(2.1.1) Let $k$ be a field of characteristic zero. We set $R=k[\negthinspace[t]\negthinspace]$ and $K=k(\negthinspace(t)\negthinspace)$, and we fix a $t$-adic absolute value $|\cdot|_{K}$ on $K$ by setting $|t|_{K}=1/e$. For every $K$-scheme of finite type $Y$, we denote by $Y^{\mathrm{an}}$ the associated $K$-analytic space. For every separated $R$-scheme of finite type $\mathscr{Y}$ we set $\mathscr{Y}_{k}=\mathscr{Y}\times_{R}k$ and $\mathscr{Y}_{K}=\mathscr{Y}\times_{R}K$. Moreover, we will denote by $\widehat{\mathscr{Y}}$ the $t$-adic completion of $\mathscr{Y}$, by $\widehat{\mathscr{Y}}_{\eta}$ the generic fiber of $\widehat{\mathscr{Y}}$ in the category of $K$-analytic spaces and by

the canonical reduction map. The generic fiber $\widehat{\mathscr{Y}}_{\eta}$ is an analytic domain in $\mathscr{Y}_{K}^{\mathrm{an}}$, and it is equal to $\mathscr{Y}_{K}^{\mathrm{an}}$ if and only if $\mathscr{Y}$ is proper over $R$.

(2.1.2) Let $\mathscr{C}$ be a connected smooth algebraic curve over $k$. Let $s$ be a $k$-rational point on $\mathscr{C}$ and set $C=\mathscr{C}\setminus\{s\}$. We fix a uniformizer $t$ in $\mathcal{O}_{\mathscr{C},s}$. This choice determines an isomorphism of $k$-algebras $R\to\widehat{\mathcal{O}}_{\mathscr{C},s}$ and thus a morphism of $k$-schemes $\mathrm{Spec}\,R\to\mathscr{C}$.

(2.1.3) Let $X$ be a smooth and proper scheme over $C$ with geometrically connected fibers. A model of $X$ over $\mathscr{C}$ is a flat separated $\mathscr{C}$-scheme of finite type $\mathscr{X}$ endowed with an isomorphism of $C$-schemes $\mathscr{X}\times_{\mathscr{C}}C\to X$. Note that we do not require $\mathscr{X}$ to be proper over $\mathscr{C}$. Morphisms of models are defined in the usual way. We denote by $\mathscr{X}_{s}$ the fiber of $\mathscr{X}$ over $s$, by $\mathscr{X}_{R}$ the base change of $\mathscr{X}$ to $\mathrm{Spec}\,R$ and by $X_{K}$ the base change of $X$ to $\mathrm{Spec}\,K$. We denote by $K_{X}$ a relative canonical divisor for $X$ over $C$, and for every normal model $\mathscr{X}$ of $X$, we denote by $K_{\mathscr{X}}$ a relative canonical divisor for $\mathscr{X}$ over $\mathscr{C}$.

(2.1.4) For every $\mathscr{C}$-model $\mathscr{X}$ of $X$, we denote by $\mathscr{X}^{\mathrm{snc}}$ the subset of $\mathscr{X}$ consisting of the points where $\mathscr{X}$ is regular and $\mathscr{X}_{s}$ is a divisor with strict normal crossings (some authors use the terminology “simple normal crossings” instead). Thus $\mathscr{X}^{\mathrm{snc}}$ is the union of $X$ with the set of points $x$ of $\mathscr{X}_{s}$ such that $\mathcal{O}_{\mathscr{X},x}$ is regular and there exist a unit $u$ and a regular system of local parameters $(z_{1},\ldots,z_{n})$ in $\mathcal{O}_{\mathscr{X},x}$ and non-negative integers $N_{1},\ldots,N_{n}$ such that

The subset $\mathscr{X}^{\mathrm{snc}}$ is an open subscheme of $\mathscr{X}$ and it is again a $\mathscr{C}$-model of $X$. Moreover, if $\mathscr{X}$ is normal, then $\mathscr{X}^{\mathrm{snc}}_{s}$ is dense in $\mathscr{X}_{s}$. We say that $\mathscr{X}$ is an $snc$-model of $X$ if $\mathscr{X}=\mathscr{X}^{\mathrm{snc}}$, that is, if $\mathscr{X}$ is regular and $\mathscr{X}_{s}$ is a divisor with strict normal crossings. If $\mathscr{X}$ is a model of $X$ over $\mathscr{C}$, then a log resolution of $(\mathscr{X},\mathscr{X}_{s})$ is a proper morphism of $\mathscr{C}$-models $h:\mathscr{Y}\to\mathscr{X}$ such that $\mathscr{Y}$ is an $snc$-model of $X$.

(2.1.5) Let $h:\mathscr{Y}\to\mathscr{X}$ be a proper morphism of normal $\mathscr{C}$-models of $X$. Assume that $K_{\mathscr{X}}+(\mathscr{X}_{s})_{\mathrm{red}}$ is $\mathbb{Q}$-Cartier. Then the log pullback of $(\mathscr{X}_{s})_{\mathrm{red}}$ to $\mathscr{Y}$ is the unique $\mathbb{Q}$-Weil divisor $\Delta$ on $\mathscr{Y}$ such that $K_{\mathscr{Y}}+\Delta$ is $\mathbb{Q}$-linearly equivalent to

and $f_{*}\Delta=(\mathscr{X}_{s})_{\mathrm{red}}$.

