All schemes considered in this paper are separated and Noetherian, and all ideal sheaves are coherent. Let $R$ be a complete discrete valuation ring with fraction field $K$ and residue field $k$. We shall assume that $k$ has characteristic zero (but we don’t require it to be algebraically closed). Let $\varpi\in R$ be a uniformizing parameter and normalize the corresponding absolute value on $K$ by $\log|\varpi|^{-1}=1$. Each choice of a field of representatives of $k$ in $R$ then induces an isomorphism $R\simeq k[\![t]\!]$ by Cohen’s structure theorem. Write $S:=\spec R$.

We will use the following terminology. An $S$-variety is a flat integral $S$-scheme $\mathcal{X}$ of finite type. We denote by $\mathcal{X}_{0}$ its special fiber and by $\mathcal{X}_{K}$ its generic fiber, and we write $\kappa(\xi)$ for the residue field of a point $\xi\in\mathcal{X}$. An ideal sheaf $\mathfrak{a}$ on $\mathcal{X}$ is vertical if it is co-supported on the special fiber, and a fractional ideal sheaf $\mathfrak{a}$ is vertical if $\varpi^{m}\mathfrak{a}$ is a vertical ideal sheaf for some positive integer $m$. A vertical blow-up $\mathcal{X}^{\prime}\to\mathcal{X}$ is the blow-up of a vertical (fractional) ideal sheaf.

Except for Appendix B, we will use additive notation for Picard groups, and we write $\mathcal{L}+\mathcal{M}:=\mathcal{L}\otimes\mathcal{M}$, and $m\mathcal{L}:=\mathcal{L}^{\otimes m}$ for $\mathcal{L},\mathcal{M}\in\Pic(\mathcal{X})$. We denote by $\Div_{0}(\mathcal{X})$ the group of vertical Cartier divisors of $\mathcal{X}$, i.e. those Cartier divisors on $\mathcal{X}$ that are supported on the special fiber. When $\mathcal{X}$ is normal, it is easy to see that $\Div_{0}(\mathcal{X})$ is a free $\mathbf{Z}$-module of finite rank and that the natural sequence

$$0\to\mathbf{Z}\mathcal{X}_{0}\to\Div_{0}(\mathcal{X})\to\Pic(\mathcal{X})\to\Pic(\mathcal{X}_{K})$$

is exact. The last arrow to the right is surjective if $\mathcal{X}$ is for instance regular.

Given an $S$-variety $\mathcal{X}$ let $(E_{i})_{i\in I}$ be the (finite) set of irreducible components of its special fiber $\mathcal{X}_{0}$. For each subset $J\subset I$ set $E_{J}:=\bigcap_{j\in J}E_{j}$.

Note that (i) implies that $\mathcal{X}$ is regular. Given a point $\xi$ of $\mathcal{X}_{0}$, let $I_{\xi}\subset I$ be the set of components $E_{i}$ containing $\xi$, and pick a local equation $z_{i}\in\mathcal{O}_{\mathcal{X},\xi}$ of $E_{i}$ at $\xi$. Condition (i) means that $\{z_{i},\,i\in I_{\xi}\}$ can be completed to a uniformizing system of parameters of $\mathcal{O}_{\mathcal{X},\xi}$. Condition (ii) is not imposed in the usual definition of a simple normal crossing divisor, but can always be achieved from (i) by further blowing-up along components of the possibly non-connected $E_{J}$’s. Since $k$ has characteristic zero, each $S$-variety is a $\mathbf{Q}$-scheme, which is furthermore excellent since it has finite type over $S$. It therefore follows from  [Tem06] that for any $S$-variety $\mathcal{X}$ with smooth generic fiber there exists a vertical blow-up $\mathcal{X}^{\prime}\to\mathcal{X}$ such that $\mathcal{X}^{\prime}$ is SNC.

Let $\mathcal{X}$ be a normal projective $S$-variety.

We will then simply say that $\mathcal{L}$ is nef.

We recall the following standard notions.

Note that the $\mathbf{R}$-vector space $N^{1}(\mathcal{X}/S)$ is finite dimensional. Indeed Lemma 1.2 shows that the restriction map $N^{1}(\mathcal{X}/S)\to N^{1}(\mathcal{X}_{0}/k)$ is injective, and the latter space is finite dimensional since $\mathcal{X}_{0}$ is projective over $k$. Observe also that $\Nef(\mathcal{X}/S)$ is a closed convex cone of $N^{1}(\mathcal{X}/S)$. Lemma 1.2 implies that $\Nef(\mathcal{X}/S)=\Nef(\mathcal{X}_{0}/k)\cap N^{1}(\mathcal{X}/S)$ under the injection $N^{1}(\mathcal{X}/S)\to N^{1}(\mathcal{X}_{0}/k)$.

We have the following standard fact:

Recall that an $\mathbf{R}$-line bundle on $\mathcal{X}$ (resp. $\mathcal{X}_{K}$) is ample if it can be written as a positive linear combination of ample line bundles. As a direct consequence of Lemma 1.4 we get:

We shall use the following version of the Negativity Lemma, cf.  [KM98, Lemma 3.39]. The proof we give is a variant of the argument used in [BdFF10, Proposition 2.11].