIX]{x1D6E4}_{x}(f)^{\prime }$ and $f_{x}=f_{x}^{\prime }$ .

Point (ii) follows from the fact that the coefficients $c_{\unicode[STIX]{x1D6FE}}$ of $f_{x}$ are independent of the chosen section up to multiplication by a unit in ${\mathcal{O}}_{\mathscr{X},x}$ , so that the support $\unicode[STIX]{x1D6E4}_{x}(f)$ of $f_{x}$ only depends on $f$ and $x$ .◻

Proof. We will prove that the map

(3.2.11) $$\begin{eqnarray}v:{\mathcal{O}}_{\mathscr{X},x}\setminus \{0\}\rightarrow \mathbb{R}:f\mapsto \min \{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FE})\,|\,\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{x}(f)\}\end{eqnarray}$$

satisfies the requirements in the statement. We fix a section

$$\begin{eqnarray}{\mathcal{C}}_{\mathscr{X},x}\rightarrow {\mathcal{M}}_{\mathscr{X},x}.\end{eqnarray}$$

It is straightforward to check that $(f\cdot g)_{x}=f_{x}\cdot g_{x}$ for all $f$ and $g$ in ${\mathcal{O}}_{\mathscr{X},x}$ . This implies that $v$ is a valuation. It is obvious that $v(m)=\unicode[STIX]{x1D6FC}(\overline{m})$ for all $m$ in ${\mathcal{M}}_{\mathscr{X},x}$ , since we can write $m$ as the product of an element of ${\mathcal{C}}_{\mathscr{X},x}$ and a unit in ${\mathcal{O}}_{\mathscr{X},x}$ .

Now we prove minimality. Consider any real valuation

$$\begin{eqnarray}w:{\mathcal{O}}_{\mathscr{X},x}\rightarrow \mathbb{R}\end{eqnarray}$$

such that $w(m)=\unicode[STIX]{x1D6FC}(\overline{m})$ for each element $m$ of ${\mathcal{M}}_{\mathscr{X},x}$ , and let $f$ be an element of ${\mathcal{O}}_{\mathscr{X},x}$ . We must show that $w(f)\geqslant v(f)$ .

We set

$$\begin{eqnarray}C_{\unicode[STIX]{x1D6FC}}={\mathcal{C}}_{\mathscr{X},x}\setminus \unicode[STIX]{x1D6FC}^{-1}(0).\end{eqnarray}$$

We denote by $I$ the ideal in ${\mathcal{O}}_{\mathscr{X},x}$ generated by $C_{\unicode[STIX]{x1D6FC}}$ and by $A$ the $I$ -adic completion of ${\mathcal{O}}_{\mathscr{X},x}$ . By Lemma 3.2.3, we see that we can write $f$ in $A$ as

(3.2.12) $$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D6FD}\in C_{\unicode[STIX]{x1D6FC}}\cup \{1\}}d_{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6FD},\end{eqnarray}$$

where $d_{\unicode[STIX]{x1D6FD}}$ is either zero or contained in the complement of $I$ in ${\mathcal{O}}_{\mathscr{X},x}$ .

Since $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FD})>0$ for every $\unicode[STIX]{x1D6FD}\in C_{\unicode[STIX]{x1D6FC}}$ , we can find an integer $N>0$ such that $w(g)>w(f)$ for every element $g$ in $I^{N}$ . So we have

$$\begin{eqnarray}w(f)\geqslant \min \{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FD})\,|\,d_{\unicode[STIX]{x1D6FD}}\neq 0\},\end{eqnarray}$$

recalling that $w(\unicode[STIX]{x1D6FD})=\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FD})$ for all $\unicode[STIX]{x1D6FD}$ in ${\mathcal{C}}_{\mathscr{X},x}$ .

We consider the coefficients in the expansion (3.2.12) of $f$ . Applying Lemma 3.2.3 as in paragraph (3.2.5), we can write admissible expansions of these coefficients in $\widehat{{\mathcal{O}}}_{\mathscr{X},x}$ as

$$\begin{eqnarray}d_{\unicode[STIX]{x1D6FD}}=\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}}c_{\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6FE},\quad c_{\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FD}}\in {\mathcal{O}}_{\mathscr{X},x}^{\times }\cup \{0\},\end{eqnarray}$$

with $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FE})=0$ in the expansions of $d_{\unicode[STIX]{x1D6FD}}$ that belong to $\mathfrak{m}_{x}\setminus I$ .

