Let $I$ be the set of vertices in $\Delta$, let $L\subset I$ be the set of vertices contained in $\sta_{\Delta}(\sigma)$ and $J\subset L$ the set of vertices of $\sigma$. Thus $\sigma=\sigma_{J}$.

Consider the simplicial projective subdivision $\Delta^{\prime}=\Delta^{\prime}(\varepsilon)$ constructed in §6.2. For $j\in L$, $e^{\prime}_{j}:=\varepsilon e_{j}+(1-\varepsilon)v$ is a vertex of $\Delta^{\prime}$. Recall that $\sigma^{\prime}=\sigma^{\prime}_{J}$ is the face of $\Delta^{\prime}$ containing $v$ in its relative interior. Since $\varphi|_{\sigma}$ is assumed affine in a neighborhood of $v$, we may choose $\varepsilon>0$ small enough that:

• •

  $\varphi$ is affine on $\sigma^{\prime}\subset\sigma$

• •

  $\varphi$ is affine on each segment $[v,e^{\prime}_{j}]$, $j\in L$.

Let $\rho:\mathcal{X}^{\prime}\to\mathcal{X}$ be the vertical blow-up corresponding to the subdivision $\Delta^{\prime}$ of $\Delta$ as in Theorem 3.11. Note that $\rho$ induces a generically finite map $E^{\prime}_{J}\to E_{J}$ of projective $k$-varieties. Indeed, $E_{J}$ (resp. $E^{\prime}_{J}$) is the closure of the center of $v$ on $\mathcal{X}$ (resp. $\mathcal{X}^{\prime}$), and both have codimension $|J|$ by Theorem 3.11.

Recall that $\varphi=\varphi_{G}$ for some $G\in\Div_{0}(\mathcal{Y})$. We may assume that the determination $\mathcal{Y}$ of $\varphi$ dominates $\mathcal{X}^{\prime}$, so that $\pi$ factors as $\pi=\rho\circ\mu$ with $\mu:\mathcal{Y}\to\mathcal{X}^{\prime}$. As we shall see shortly, a first computation shows:

The key observation is now the following positivity property:

Grant this result for the moment. Lemma 6.4 and the projection formula yield

$$\deg(\rho|_{E^{\prime}_{J}})E_{J}\cdot\left(\theta_{\mathcal{X}}+\varphi(v)\,\mathcal{X}_{0}+\sum_{j\in L}D_{v}\varphi(e_{j})\,b_{j}E_{j}\right)\cdot\mathcal{A}^{n-p-1}=E^{\prime}_{J}\cdot(\rho^{*}\theta_{\mathcal{X}}+\mu_{*}G)\cdot\rho^{*}\mathcal{A}^{n-p-1}.$$

Here the right-hand side is non-negative by Lemma 6.5, since $\rho^{*}\mathcal{A}$ is nef, and we get

(6.3)

$$\sum_{j\in L}D_{v}\varphi(e_{j})\,b_{j}\left(E_{J}\cdot E_{j}\cdot\mathcal{A}^{n-p-1}\right)\geq-(E_{J}\cdot\theta_{\mathcal{X}}\cdot\mathcal{A}^{n-p-1})-\varphi(v)(E_{J}\cdot\mathcal{X}_{0}\cdot\mathcal{A}^{n-p-1}).$$

By induction, the $C^{0,1}$-norm of $\varphi|_{\sigma}$ is under control. Since $v$ belongs to $\sigma=\sigma_{J}$, this gives

$$|\varphi(v)|\leq C\text{ and }\max_{j\in J}|D_{v}\varphi(e_{j})|\leq C,$$

and (6.3) yields a lower bound

$$\sum_{j\in L\setminus J}D_{v}\varphi(e_{j})\,b_{j}\left(E_{J}\cdot E_{j}\cdot\mathcal{A}^{n-p-1}\right)\geq-C.$$

