1 Introduction
If 
$f:X\rightarrow T$ is a smooth projective morphism of complex quasi-projective varieties, then by a celebrated theorem of Siu [Reference Siu13], [Reference Siu14], it is known that the plurigenera of the fibers 
$P_{m}(X_{t}):=h^{0}(mK_{X_{t}})$ are independent of the point 
$t\in T$. This result (and its generalizations to log pairs) is a fundamental fact of great importance in higher dimensional birational geometry. It plays a fundamental role in the construction of moduli spaces of varieties of log general type. Unluckily, over an algebraically closed field of characteristic 
$p>0$, this result does not generalize even to families of surfaces over a curve (or a discrete valuation ring (DVR)). In [Reference Lang12], it is shown that 
$P_{1}$ is not deformation invariant for Enriques surfaces in characteristic 2. In [Reference Katsura and Ueno11], it is shown that, in fact, the deformation invariance of plurigenera does not hold for certain elliptic surfaces, and in [Reference Suh15], there are examples of smooth families of surfaces of general type over any DVR of mixed characteristic for which 
$P_{1}$ is nonconstant (and, in fact, its value can jump by an arbitrarily big amount). On the positive side, in [Reference Katsura and Ueno11], it is shown that if 
$X\rightarrow \text{Spec}(R)$ is a smooth family of surfaces over a DVR in positive or mixed characteristic, then one can run the minimal model program (MMP) for 
$X$ (over an extension of 
$R$). As a consequence of this, it is observed that 
$\unicode[STIX]{x1D705}(X_{K})=\unicode[STIX]{x1D705}(X_{k})$, where 
$k$ is the residue field and 
$K$ is the fraction field of 
$R$. It should be noted that the minimal model program is established for semistable families of surfaces in positive or mixed characteristic (see [Reference Kawamata9]), for log canonical surfaces over excellent base schemes (see [Reference Tanaka19]) and for 3 folds over a field 
$k$ of characteristic 
$p\geqslant 7$ (see [Reference Hacon and Xu8] and [Reference Birkar3]). In this paper (Theorems 3.1 and 3.4), we generalize the result of Katsura and Ueno to log surfaces (smooth over a DVR) and we show the deformation invariance of certain plurigenera.
Theorem 1.1. Let 
$(X,B)$ be a klt pair which is log smooth, projective of dimension 
$2$ over an irreducible integral Noetherian scheme 
$S$, then 
$\unicode[STIX]{x1D705}(K_{X_{s}}+B_{s})$ is independent of 
$s\in S$. If, moreover, 
$K_{X}+B$ is big over 
$S$, then there exists an integer 
$m_{0}>0$ such that for any positive integer 
$m\in m_{0}\mathbb{N}$, we have
$$\begin{eqnarray}h^{0}(m(K_{X_{s}}+B_{s}))=h^{0}(m(K_{X_{s^{\prime }}}+B_{s^{\prime }}))\quad \forall s,s^{\prime }\in S\end{eqnarray}$$ and the log canonical model for 
$(X,B)$ over 
$S$ exists.
When 
$S$ is the spectrum of a DVR, we obtain a more precise result (see Theorems 3.1 and 3.4). The strategy is to reduce the proof of the above theorem to the case when 
$(X_{k},B_{k})$ is terminal and 
$\mathbf{B}(K_{X_{k}}+B_{k})$ contains no components of the support of 
$B_{k}$. In this case, we observe that the steps of a 
$K_{X_{k}}+B_{k}$ MMP are also steps of a 
$K_{X_{k}}$ MMP, and we are thus able to deduce the result from [Reference Katsura and Ueno11].
Remark 1.2. Many results and techniques in this paper were developed in the first author’s Ph.D. thesis [Reference Egbert4].
2 Preliminaries
Let 
$X$ be a normal quasi-projective variety over an algebraically closed field 
$k$ and 
$\text{WDiv}(X)$ the group of Weil divisors. If 
$B=\sum b_{i}B_{i}\in \text{WDiv}_{\mathbb{Q}}(X)$ is a 
$\mathbb{Q}$-divisor on 
$X$, then 
$\lfloor B\rfloor =\sum \lfloor b_{i}\rfloor B_{i}$, where 
$\lfloor b_{i}\rfloor =\text{max}\{n\in \mathbb{Z}|n\leqslant b_{i}\}$. We denote 
$\{B\}=B-\lfloor B\rfloor$ and 
$|B|=|\lfloor B\rfloor |+\{B\}$, where
$$\begin{eqnarray}|\lfloor B\rfloor |=\{D\in \text{WDiv}(X)|D\geqslant 0,~D-\lfloor B\rfloor =(f),~f\in K(X)\}.\end{eqnarray}$$ The stable base locus of 
$B$ is 
$\mathbf{B}(D)=\bigcap _{m\in \mathbb{N}}\text{Bs}(mD)$. Let 
$(X,B)$ be a pair so that 
$X$ is normal, 
$0\leqslant B$ is a 
$\mathbb{Q}$-divisor and 
$K_{X}+B$ is 
$\mathbb{Q}$-Cartier. If 
$\unicode[STIX]{x1D708}:X^{\prime }\rightarrow X$ is a proper birational morphism, then we write 
$K_{X^{\prime }}+B_{X^{\prime }}=\unicode[STIX]{x1D708}^{\ast }(K_{X}+B)$. We say that 
$(X,B)$ is Kawamata log terminal or klt (resp. terminal) if for any proper birational morphism 
$\unicode[STIX]{x1D708}:X^{\prime }\rightarrow X$, we have 
$\lfloor B_{X^{\prime }}\rfloor \leqslant 0$ (resp. 