Therefore we obtain an admissible expansion of $f$

$$\begin{eqnarray}f=\mathop{\sum }_{\substack{ \unicode[STIX]{x1D6FD}\in C_{\unicode[STIX]{x1D6FC}}\cup \{1\}\unicode[STIX]{x1D6FE}\in {\mathcal{C}}_{\mathscr{X},x}}}c_{\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FD}}\,\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD}\end{eqnarray}$$

and we have $v(f)=\min \{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD})\,|\,c_{\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FD}}\neq 0\}=\min \{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FD})\,|\,d_{\unicode[STIX]{x1D6FD}}\neq 0\}\leqslant w(f)$ .◻

3.2.14. We will denote the valuation $v$ from Proposition 3.2.10 by $v_{x,\unicode[STIX]{x1D6FC}}$ . Since $v_{x,\unicode[STIX]{x1D6FC}}$ induces a real valuation on the function field of $\mathscr{X}_{K}$ that extends the discrete valuation $v_{K}$ on $K$ , it defines a point of the $K$ -analytic space $\mathscr{X}_{K}^{\text{an}}$ , which we will denote by the same symbol $v_{x,\unicode[STIX]{x1D6FC}}$ . We now show that the characterization of $v_{x,\unicode[STIX]{x1D6FC}}$ in Proposition 3.2.10 implies that

$$\begin{eqnarray}v_{y,\unicode[STIX]{x1D6FC}^{\prime }}=v_{x,\unicode[STIX]{x1D6FC}^{\prime }\circ \unicode[STIX]{x1D70F}_{x,y}}\end{eqnarray}$$

for every $y$ in $F(x)$ and every $\unicode[STIX]{x1D6FC}^{\prime }$ in $\unicode[STIX]{x1D70E}_{y}$ .

First we note that ${\mathcal{O}}_{\mathscr{X},y}$ is the localization of ${\mathcal{O}}_{\mathscr{X},x}$ with respect to the elements $m\in {\mathcal{M}}_{\mathscr{X},x}$ in the kernel of $\unicode[STIX]{x1D70F}_{x,y}$ . Indeed, by construction of $\unicode[STIX]{x1D70F}_{x,y}$ , the kernel is given by

$$\begin{eqnarray}\ker (\unicode[STIX]{x1D70F}_{x,y})=\{s\in {\mathcal{C}}_{\mathscr{X},x}\mid s(y)\neq 0\}.\end{eqnarray}$$

To obtain ${\mathcal{O}}_{\mathscr{X},y}$ from ${\mathcal{O}}_{\mathscr{X},x}$ , we localize by

$$\begin{eqnarray}S=\{a\in {\mathcal{O}}_{\mathscr{X},x}\mid a(y)\neq 0\}.\end{eqnarray}$$

Therefore we can identify the set of elements of ${\mathcal{M}}_{\mathscr{X},x}$ whose reduction is in $\ker (\unicode[STIX]{x1D70F}_{x,y})$ with the set $S$ , recalling that, for points in the Kato fan, ${\mathcal{C}}_{\mathscr{X},x}\setminus \{1\}$ generates the maximal ideal of ${\mathcal{O}}_{\mathscr{X},x}$ . Therefore we are dealing with these two morphisms:

$$\begin{eqnarray}\displaystyle {\mathcal{O}}_{\mathscr{X},x}{\hookrightarrow}S^{-1}{\mathcal{O}}_{\mathscr{X},x} & = & \displaystyle {\mathcal{O}}_{\mathscr{X},y},\nonumber\\ \displaystyle {\mathcal{C}}_{\mathscr{X},x}{\twoheadrightarrow}{\mathcal{C}}_{\mathscr{X},x}/S & = & \displaystyle {\mathcal{C}}_{\mathscr{X},y}.\nonumber\end{eqnarray}$$

Let $f$ be an element of ${\mathcal{O}}_{\mathscr{X},x}$ . In the notation of Lemma 3.2.3, we apply the lemma to $A={\mathcal{O}}_{\mathscr{X},y}$ and $B={\mathcal{O}}_{\mathscr{X},x}$ , choosing a system of generators of $\mathfrak{m}_{y}$ in ${\mathcal{C}}_{\mathscr{X},x}$ : we can find an admissible expansion of $f$ of the form

$$\begin{eqnarray}f=\mathop{\sum }_{\unicode[STIX]{x1D6FF}\in {\mathcal{C}}_{\mathscr{X},y}}d_{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6FF}\quad \text{with }d_{\unicode[STIX]{x1D6FF}}\in ({\mathcal{O}}_{\mathscr{X},x}\cap {\mathcal{O}}_{\mathscr{X},y}^{\times })\cup \{0\}.\end{eqnarray}$$