(2.1.6) We will use the following notations from [MN13]. If $\mathscr{X}$ is a normal model of $X$ over $\mathscr{C}$, $x$ is a point of $\widehat{\mathscr{X}}_{\eta}$ and $D$ is a divisor on $\mathscr{X}$ that is supported on $\mathscr{X}_{s}$ and Cartier at $\mathrm{red}_{\mathscr{X}}(x)$, then we set

where $f$ is any element of the local ring of $\mathscr{X}$ at $\mathrm{red}_{\mathscr{X}}(x)$ such that $D=\mathrm{div}(f)$ locally at $\mathrm{red}_{\mathscr{X}}(x)$. It is clear that $v_{x}(D)$ is linear in $D$. If $\mathscr{X}$ is regular and $\omega$ is a non-zero rational section of $\omega_{\mathscr{X}_{R}/R}^{\otimes m}$, for some $m>0$ (for instance, an $m$-pluricanonical form on $X_{K}$) then we denote by $\mathrm{div}_{\mathscr{X}}(\omega)$ the corresponding divisor on $\mathscr{X}_{R}$.

(2.2.1) A $dlt$-model of $X$ is a normal proper $\mathscr{C}$-model $\mathscr{X}$ of $X$ such that $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ is a $dlt$-pair. This means that $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ is log canonical and that each log canonical center of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ has non-empty intersection with $\mathscr{X}^{\mathrm{snc}}$. In particular, every proper $snc$-model of $X$ is a $dlt$-model. An equivalent formulation of the definition is the following: $K_{\mathscr{X}}+(\mathscr{X}_{s})_{\mathrm{red}}$ is $\mathbb{Q}$-Cartier, and for every log resolution $h:\mathscr{Y}\to\mathscr{X}$ of $(\mathscr{X},\mathscr{X}_{s})$ and every irreducible component $E$ of $\mathscr{Y}_{s}$, the multiplicity of $E$ in the log pullback $\Delta$ of $(\mathscr{X}_{s})_{\mathrm{red}}$ to $\mathscr{Y}$ is at most $1$. Moreover, if it is equal to $1$, then $h(E)$ must have non-empty intersection with $\mathscr{X}^{\mathrm{snc}}$. In practice, we will apply the $dlt$ property via Lemma 3.2.3 below.

(2.2.2) We say that a $dlt$-model $\mathscr{X}$ of $X$ is a good minimal model if $\mathscr{X}$ is $\mathbb{Q}$-factorial and $K_{\mathscr{X}}+(\mathscr{X}_{s})_{\mathrm{red}}$ is semi-ample over $\mathscr{C}$.

(2.2.3) For every $dlt$-model $\mathscr{X}$ of $X$, we can define the dual complex $\mathcal{D}((\mathscr{X}_{s})_{\mathrm{red}})$ for the $dlt$-pair $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ by gluing cells corresponding to irreducible components of intersections of irreducible components of $\mathscr{X}_{s}$, as in Definition 8 in [dFKX12]. When $k$ is not algebraically closed, we note that we only glue cells corresponding to irreducible components (instead of geometrically irreducible components). In other words, $\mathcal{D}((\mathscr{X}_{s})_{\mathrm{red}})$ is the quotient of the $\mathrm{Gal}(\bar{k}/k)$-equivariant dual complex constructed in [dFKX12, §31].

(2.2.4) For every $dlt$-model $\mathscr{X}$ of $X$, the log canonical centers of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$ are the irreducible components of intersections of irreducible components of $(\mathscr{X}_{s})_{\mathrm{red}}$, by [Ko13, 4.16]. These are also precisely the closures in $\mathscr{X}_{s}$ of the connected components of intersections of irreducible components of $\mathscr{X}^{\mathrm{snc}}_{s}$ (since these connected components are the log canonical centers of $(\mathscr{X}^{\mathrm{snc}},(\mathscr{X}_{s}^{\mathrm{snc}})_{\mathrm{red}})$). Thus, the dual intersection complex $\mathcal{D}((\mathscr{X}_{s})_{\mathrm{red}})$ is the same as the dual intersection complex of the strict normal crossings divisor $\mathscr{X}^{\mathrm{snc}}_{s}$, and the cells of this complex correspond bijectively to the log canonical centers of $(\mathscr{X},(\mathscr{X}_{s})_{\mathrm{red}})$. See Section 2 in [dFKX12] for more background.

(2.2.5) Let $\mathscr{X}_{1}$ and $\mathscr{X}_{2}$ be two $dlt$-models of $X$ over $\mathscr{C}$. We say that $\mathscr{X}_{1}$ and $\mathscr{X}_{2}$ are crepant birational if there exist a normal proper $\mathscr{C}$-model $\mathscr{Y}$ of $X$ and morphisms of $\mathscr{C}$-models $f_{i}:\mathscr{Y}\to\mathscr{X}_{i}$ for $i=1,2$ such that the log pullbacks of $(\mathscr{X}_{1,s})_{\mathrm{red}}$ and $(\mathscr{X}_{2,s})_{\mathrm{red}}$ coincide (see [Ko13, 2.23]). Note that we can always assume that $\mathscr{Y}$ is an $snc$-model, by taking a log resolution of $(\mathscr{Y},\mathscr{Y}_{s})$. The following theorem collects two fundamental results from the Minimal Model Program.