The map $\ev$ is well-defined by Theorem 3.1 (i). It is a homeomorphism onto its image by Corollary 2.5 and the fact that any model is dominated by an SNC model. As $X$ is compact, we only need to show that $\ev(X)$ is dense in $\varprojlim\Delta_{\mathcal{X}}$. Pick $s=(s_{\mathcal{X}})_{\mathcal{X}}\in\varprojlim\Delta_{\mathcal{X}}$ and fix an SNC model $\mathcal{X}$. If $\mathcal{Y}$ is an SNC model dominated by $\mathcal{X}$, then $\ev_{\mathcal{X}}\circ\emb_{\mathcal{X}}=\id$ yields $\ev_{\mathcal{Y}}(\emb_{\mathcal{X}}(s_{\mathcal{X}}))=s_{\mathcal{Y}}$. Hence $s=\lim_{\mathcal{X}}\ev(\emb_{\mathcal{X}}(s_{\mathcal{X}}))\in\overline{\ev(X)}$. ∎

By definition of $c_{\mathcal{X}}$ we have $c_{\mathcal{X}}(x)\in E_{i}$ for a given $i\in I$ iff $\langle\ev_{\mathcal{X}}(x),E_{i}\rangle>0$, and it follows that $\ev_{\mathcal{X}}(x)$ lies in the relative interior of the simplex $\sigma_{J}$ for the maximal $J\subset I$ such that $c_{\mathcal{X}}(x)\in E_{J}$. Property (b) in Theorem 3.1 then shows that $c_{\mathcal{X}}(p_{\mathcal{X}}(x))$ is the generic point of $E_{J}$, which proves (i).

Let us prove (ii). For each $x\in X$ we have

$$\varphi_{D}(p_{\mathcal{X}}(x))=\langle D,\ev_{\mathcal{X}}(p_{\mathcal{X}}(x))\rangle=\langle D,\ev_{\mathcal{X}}\circ\emb_{\mathcal{X}}\circ\ev_{\mathcal{X}}(x)\rangle=\langle D,\ev_{\mathcal{X}}(x)\rangle=\varphi_{D}(x),$$

using the identity $\ev_{\mathcal{X}}\circ\emb_{\mathcal{X}}=\id$.

∎

(i) amounts to the fact that $\varphi_{D}\circ p_{\mathcal{X}}=\varphi_{D}$ for all $D\in\Div_{0}(\mathcal{Y})$, which is a special case of Lemma 3.4.

Let us now prove (ii). The map $\emb^{\prime}_{\mathcal{Y}}:=p_{\mathcal{X}}\circ\emb_{\mathcal{Y}}:\Delta_{\mathcal{Y}}\to X$ is continuous, and the previous identity implies that $\ev_{\mathcal{Y}}\circ\emb^{\prime}_{\mathcal{Y}}=\ev_{\mathcal{Y}}\circ\emb_{\mathcal{Y}}=\id$. By the uniqueness part of Theorem 3.1 it suffices to prove that $c_{\mathcal{Y}}\circ\emb^{\prime}_{\mathcal{Y}}=c_{\mathcal{Y}}\circ\emb_{\mathcal{Y}}$ on $\Delta_{\mathcal{Y}}$. Pick $s\in\Delta_{\mathcal{Y}}$ and set $x:=\emb_{\mathcal{Y}}(s)$, $x^{\prime}:=\emb^{\prime}_{\mathcal{Y}}(s)$. On the one hand (i) shows that

$$p_{\mathcal{Y}}(x^{\prime})=p_{\mathcal{Y}}\circ p_{\mathcal{X}}\circ\emb_{\mathcal{Y}}(s)=p_{\mathcal{Y}}\circ\emb_{\mathcal{Y}}(s)=x,$$

so $c_{\mathcal{Y}}(x^{\prime})\in\overline{\{c_{\mathcal{Y}}(x)\}}$ by (i) of Lemma 3.4. On the other hand $p_{\mathcal{X}}(x)=x^{\prime}$ by definition, so $c_{\mathcal{X}}(x)\in\overline{\{c_{\mathcal{X}}(x^{\prime})\}}$ and hence $c_{\mathcal{Y}}(x)\in\overline{\{c_{\mathcal{Y}}(x^{\prime})\}}$ by continuity of the map $\mathcal{X}\to\mathcal{Y}$ for the Zariski topology. ∎