Admissible expansions of coefficients $d_{\unicode[STIX]{x1D6FF}}$ induce an admissible expansion for $f$ by

$$\begin{eqnarray}f=\mathop{\sum }_{\unicode[STIX]{x1D6FF}\in {\mathcal{C}}_{\mathscr{X},y}}\bigg(\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in S}c_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6FE}\bigg)\unicode[STIX]{x1D6FF}\quad \text{with }c_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF}}\in {\mathcal{O}}_{\mathscr{X},x}^{\times }\cup \{0\},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}$ runs through the set $S$ since $d_{\unicode[STIX]{x1D6FF}}\in {\mathcal{O}}_{\mathscr{X},y}^{\times }$ . Thus we have

$$\begin{eqnarray}\displaystyle v_{y,\unicode[STIX]{x1D6FC}^{\prime }}(f) & = & \displaystyle \min \{\unicode[STIX]{x1D6FC}^{\prime }(\unicode[STIX]{x1D6FF})\,|\,\unicode[STIX]{x1D6FF}\in \unicode[STIX]{x1D6E4}_{y}(f)\}\nonumber\\ \displaystyle & = & \displaystyle \min \{\unicode[STIX]{x1D6FC}^{\prime }\circ \unicode[STIX]{x1D70F}_{x,y}(\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF})\,|\,\unicode[STIX]{x1D6FF}\in \unicode[STIX]{x1D6E4}_{y}(f),\unicode[STIX]{x1D6FE}\in S\}\nonumber\\ \displaystyle & = & \displaystyle \min \{\unicode[STIX]{x1D6FC}^{\prime }\circ \unicode[STIX]{x1D70F}_{x,y}(\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF})\,|\,\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF}\in \unicode[STIX]{x1D6E4}_{x}(f)\}\nonumber\\ \displaystyle & = & \displaystyle v_{x,\unicode[STIX]{x1D6FC}^{\prime }\circ \unicode[STIX]{x1D70F}_{x,y}}(f).\nonumber\end{eqnarray}$$

Hence, we obtain a well-defined map

$$\begin{eqnarray}\unicode[STIX]{x1D704}:\text{Sk}(\mathscr{X}^{+})\rightarrow \mathscr{X}_{K}^{\text{an}}\end{eqnarray}$$

by sending $\unicode[STIX]{x1D6FC}$ to $v_{x,\unicode[STIX]{x1D6FC}}$ for every point $x$ of $F$ and every $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D70E}_{x}$ .

Proof. First, we show that $\unicode[STIX]{x1D704}$ is injective. Let $x$ be a point of $F$ and $\unicode[STIX]{x1D6FC}$ an element of $\unicode[STIX]{x1D70E}_{x}$ . Let $y$ be the point of $F(x)$ corresponding to the face ${\mathcal{C}}_{\mathscr{X},x}\setminus \unicode[STIX]{x1D6FC}^{-1}(0)$ of ${\mathcal{C}}_{\mathscr{X},x}$ . Then $\unicode[STIX]{x1D6FC}$ factors through an element

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}^{\prime }:{\mathcal{C}}_{\mathscr{ X},y}\rightarrow \mathbb{R}_{{\geqslant}0}\end{eqnarray}$$

of $\unicode[STIX]{x1D70E}_{y}$ . Note that $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}^{\prime }$ in $\text{Sk}(\mathscr{X}^{+})$ because $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}^{\prime }\circ \unicode[STIX]{x1D70F}_{x,y}$ . Moreover, since $(\unicode[STIX]{x1D6FC}^{\prime })^{-1}(0)=\{1\}$ , the center of the valuation $v_{y,\unicode[STIX]{x1D6FC}^{\prime }}$ is the point $y$ , so that $\text{red}_{\mathscr{X}}(v_{y,\unicode[STIX]{x1D6FC}^{\prime }})=y$ . Thus we can recover $y$ from $v_{y,\unicode[STIX]{x1D6FC}^{\prime }}$ . Then we can also reconstruct $\unicode[STIX]{x1D6FC}^{\prime }$ by looking at the values of $v_{y,\unicode[STIX]{x1D6FC}^{\prime }}$ at the elements of ${\mathcal{M}}_{\mathscr{X},y}$ . We conclude that $\unicode[STIX]{x1D704}$ is injective.

Now, we show that $\unicode[STIX]{x1D704}$ is a homeomorphism onto its image. For any valuation $v$ in $\text{Sk}(\mathscr{X}^{+})$ and any small open neighborhood $U$ of $\unicode[STIX]{x1D704}(v)$ in $\mathscr{X}_{K}^{\text{an}}$ , there exists a closed subset $C$ in $\text{Sk}(\mathscr{X}^{+})$ such that $U\cap \unicode[STIX]{x1D704}(\text{Sk}(\mathscr{X}^{+}))\subseteq \unicode[STIX]{x1D704}(C)$ and, up to subdivisions, we can assume that the $C$ is a closed cell of $\text{Sk}(\mathscr{X}^{+})$ . Therefore, it suffices to prove that the restriction of $\unicode[STIX]{x1D704}$ to any closed cell $\unicode[STIX]{x1D70E}_{x}$ of $\text{Sk}(\mathscr{X}^{+})$ is a homeomorphism. The restriction $\unicode[STIX]{x1D704}_{|\unicode[STIX]{x1D70E}_{x}}$ is an injective map from a compact set to the Hausdorff space $\mathscr{X}_{K}^{\text{an}}$ , so we reduce to showing that $\unicode[STIX]{x1D704}_{|\unicode[STIX]{x1D70E}_{x}}$ is continuous, to conclude that $\unicode[STIX]{x1D704}_{|\unicode[STIX]{x1D70E}_{x}}$ is a homeomorphism. By definition of the Berkovich topology, it is enough to prove that the map