Now the convexity of $\varphi$ and the normalization $\sup_{X}\varphi=0$ show that

$$\max_{j\in L\setminus J}D_{v}\varphi(e_{j})\leq\max_{j\in L\setminus J}(\varphi(e_{j})-\varphi(v))\leq-\varphi(v)\leq C.$$

Here $E_{j}|_{E_{J}}$ is a non-zero effective divisor for $j\notin J$, hence $E_{J}\cdot E_{j}\cdot\mathcal{A}^{n-p-1}>0$ since $\mathcal{A}$ is ample. The previous inequality therefore implies, as desired, that $D_{v}\varphi(e)\geq-C$, since $e=e_{j}$ for some $j\in L\setminus J$. ∎

We write $v=\sum_{j\in J}s_{j}e_{j}$ with $s_{j}>0$ rational and $\sum_{j\in J}s_{j}=1$. Set $s_{i}=0$ for $i\in I\setminus J$. For $i\in I$ let $\varphi_{i}$ be the model function induced by the vertical divisor $b_{i}E_{i}\in\Div_{0}(\mathcal{X})$. This function is affine on each face of $\Delta$ and satisfies $\varphi_{i}(e_{j})=\delta_{ij}$ for all $j\in I$. Since $e^{\prime}_{j}=\varepsilon e_{j}+(1-\varepsilon)v$ for $j\in L$ we get:

$$\varphi_{i}(e^{\prime}_{j})=\begin{cases}\varepsilon+(1-\varepsilon)s_{i}&\text{if $i=j\in J$}\\ (1-\varepsilon)s_{i}&\text{if $i\neq j\in J$}\\ \varepsilon&\text{if $i=j\in L\setminus J$}\\ 0&\text{if $i\neq j\in L\setminus J$}\end{cases}$$

By Theorem 3.11, $E^{\prime}_{j}$ intersects $E^{\prime}_{J}$ iff $j\in L$. We thus have

$$\rho^{*}(b_{i}E_{i})|_{E^{\prime}_{J}}=\sum_{j\in L}\varphi_{i}(e^{\prime}_{j})b^{\prime}_{j}\,E^{\prime}_{j}|_{E^{\prime}_{J}}\ \text{for all $i\in I$}$$

and

$$(\mu_{*}G-\varphi(v)\rho^{*}\mathcal{X}_{0})|_{E^{\prime}_{J}}=\sum_{j\in L}(\varphi(e^{\prime}_{j})-\varphi(v))b^{\prime}_{j}\,E^{\prime}_{j}|_{E^{\prime}_{J}}$$

in $\Pic(E^{\prime}_{J})_{\mathbf{Q}}$, where we have set $b^{\prime}_{j}:=\ord_{E^{\prime}_{j}}(t)$.

Recall also that $\varphi$ is affine on each segment $[v,e^{\prime}_{i}]$, so that $D_{v}\varphi(e_{i})=\varepsilon^{-1}\left(\varphi(e^{\prime}_{i})-\varphi(v)\right)$ for $i\in L$. We can now compute in $\Pic(E^{\prime}_{J})_{\mathbf{Q}}$

$$\left.\rho^{*}\left(\sum_{i\in L}D_{v}\varphi(e_{i})\,b_{i}E_{i}\right)\right|_{E^{\prime}_{J}}=\sum_{i\in L}\varepsilon^{-1}\left(\varphi(e^{\prime}_{i})-\varphi(v)\right)\left(\sum_{j\in L}\varphi_{i}(e^{\prime}_{j})\,b^{\prime}_{j}\,E^{\prime}_{j}|_{E^{\prime}_{J}}\right)=\\ =\sum_{i\in J}\varepsilon^{-1}(\varphi(e^{\prime}_{i})-\varphi(v))\left(\varepsilon\,b^{\prime}_{i}\,E^{\prime}_{i}|_{E^{\prime}_{J}}+s_{i}\sum_{j\in J}(1-\varepsilon)\,b^{\prime}_{j}\,E^{\prime}_{j}|_{E^{\prime}_{J}}\right)+\\ +\sum_{i\in L\setminus J}\varepsilon^{-1}(\varphi(e^{\prime}_{i})-\varphi(v))\,\varepsilon\,b^{\prime}_{i}\,E^{\prime}_{i}|_{E^{\prime}_{J}}=\\ =\sum_{i\in L}(\varphi(e^{\prime}_{i})-\varphi(v))\,b^{\prime}_{i}\,E^{\prime}_{i}|_{E^{\prime}_{J}}+\varepsilon^{-1}(1-\varepsilon)\left(\sum_{i\in J}s_{i}(\varphi(e^{\prime}_{i})-\varphi(v))\right)\left(\sum_{j\in J}\,b^{\prime}_{j}\,{E^{\prime}_{j}}|_{E^{\prime}_{J}}\right)\\ =(\mu_{*}G-\varphi(v)\rho^{*}\mathcal{X}_{0})|_{E^{\prime}_{J}}.$$