By Corollary 3.2 it suffices to show that $\lim_{\mathcal{X}}\ev\circ p_{\mathcal{X}}=\ev$, which amounts to proving $\lim_{\mathcal{X}}\ev_{\mathcal{Y}}\circ p_{\mathcal{X}}=\ev_{\mathcal{Y}}$ for each $\mathcal{Y}$. This follows from (i) of Proposition 3.5. ∎

Upon multiplying by $\varpi^{m}$ with $m\gg 1$, we may assume that $\mathfrak{a}\subset\mathcal{O}_{\mathcal{X}}$ is a vertical ideal sheaf. Pick $J\subset I$ such that $E_{J}$ is non-empty, choose a point $\xi\in E_{J}$ and let $f_{1},\dots,f_{m}$ be generators of $\mathfrak{a}$ at $\xi$. With the notation introduced in the proof of Theorem 3.1 we then have

(3.5)

$$\log|\mathfrak{a}|(\emb_{\mathcal{X}}(s))=\max\left\{-\val_{\mathcal{X},s}(f_{i}),\,i=1,\dots,m\right\}.$$

By (3.4) each function $s\mapsto-\val_{\mathcal{X},s}(f_{i})$ is piecewise affine and convex on $\sigma_{J}$, proving (i). To prove (ii), pick any $x\in X$, set $\xi:=c_{\mathcal{X}}(x)$ and let $J\subset I$ be the set of indices $j\in I$ such that $\xi\in E_{j}$. Arguing similarly with generators of $\mathfrak{a}$, it is enough to show that $|f(x)|\leq|f(p_{\mathcal{X}}(x))|$ for each $f\in\mathcal{O}_{\mathcal{X},\xi}$. Note that the seminorm $f\mapsto|f(x)|$ extends by continuity to $\widehat{\mathcal{O}}_{\mathcal{X},\xi}$ since $\xi=c_{\mathcal{X}}(x)$. Writing in the notation of Remark 3.8 $f=\sum_{\alpha\in\mathbf{N}^{L}}f_{\alpha}z^{\alpha}\in\widehat{\mathcal{O}}_{\mathcal{X},\xi}$ we then have

$$|f(x)|\leq\sup_{f_{\alpha}\neq 0}\prod_{j\in J}|z_{j}(x)|^{\alpha_{j}}$$

by the ultrametric property, using that $|f_{\alpha}(x)|=1$ since each non-zero $f_{\alpha}\in\mathcal{O}_{\mathcal{X},\xi}$ is a unit. On the other hand, if we set $s_{j}:=-\log|z_{j}(x)|$ for $j\in J$ then we have by definition $p_{\mathcal{X}}(x)=\emb_{\mathcal{X}}(s)$, hence

$$\sup_{f_{\alpha}\neq 0}\prod_{j\in J}|z_{j}(x)|^{\alpha_{j}}=|f(p_{\mathcal{X}}(x))|$$

and the result follows. ∎

Step 1. Given a finite set $L$ and a field $\kappa$, we rely on basic toric geometry (cf. [KKMS, Ful93, Oda88]) to show that $Z:=\mathbf{A}^{L}_{\kappa}=\Spec\kappa[t_{i},\,i\in L]$ and its coordinate hyperplanes $(H_{i})_{i\in L}$ satisfy an analogue of (i)-(iv). Set $T:=(\mathbf{G}_{m,\kappa})^{L}$ to be the multiplicative split torus of dimension $L$ over $\kappa$. The fan $\Sigma$ of the toric $\kappa$-variety $Z$ consists of the cones $\hat{\sigma}_{J}=\sum_{j\in J}\mathbf{R}_{+}e_{j}$, $J\subset L$. For each $s\in\mathbf{R}_{+}^{L}$ let

$$\val_{Z,s}:\kappa[[t_{i},\,i\in L]]\to\mathbf{R}_{+}$$

be the monomial valuation with $\val_{Z,s}(t_{i})=s_{i}$ for $i\in L$, so that the center of $\val_{Z,s}$ on $Z$ is the generic point of $H_{J}:=\bigcap_{j\in J}H_{j}$ for all $s$ in the relative interior of $\hat{\sigma}_{J}$.