$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{x}\rightarrow \mathbb{R}:\unicode[STIX]{x1D6FC}\mapsto v_{x,\unicode[STIX]{x1D6FC}}(f)\end{eqnarray}$$

is continuous for every $f$ in ${\mathcal{O}}_{\mathscr{X},x}$ . This is obvious from the formula (3.2.11).◻

3.2.16. From now on, we will view $\text{Sk}(\mathscr{X}^{+})$ as a topological subspace of $\mathscr{X}_{K}^{\text{an}}$ by means of the embedding $\unicode[STIX]{x1D704}$ in Proposition 3.2.15. If $\mathscr{X}$ is regular over $R$ and $\mathscr{X}_{k}$ is a divisor with strict normal crossings, the skeleton $\text{Sk}(\mathscr{X}^{+})$ was described in [Reference Mustaţă and NicaiseMN15, §3.1].

3.3.1. We denote by $D_{\mathscr{X},\text{hor}}$ the component of $D_{\mathscr{X}}$ not contained in the special fiber $\mathscr{X}_{k}$ . The inclusion $\unicode[STIX]{x1D704}:\text{Sk}(\mathscr{X}^{+})\rightarrow \mathscr{X}_{K}^{\text{an}}$ is actually an inclusion in $(\mathscr{X}_{K}\setminus D_{\mathscr{X},\text{hor}})^{\text{an}}$ and it admits a continuous retraction

$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{\mathscr{X}}:(\mathscr{X}_{K}\setminus D_{\mathscr{X},\text{hor}})^{\text{an}}\rightarrow \text{Sk}(\mathscr{X}^{+})\end{eqnarray}$$

constructed as follows. Let $x$ be a point of $(\mathscr{X}_{K}\setminus D_{\mathscr{X},\text{hor}})^{\text{an}}$ and consider the reduction map

$$\begin{eqnarray}\text{red}_{\mathscr{X}}:(\mathscr{X}_{K}\setminus D_{\mathscr{X},\text{hor}})^{\text{an}}\rightarrow \mathscr{X}_{k}.\end{eqnarray}$$

Let $E_{1},\ldots ,E_{r}$ be the irreducible components of $D_{\mathscr{X}}$ passing through the point $\text{red}_{\mathscr{X}}(x)$ . We denote by $\unicode[STIX]{x1D709}$ the generic point of the connected component of $E_{1}\cap \cdots \cap E_{r}$ that contains $\text{red}_{\mathscr{X}}(x)$ . By Lemma 2.2.3, $\unicode[STIX]{x1D709}$ is a point in the associated Kato fan $F$ . We set $\unicode[STIX]{x1D6FC}$ to be the morphism of monoids

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}:{\mathcal{C}}_{\mathscr{X},\unicode[STIX]{x1D709}}\rightarrow \mathbb{R}_{{\geqslant}0}\end{eqnarray}$$

such that $\unicode[STIX]{x1D6FC}(\overline{m})=v_{x}(m)$ for any element $m$ of ${\mathcal{M}}_{\mathscr{X},\unicode[STIX]{x1D709}}$ . In particular, $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70B})=v_{x}(\unicode[STIX]{x1D70B})=1$ as we assumed the normalization of all valuations in the Berkovich space. Then $\unicode[STIX]{x1D70C}_{\mathscr{X}}(x)$ is the point of $\text{Sk}(\mathscr{X}^{+})$ corresponding to the couple $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D6FC})$ . By construction $\unicode[STIX]{x1D70C}_{\mathscr{X}}$ is continuous and right inverse to the inclusion $\unicode[STIX]{x1D704}$ .

3.3.2. Given a dominant morphism $f:\mathscr{X}^{+}\rightarrow \mathscr{Y}^{+}$ of integral flat separated log-regular log schemes over $S$ , it induces a map between the set of birational points $\text{Bir}(\mathscr{X}_{K})\rightarrow \text{Bir}(\mathscr{Y}_{K})$ . As $\text{Bir}(\mathscr{X}_{K})\subseteq (\mathscr{X}_{K}\setminus D_{\mathscr{X},\text{hor}})^{\text{an}}$ , we can employ the retraction $\unicode[STIX]{x1D70C}$ to define a map of skeletons as follows.