The last equality follows from the fact that $\varphi$ is affine on the simplex $\sigma^{\prime}_{J}$ of $\Delta^{\prime}$ so that $\sum_{i\in J}s_{i}\varphi(e^{\prime}_{i})=\varphi(v)=\sum_{i\in J}s_{i}\varphi(v)$. This concludes the proof. ∎

Set $F:=\mu^{*}\mu_{*}G-G\in\Div_{0}(\mathcal{Y})_{\mathbf{Q}}$. The divisor $G$ is $\mu$-nef since $\mu^{*}(\rho^{*}\theta_{\mathcal{X}})+G$ is nef by assumption, and Lemma 1.6 therefore implies that $F$ is effective.

Let $W$ be the closure of the center of $v$ on $\mathcal{Y}$. Since the center of $v$ on $\mathcal{X}^{\prime}$ is the generic point of $E^{\prime}_{J}$, we must have $\mu(W)=E^{\prime}_{J}$. Note, however, that we do not claim $\dim W=\dim E^{\prime}_{J}$.

By Theorem 3.11, the function $\varphi_{\mu_{*}G}$ is affine on the face $\sigma^{\prime}_{J}$ of $\Delta^{\prime}$. But $\varphi_{G}$ is also affine on $\sigma^{\prime}_{J}$ by assumption, and we have

$$\varphi_{G}(e^{\prime}_{j})=b^{\prime}_{j}\ord_{E^{\prime}_{j}}(G)=\varphi_{\mu_{*}G}(e^{\prime}_{j})\quad\text{for all $j\in J$}.$$

It follows that $\varphi_{F}\equiv 0$ on $\sigma^{\prime}_{J}$, and in particular $v(F)=0$. But this means precisely that $W$ is not contained in $\supp F$, so that $F|_{W}$ is an effective $\mathbf{Q}$-Cartier divisor. Hence

$$\mu^{*}(\rho^{*}\theta_{\mathcal{X}}+\mu_{*}G)|_{E^{\prime}_{J}}=(\pi^{*}\theta_{\mathcal{X}}+\mu^{*}\mu_{*}G)|_{W}=(\pi^{*}\theta_{\mathcal{X}}+G)|_{W}+F|_{W}$$

is the sum of a nef class and an effective class. We conclude by Lemma 6.6 below. ∎

Let $k^{a}$ be an algebraic closure of $k$ and let $V^{\prime}$ be an irreducible component of (the reduction of) $V^{a}:=V\otimes k^{a}$. There exists a component $W^{\prime}$ of $W^{a}$ dominating $V^{\prime}$. Upon replacing $W$ and $V$ by $W^{\prime}$ and $V^{\prime}$ we are reduced to the case where $k$ is algebraically closed. Upon taking successive hyperplane sections of $W$ not containing any component of $F$ and choosing an irreducible component dominating $V$ we may then assume that $\mu$ is generically finite. In that case we have

$$(\deg\mu)\left(\alpha\cdot\beta^{\dim V-1}\right)_{V}=\left(\mu^{*}\alpha\cdot\mu^{*}\beta^{\dim V-1}\right)_{W}$$

and the result is then clear since $\mu^{*}\beta$ is nef. ∎