Let $\Sigma^{\prime}$ be a simplicial fan decomposition of $\Sigma$. The toric $\kappa$-variety $Z^{\prime}$ attached to $\Sigma^{\prime}$ comes with a $T$-equivariant proper birational morphism $\rho:Z^{\prime}\to Z$ satisfying the following properties:

• (a)

  $Z^{\prime}$ is normal (because it is toric), and all toric Weil divisors of $Z^{\prime}$ are $\mathbf{Q}$-Cartier (since $\Sigma^{\prime}$ is simplicial).

• (b)

  There is a bijection between the set of rays $(R_{i})_{i\in L^{\prime}}$ of $\Sigma^{\prime}$ and the toric prime divisors $(H^{\prime}_{i})_{i\in L^{\prime}}$ of $Z^{\prime}$, in such a way that for each $s\in R_{i}\setminus\{0\}$ the center of $\val_{Z,s}$ on $Z^{\prime}$ is the generic point of $H^{\prime}_{i}$.

• (c)

  For each $J^{\prime}\subset L^{\prime}$ the intersection $H^{\prime}_{J^{\prime}}:=\bigcap_{j\in J^{\prime}}H^{\prime}_{j}$ is normal, irreducible, and non-empty iff $\hat{\sigma}^{\prime}_{J^{\prime}}=\sum_{j\in J^{\prime}}R_{j}$ is a cone of $\Sigma^{\prime}$. In this case $H^{\prime}_{J^{\prime}}$ has codimension $|J^{\prime}|$, and its generic point is the center of $\val_{Z,s}$ on $Z^{\prime}$ for all $s$ in the relative interior of $\hat{\sigma}^{\prime}_{J^{\prime}}$.

• (d)

  For each toric divisor $G$ of $Z^{\prime}$, the map $s\mapsto\val_{Z,s}(G)$ is linear on each cone of $\Sigma^{\prime}$.

With the notation of (c), assume that $H^{\prime}_{J^{\prime}}$ is non-empty and let $\hat{\sigma}_{J}$ be the smallest cone of $\Sigma$ containing $\hat{\sigma}^{\prime}_{J^{\prime}}$. We then have $\rho(H^{\prime}_{J^{\prime}})=H_{J}$, and we claim that

(3.7)

$$\rho_{*}\mathcal{O}_{H^{\prime}_{J^{\prime}}}=\mathcal{O}_{H_{J}}.$$

Indeed, denote by $\zeta^{\prime}_{J^{\prime}}$ and $\zeta_{J}$ the generic points of $H^{\prime}_{J^{\prime}}$ and $H_{J}$ respectively. Since $H_{J}$ is normal, (3.7) amounts to the fact $\kappa(\zeta_{J})$ is algebraically closed in $\kappa(\zeta^{\prime}_{J^{\prime}})$ (cf. [EGA, III.4.3.12]). But $H^{\prime}_{J^{\prime}}$ is the closure of a $T$-orbit $(H^{\prime}_{J^{\prime}})^{0}$ in $Z^{\prime}$, mapping to the $T$-orbit $H_{J}^{0}:=(\bigcap_{j\in J}H_{j})\setminus(\bigcup_{j\notin J}H_{j})$ in $Z$. The stabilizer of $H_{J}^{0}$ in $T$ is $(\mathbf{G}_{m,\kappa})^{J}$, so the $T$-equivariant morphism $(H^{\prime}_{J^{\prime}})^{0}\to H_{J}^{0}$ has geometrically integral fibers. In particular $\zeta^{\prime}_{J^{\prime}}$ is the generic point of the fiber over $\zeta_{J}$, and it follows as desired that $\kappa(\zeta_{J})$ is algebraically closed in $\kappa(\zeta^{\prime}_{J^{\prime}})$ (cf. [EGA, IV.4.5.9]).