3.4.1. Let $\mathscr{X}^{+}$ and $\mathscr{Y}^{+}$ be log-smooth log schemes over $S^{+}$ , and let $\mathscr{Z}^{+}$ be their fs fibred product. Let

$$\begin{eqnarray}\text{Sk}(\mathscr{Z}^{+})\rightarrow \text{Sk}(\mathscr{X}^{+})\times \text{Sk}(\mathscr{Y}^{+})\end{eqnarray}$$

be the continuous map of skeletons functorially associated to the projections $\text{pr}_{\mathscr{X}}:\mathscr{Z}^{+}\rightarrow \mathscr{X}^{+}$ and $\text{pr}_{\mathscr{Y}}:\mathscr{Z}^{+}\rightarrow \mathscr{Y}^{+}$ . We denote this map by $(\text{pr}_{\text{Sk}(\mathscr{X})},\text{pr}_{\text{Sk}(\mathscr{Y})})$ and we recall that it is constructed considering the following diagram.

Proof. We first provide an explicit description of the map $\text{pr}_{\text{Sk}(\mathscr{X})}$ (respectively, of $\text{pr}_{\text{Sk}(\mathscr{Y})}$ ) in diagram (3.4.2). Then we show that the map $(\text{pr}_{\text{Sk}(\mathscr{X})},\text{pr}_{\text{Sk}(\mathscr{Y})})$ is injective and surjective.

Let $v_{z,\unicode[STIX]{x1D700}}$ be the valuation in $\text{Sk}(\mathscr{Z}^{+})$ corresponding to a couple $(z,\unicode[STIX]{x1D700})$ with $z\in F_{\mathscr{Z}}\cap \mathscr{Z}_{k}$ and $\unicode[STIX]{x1D700}\in \unicode[STIX]{x1D70E}_{z}$ . We consider the morphism of associated Kato fans

$$\begin{eqnarray}F_{\mathscr{Z}}\rightarrow F_{\mathscr{X}}\times _{F_{S}}F_{\mathscr{Y}}\end{eqnarray}$$

as established in Proposition 2.4.8. We denote respectively by $\text{pr}_{F_{\mathscr{X}}}$ and $\text{pr}_{F_{\mathcal{Y}}}$ the projection to the first and second factor. Then $\text{pr}_{F_{\mathscr{X}}}(z)$ is a point in the associated Kato fan $F_{\mathscr{X}}$ , which we denote by $x$ . We consider the morphism of monoids

$$\begin{eqnarray}i_{x}:{\mathcal{C}}_{\mathscr{X},x}\rightarrow {\mathcal{C}}_{\mathscr{Z},z}\end{eqnarray}$$

and the composition

It trivially satisfies $\unicode[STIX]{x1D700}\circ i_{x}(\unicode[STIX]{x1D70B})=1$ . In order to conclude that it correctly defines a point in the skeleton $\text{Sk}(\mathscr{X}^{+})$ , we need to check the compatibility with respect to the equivalence relation ${\sim}$ . Indeed, suppose that $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}^{\prime }\circ \unicode[STIX]{x1D70F}_{z,z^{\prime }}$ for some $z\in \overline{\{z^{\prime }\}}$ . We denote by $x^{\prime }$ the projection of $z^{\prime }$ under the local isomorphism of associated Kato fans. The diagram

is commutative as it is made up of a commutative square and a commutative triangle of arrows. Therefore, by commutativity

$$\begin{eqnarray}\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})=\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700}^{\prime })\circ \unicode[STIX]{x1D70F}_{x,x^{\prime }}\end{eqnarray}$$

and this implies that $\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})$ defines a well-defined point $v_{x,\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})}$ of $\text{Sk}(\mathscr{X}^{+})$ .

We claim that $v_{x,\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})}$ is indeed the image of $v_{z,\unicode[STIX]{x1D700}}$ under the map $\text{pr}_{\text{Sk}(\mathscr{X})}$ . We recall that the projection $\widehat{\text{pr}_{\mathscr{X}}}$ , in diagram (3.4.2), is such that a valuation $v$ on the function field $K(\mathscr{Z}_{K})$ maps to the composition $v\circ i$ where $i:K(\mathscr{X}_{K}){\hookrightarrow}K(\mathscr{Z}_{K})$ . Thus, we need to prove that the equality in the following inner diagram holds.

We denote $\unicode[STIX]{x1D70C}_{\mathscr{X}}(v_{z,\unicode[STIX]{x1D700}}\circ i)$ by $(x,\unicode[STIX]{x1D6FC})$ as a point of $\text{Sk}(\mathscr{X}^{+})$ . By definition of the retraction $\unicode[STIX]{x1D70C}_{\mathscr{X}}$ , the morphism $\unicode[STIX]{x1D6FC}$ is characterized by the fact that $\unicode[STIX]{x1D6FC}(\overline{m})=(v_{z,\unicode[STIX]{x1D700}}\circ i)(m)$ for any $m$ in ${\mathcal{M}}_{\mathscr{X},x}$ and then we have

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}(\overline{m})=(v_{z,\unicode[STIX]{x1D700}}\circ i)(m)=v_{z,\unicode[STIX]{x1D700}}(m)=\unicode[STIX]{x1D700}(\overline{m}).\end{eqnarray}$$

On the other hand, for any $m$ in ${\mathcal{M}}_{\mathscr{X},x}$ ,

$$\begin{eqnarray}v_{x,\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})}(m)=\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})(\overline{m})=\unicode[STIX]{x1D700}(\overline{m})\end{eqnarray}$$

hence we obtain that $\unicode[STIX]{x1D6FC}$ coincides with the morphism $\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})$ . This means that their associated points $\unicode[STIX]{x1D70C}_{\mathscr{X}}(v_{z,\unicode[STIX]{x1D700}}\circ i)$ and $v_{x,\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})}$ coincide in $\text{Sk}(\mathscr{X}^{+})$ .

We now prove injectivity and surjectivity of the map $(\text{pr}_{\text{Sk}(\mathscr{X})},\text{pr}_{\text{Sk}(\mathscr{Y})})$ . Given a pair of points in $\text{Sk}(\mathscr{X}^{+})\times \text{Sk}(\mathscr{Y}^{+})$ , they are of the form $(v_{x,\unicode[STIX]{x1D6FC}},v_{y,\unicode[STIX]{x1D6FD}}),$ with $x\in F_{\mathscr{X}}\cap \mathscr{X}_{k}$ , $y\in F_{\mathscr{Y}}\cap \mathscr{Y}_{k}$ , $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D70E}_{x}$ , and $\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D70E}_{y}$ . The assumptions of semistability of $\mathscr{X}^{+}$ and algebraic closedness of $k$ guarantee that there is a unique $z$ in $F_{\mathscr{Z}}$ in the fiber of $x$ and $y$ , by Proposition 2.5.4 and Remark 3.1.4. We claim that we can construct a unique $\unicode[STIX]{x1D700}\in \unicode[STIX]{x1D70E}_{z}$ such that $(\text{pr}_{\text{Sk}(\mathscr{X})},\text{pr}_{\text{Sk}(\mathscr{Y})})(v_{z,\unicode[STIX]{x1D700}})=(v_{x,\unicode[STIX]{x1D6FC}},v_{y,\unicode[STIX]{x1D6FD}}).$ We set

this is well defined as $\unicode[STIX]{x1D700}([\unicode[STIX]{x1D70B},1])=\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70B})=1=\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D70B})=\unicode[STIX]{x1D700}([1,\unicode[STIX]{x1D70B}])$ . By construction, the image of $v_{z,\unicode[STIX]{x1D700}}$ is $(v_{x,\unicode[STIX]{x1D6FC}},v_{y,\unicode[STIX]{x1D6FD}})$ , so the map $(\text{pr}_{\text{Sk}(\mathscr{X})},\text{pr}_{\text{Sk}(\mathscr{Y})})$ is surjective. Moreover, we can uniquely reconstruct $\unicode[STIX]{x1D700}$ by looking at the values of $v_{x,\text{pr}_{\mathscr{X}}(\unicode[STIX]{x1D700})}$ at the elements of ${\mathcal{M}}_{\mathscr{X},x}$ , and of $v_{y,\text{pr}_{\mathcal{Y}}(\unicode[STIX]{x1D700})}$ at the elements of ${\mathcal{M}}_{\mathscr{Y},y}$ . Thus $(\text{pr}_{\text{Sk}(\mathscr{X})},\text{pr}_{\text{Sk}(\mathscr{Y})})$ is injective.◻

4.1.1. Let $X$ be a connected, smooth and proper $K$ -variety of dimension $n$ . We introduce the following notation: for any log-regular model $\mathscr{X}^{+}$ of $X$ , for any point $x=(\unicode[STIX]{x1D709}_{x},|\cdot |_{x})\in X^{\text{an}}$ , and for any $\mathbb{Q}$ -Cartier divisor $D$ on $\mathscr{X}^{+}$ whose support does not contain $\unicode[STIX]{x1D709}_{x}$ , we set

$$\begin{eqnarray}v_{x}(D)=-\ln |f(x)|^{1/m},\end{eqnarray}$$

where $f$ is any element of $K(X)^{\times }$ such that $mD=\text{div}(f)$ locally at $\text{red}_{\mathscr{X}}(x)$ , and $m\in \mathbb{Z}_{{>}0}$ is such that $mD$ is Cartier.