Step 2. Let $h\in\PA(\Delta_{\mathcal{X}})_{\mathbf{Z}}$ be a strictly convex support function for $\Delta^{\prime}$. We define $\mathcal{X}^{\prime}$ as the blow-up of $\mathcal{X}$ along the vertical fractional ideal sheaf $\mathfrak{a}_{h}$ given in Definition 3.10.

Let $\xi\in\mathcal{X}_{0}$ be a given point and use the notation of Remark 3.8. Since $\mathcal{X}$ and $Z:=\mathbf{A}^{L}_{\kappa(\xi)}$ are excellent we get a diagram

where $p$ and $q$ are regular, i.e. flat and with (geometrically) regular fibers (but a priori not of finite type, as opposed to a smooth morphism). By Remark 3.8 we have

(3.8)

$$p_{*}\val_{\widehat{\mathcal{X}}_{\xi},s}=\val_{\mathcal{X},s}\text{ and }q_{*}\val_{\widehat{\mathcal{X}}_{\xi},s}=\val_{Z,s}$$

for all $s\in\sigma_{I_{\xi}}$. The subdivision of $\sigma_{I_{\xi}}$ defined by $\Delta^{\prime}$ induces a simplicial fan decomposition $\Sigma^{\prime}$ of $\mathbf{R}_{+}^{L}$, to which the results of Step 1 apply. Since $h$ is a support function of $\Delta^{\prime}$, the toric $\kappa(\xi)$-variety $Z^{\prime}$ attached to $\Sigma^{\prime}$ coincides in fact with the blow-up of $Z$ along the toric fractional ideal sheaf

$$\mathfrak{b}_{h}:=\sum\{\mathcal{O}_{Z}(H_{m}),\,m\in\mathbf{Z}^{I_{\xi}},\,\langle m,\cdot\rangle\leq h\text{ on }\sigma_{\xi}\},$$

where we have set $H_{m}:=\sum_{i\in I_{\xi}}m_{i}H_{i}$. Comparing with (3.6), we see that

$$p^{-1}\mathfrak{a}_{h}\cdot\widehat{\mathcal{O}}_{\mathcal{X},\xi}=q^{-1}\mathfrak{b}_{h}\cdot\widehat{\mathcal{O}}_{\mathcal{X},\xi}.$$

Since blow-ups commute with flat base change (cf. [Liu, 8.1.12]), $\widehat{\mathcal{X}}^{\prime}_{\xi}:=\mathcal{X}^{\prime}\times_{\mathcal{X}}\spec\widehat{\mathcal{X}}_{\xi}$ sits in a commutative diagram

(3.9)

where the two squares are Cartesian. The morphisms $p^{\prime}$ and $q^{\prime}$ are also regular, since the latter property is preserved under finite type base change (cf. [EGA, IV.6.8.3]).

Let $(e^{\prime}_{i})_{i\in I^{\prime}_{\xi}}$ be the set of vertices of $\Delta^{\prime}$ contained in $\sigma_{I_{\xi}}$, so that each ray $\mathbf{R}_{+}e^{\prime}_{i}$ belongs to the fan $\Sigma^{\prime}$. If we let $H^{\prime}_{i}$ be the corresponding toric prime divisor of $Z^{\prime}$ and pick $J^{\prime}\subset I^{\prime}_{\xi}$ then $H^{\prime}_{J^{\prime}}=\bigcap_{j\in J^{\prime}}H_{j}$ is normal, irreducible, and non-empty iff the $e^{\prime}_{j}$, $j\in J^{\prime}$ span a face $\sigma^{\prime}_{J^{\prime}}$ of $\Delta^{\prime}$, by property (c). Since $q^{\prime}$ is regular, if follows that $q^{\prime-1}(H^{\prime}_{J^{\prime}})$ is normal and is either empty or of codimension $|J^{\prime}|$. It is furthermore irreducible, by (3.7) and Lemma 3.12 below. In particular, $(q^{\prime-1}(H^{\prime}_{i}))_{i\in I^{\prime}_{\xi}}$ is exactly the set of irreducible components of the special fiber of $\widehat{\mathcal{X}}^{\prime}_{\xi}$. We then easily obtain the analogue of (i)-(iv) of Theorem 3.11 with $\widehat{\mathcal{X}}^{\prime}_{\xi}$, $\sigma_{I_{\xi}}$ and $\val_{\widehat{\mathcal{X}}_{\xi}}$ in place of $\mathcal{X}^{\prime}$, $\Delta_{\mathcal{X}}$ and $\val_{\mathcal{X}}$.