4.1.2. Let $(X,\unicode[STIX]{x1D6E5}_{X})$ be a fractional snc pair as in § 1.5, so that $\unicode[STIX]{x1D6E5}_{X}$ is an effective $\mathbb{Q}$ -divisor such that $\unicode[STIX]{x1D6E5}_{X}=\sum a_{i}\unicode[STIX]{x1D6E5}_{X,i}$ has $0\leqslant a_{i}\leqslant 1$ , and the round-up $X^{+}=(X,\lceil \unicode[STIX]{x1D6E5}_{X}\rceil )$ is snc. Let $\unicode[STIX]{x1D714}$ be a regular $m$ -pluricanonical form on $X^{+}$ with poles of order at most $ma_{i}$ along $\unicode[STIX]{x1D6E5}_{X,i}$ , for some $m$ such that $ma_{i}\in \mathbb{N}$ for any $i$ . Thus, such a form is a section of ${\mathcal{O}}_{X}(m(K_{X}+\unicode[STIX]{x1D6E5}_{X}))$ , and we call it a $\unicode[STIX]{x1D6E5}_{X}$ -logarithmic $m$ -pluricanonical form.

Given a log-regular model $\mathscr{X}^{+}$ of $X^{+}$ , where $D_{\mathscr{X}}=\overline{\lceil \unicode[STIX]{x1D6E5}_{X}\rceil }+\mathscr{X}_{k,\text{red}}$ , we can view any $\unicode[STIX]{x1D6E5}_{X}$ -logarithmic $m$ -pluricanonical forms as rational sections of the logarithmic $m$ -pluricanonical bundle $(\unicode[STIX]{x1D714}_{\mathscr{X}^{+}/S^{+}}^{\log })^{\otimes m}$ , that is, of

$$\begin{eqnarray}(\unicode[STIX]{x1D714}_{\mathscr{X}^{+}/S^{+}}^{\log })^{\otimes m}=\unicode[STIX]{x1D714}_{\mathscr{ X}/S}^{\otimes m}\otimes {\mathcal{O}}_{\mathscr{ X}}(m(\overline{\lceil \unicode[STIX]{x1D6E5}_{X}\rceil }+\mathscr{X}_{k,\text{red}}-\text{div}(\unicode[STIX]{x1D70B}))).\end{eqnarray}$$

The form $\unicode[STIX]{x1D714}$ , viewed as a rational section of $(\unicode[STIX]{x1D714}_{\mathscr{X}^{+}/S^{+}}^{\log })^{\otimes m}$ , defines a divisor $\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714})$ on $\mathscr{X}^{+}$ . Note that the multiplicity in $\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714})$ of the closure $\overline{\unicode[STIX]{x1D6E5}_{X,i}}$ in $\mathscr{X}$ of $\unicode[STIX]{x1D6E5}_{X,i}$ is at least $m(1-a_{i})$ .

4.1.3. Given a $\unicode[STIX]{x1D6E5}_{X}$ -logarithmic $m$ -pluricanonical form $\unicode[STIX]{x1D714}$ , we can consider it as a rational section of $\unicode[STIX]{x1D714}_{X/K}^{\otimes m}$ . Hence, we can associate to $\unicode[STIX]{x1D714}$ the weight function $\text{wt}_{\unicode[STIX]{x1D714}}$ as in [Reference Mustaţă and NicaiseMN15]. The following lemma gives an interpretation of the weight function associated to $\unicode[STIX]{x1D714}$ in terms of logarithmic differentials, which we will use in the sequel.

Proof. It suffices to prove the equality for the divisorial points in $\text{Sk}(\mathscr{X}^{+})$ , since they are dense in the skeleton, and both $\text{wt}_{\unicode[STIX]{x1D714}}(\cdot )$ and $v_{\cdot }(\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}))$ are continuous functions on the skeleton $\text{Sk}(\mathscr{X}^{+})$ .

Let $x$ be a divisorial point in $\text{Sk}(\mathscr{X}^{+})$ . If $x$ corresponds to a component of the special fiber $\mathscr{X}_{k}$ , then the center $\text{red}_{\mathscr{X}}(x)$ of $x$ does not contain the closure of any components of $\unicode[STIX]{x1D6E5}_{X}$ . Thus, locally around $\text{red}_{\mathscr{X}}(x)$ the log schemes $\mathscr{X}^{+}$ and $(\mathscr{X},\mathscr{X}_{k,\text{red}})$ are isomorphic and, in particular, $\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714})=\text{div}_{(\mathscr{X},\mathscr{X}_{k,\text{red}})}(\unicode[STIX]{x1D714})$ . The computation in [Reference Nicaise and XuNX16, §3.2.2] shows that $\text{wt}_{\unicode[STIX]{x1D714}}(x)=v_{x}(\text{div}_{(\mathscr{X},\mathscr{X}_{k,\text{red}})}(\unicode[STIX]{x1D714}))+m$ , so we obtain the required equality.

Otherwise, we consider the blow-up $h:{\mathscr{X}^{\prime }}^{+}\rightarrow \mathscr{X}^{+}$ at the closure of the center $\text{red}_{\mathscr{X}}(x)$ of $x$ in $\mathscr{X}^{+}$ . By [Reference Kollár and MoriKM08, Lemma 2.45],Footnote ¹ iterating this procedure a finite number of times, we obtain a log-regular model $\mathscr{Y}^{+}$ such that the $x$ corresponds to a component $E$ of the special fiber $\mathscr{Y}_{k}$ . By reduction to the previous case, it is enough to check that the value $v_{x}(\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}))$ does not change under such a blow-up $h$ .

The morphism $h$ is the blow-up of a stratum of the log boundary $\overline{\lceil \unicode[STIX]{x1D6E5}_{X}\rceil }+\mathscr{X}_{k,\text{red}}$ , hence $h$ is a log étale morphism. By [Reference KatoKat89, Proposition 3.12], we have $h^{\ast }(\unicode[STIX]{x1D714}_{\mathscr{X}^{+}/S^{+}}^{\log })=\unicode[STIX]{x1D714}_{{\mathscr{ X}^{\prime }}^{+}/S^{+}}^{\log }$ . It follows that $\text{div}_{{\mathscr{X}^{\prime }}^{+}}(\unicode[STIX]{x1D714})=h^{\ast }(\text{div}_{\mathscr{X}^{+}}(\unicode[STIX]{x1D714}))$ and this concludes the proof.◻

4.1.5. We recall from [Reference Mustaţă and NicaiseMN15, § 4.7] that the Kontsevich–Soibelman skeleton $\text{Sk}(X,\unicode[STIX]{x1D6E5}_{X},\unicode[STIX]{x1D714})$ is the closure in $\text{Bir}(X)$ of the set of divisorial points of $X^{\text{an}}$ where the weight function $\text{wt}_{\unicode[STIX]{x1D714}}$ reaches its minimal weight, namely

$$\begin{eqnarray}\text{wt}_{\unicode[STIX]{x1D714}}(X,\unicode[STIX]{x1D6E5}_{X})=\inf \{\text{wt}_{\unicode[STIX]{x1D714}}(x)\,|\,x\in \text{Div}(X)\}\in \mathbb{R}\cup \{-\infty \}.\end{eqnarray}$$

A priori the weight function associated to a rational pluricanonical form may have minimal weight $-\infty$ , hence the corresponding Kontsevich–Soibelman skeleton would be empty. We prove that this does not occur for $\unicode[STIX]{x1D6E5}_{X}$ -logarithmic pluricanonical forms.

Proof. Let $\mathscr{X}^{+}$ be a log-regular model of $X^{+}$ and let $y$ be a divisorial point of $X^{\text{an}}$ . It suffices to prove that

$$\begin{eqnarray}\text{wt}_{\unicode[STIX]{x1D714}}(y)\geqslant \text{wt}_{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D70C}_{\mathscr{X}}(y))\end{eqnarray}$$

and that the equality holds if and only if $y$ is in $\text{Sk}(\mathscr{X}^{+})$ . As in the proof of Lemma 4.1.4, we consider the blow-up of $\mathscr{X}^{+}$ at the closure of $\text{red}_{\mathscr{X}}(y)$ : iterating this procedure a finite number of times, we obtain a log-regular model $\mathscr{Y}^{+}$ such that $y\in \text{Sk}(\mathscr{Y}^{+})$ .

Let $h:\mathscr{Z}^{+}\rightarrow \mathscr{W}^{+}$ be a morphism of this sequence. If $\overline{\{\text{red}_{\mathscr{W}}(y)\}}$ is a stratum of $D_{\mathscr{W}}$ , then the morphism $h$ induces a subdivision of the skeleton $\text{Sk}(\mathscr{W}^{+})$ , so $\unicode[STIX]{x1D70C}_{\mathscr{Z}}(y)=\unicode[STIX]{x1D70C}_{\mathscr{W}}(y)$ .

Otherwise, $\overline{\{\text{red}_{\mathscr{W}}(y)\}}$ is strictly contained in a stratum $V$ of ${\mathcal{D}}_{\mathscr{W}}$ . Let $E$ be the exceptional divisor of $h$ , $r=r_{h}+r_{v}$ be the codimension of $V$ in $\mathscr{W}$ , where $r_{h}$ and $r_{v}$ are the number of irreducible components of $\overline{\lceil \unicode[STIX]{x1D6E5}_{X}\rceil }$ and respectively of the special fiber $\mathscr{W}_{k}$ , containing $V$ . Let $r+j$ be the codimension of $\overline{\{\text{red}_{\mathscr{W}}(y)\}}$ , where $j\geqslant 1$ . We denote the projections onto $S^{+}$ by $s_{\mathscr{W}}:\mathscr{W}^{+}\rightarrow S^{+}$ and $s_{\mathscr{Z}}:\mathscr{Z}^{+}\rightarrow S^{+}$ and by $\unicode[STIX]{x1D70B}$ a uniformizer in $R$ . Then we have that