On the other hand, for each irreducible component $E^{\prime}$ of $\mathcal{X}^{\prime}_{0}$ dominating $\xi$, we claim that the divisor $p^{\prime-1}(E^{\prime})$ is irreducible. Indeed, each irreducible component of the divisor $p^{\prime-1}(E^{\prime})$ is of the form $q^{\prime-1}(H^{\prime}_{i})$ for some $i\in I^{\prime}_{\xi}$. If we denote by $\xi^{\prime}$ and $\eta^{\prime}_{i}$ the generic points of $E^{\prime}$ and $p^{\prime-1}(H_{i}^{\prime})$ respectively then we have on the one hand $p^{\prime}(\eta^{\prime}_{i})=\xi^{\prime}$ since $p^{\prime}$ is flat. On the other hand, $\eta_{i}^{\prime}$ is the center of $\val_{\widehat{\mathcal{X}}_{\xi},e^{\prime}_{i}}$ on $\widehat{\mathcal{X}}^{\prime}_{\xi}$, hence $p^{\prime}(\eta_{i}^{\prime})=c_{\mathcal{X}^{\prime}}(\val_{\mathcal{X},e^{\prime}_{i}})$ thanks to (3.8). For dimension reason it follows that $\emb_{\mathcal{X}}(e^{\prime}_{i})=x_{E^{\prime}}\in X$, and the injectivity of $\emb_{\mathcal{X}}$ shows that $i$ is uniquely determined by $E^{\prime}$, which implies as desired that $p^{\prime-1}(E^{\prime})$ is irreducible.

We may thus write the irreducible components of $\mathcal{X}^{\prime}_{0}$ dominating $\xi$ as $(E^{\prime}_{i})_{i\in I^{\prime}_{\xi}}$, with the property that

$$p^{\prime-1}(E^{\prime}_{i})=q^{\prime-1}(H^{\prime}_{i}).$$

By flat descent it follows that $E^{\prime}_{i}$ is normal over $\xi$. It is also $\mathbf{Q}$-Cartier, since a Weil divisor is Cartier at a point iff its restriction to the formal neighborhood of that point is Cartier. It is now easy to conclude the proof of (i)-(iv), using the analogous properties for $\widehat{\mathcal{X}}^{\prime}_{\xi}$ together with (3.8). ∎

Note first that $U$ and $U^{\prime}$ are normal by [EGA, IV.6.5.4]. Since direct images commute with flat base change we have $f_{*}\mathcal{O}_{U^{\prime}}=\mathcal{O}_{U}$, which implies that $f$ has connected fibers as a consequence of the theorem on formal functions (cf. [EGA, III.4.3.2]). Since $U$ is connected and $f$ is closed, surjective and has connected fibers, it follows that $U^{\prime}$ is connected, hence irreducible since it is normal. ∎

If $s\in\Delta_{\mathcal{X}}$ is a rational point then Theorem 3.11 yields a vertical blow-up $\mathcal{X}^{\prime}$ such that $\emb_{\mathcal{X}^{\prime}}(s)=x_{E^{\prime}}$ for some irreducible component $E^{\prime}$ of $\mathcal{X}^{\prime}$. Conversely, it $\emb_{\mathcal{X}}(s)$ is a divisorial point then the corresponding valuation takes rational values on the local equations of the components of $\mathcal{X}_{0}$, which shows that $s$ is a rational point of $\Delta_{\mathcal{X}}$. ∎