$B_{X^{\prime }}\leqslant \unicode[STIX]{x1D708}_{\ast }^{-1}B+E$, where 
$E$ denotes the reduced exceptional divisor). We let 
$\mathbf{M}_{B}$ be the 
$b$-divisor defined by the sum of the strict transform of 
$B$ and the exceptional divisors (over 
$X$). We refer the reader to [Reference Kollár and Mori10] and [Reference Birkar, Cascini, Hacon and McKernan2] for the standard definitions of the minimal model program including extremal rays, flipping and divisorial contractions, running a minimal model program with scaling, log terminal and weak log canonical models.
Theorem 2.1. Let 
$(X,B)$ be a two-dimensional projective klt pair over an algebraically closed field 
$k$. Then
$$\begin{eqnarray}\overline{NE}(X)=\overline{NE}(X)_{K_{X}+B\geqslant 0}+\mathop{\sum }_{i\in I}\mathbb{R}_{{\geqslant}0}C_{i},\end{eqnarray}$$ where 
$I$ is countable, 
$(K_{X}+B)\cdot C_{i}<0$ and 
$C_{i}$ is rational. If 
$H$ is an ample 
$\mathbb{Q}$-divisor on 
$X$, then the set 
$\{i\in I|(K_{X}+B+H)\cdot C_{i}\leqslant 0\}$ is finite.
Proof. See [Reference Tanaka17, 3.13, 3.15] and [Reference Kollár and Mori10, 3.7]. ◻
Lemma 2.2. Let 
$X$ be a surface over an algebraically closed field 
$k$ and 
$(X,B)$ a projective klt pair. If 
$R$ is a 
$K_{X}+B$ negative extremal ray, then there exists a proper morphism 
$f:X\rightarrow X^{\prime }$ such that 
$f_{\ast }{\mathcal{O}}_{X}={\mathcal{O}}_{X^{\prime }}$ and 
$f$ contracts a curve 
$C\subset X$ if and only if 
$[C]=R$.
Proof. See [Reference Tanaka17, 3.21] and [Reference Kollár and Mori10, 3.7]. ◻
Theorem 2.3. Let 
$X$ be a projective surface over an algebraically closed field 
$k$. Assume that 
$(X,B)$ is klt. Then
(1) The ring
$R(K_{X}+B)=\bigoplus _{m\geqslant 0}H^{0}(m(K_{X}+B))$ is finitely generated.(2) If
$K_{X}+B$ is pseudo-effective, then $\unicode[STIX]{x1D705}(K_{X}+B)\geqslant 0$ and there exists a minimal model $\unicode[STIX]{x1D708}:X\rightarrow X^{\prime }$ such that $K_{X^{\prime }}+B^{\prime }=\unicode[STIX]{x1D708}_{\ast }(K_{X}+B)$ is semiample. If we write $K_{X}+B=\unicode[STIX]{x1D708}^{\ast }(K_{X^{\prime }}+B^{\prime })+F$, then $F\geqslant 0$ is $\unicode[STIX]{x1D708}$-exceptional and we have $F=N_{\unicode[STIX]{x1D70E}}(K_{X}+B)$.(3) If
$H$ is an ample $\mathbb{Q}$-divisor which is general in $N^{1}(X/X^{\prime })$ and $K_{X}+B+H$ is nef, then the MMP with scaling of $H$ yields a sequence of rational numbers $1\geqslant \unicode[STIX]{x1D706}_{1}>\unicode[STIX]{x1D706}_{2}>\cdots >\unicode[STIX]{x1D706}_{n}\geqslant 0$ and divisorial contractions $X=X_{0}\rightarrow X_{1}\rightarrow X_{2}\rightarrow \cdots \rightarrow X_{n}$, where $X_{i}=\operatorname{Proj}R(K_{X}+B+\unicode[STIX]{x1D706}_{i}H)$ and $K_{X_{i}}+B_{i}+tH_{i}$ is ample for $\unicode[STIX]{x1D706}_{i}\geqslant t>\unicode[STIX]{x1D706}_{i+1}$. If $\unicode[STIX]{x1D706}_{n}=0$, then $X\rightarrow X_{n}$ is a $K_{X}+B$ minimal model and if $\unicode[STIX]{x1D706}_{n}>0$, then $X_{n}\rightarrow Z$ is a $K_{X}+B$ Mori fiber space.
Proof. (1) and (2) follow immediately from [Reference Tanaka17]. To see (3), we proceed by induction. Assume that we have constructed 
$X\rightarrow X_{1}\rightarrow \cdots \rightarrow X_{i}$ and assume that 
$K_{X_{i}}+B_{i}+\unicode[STIX]{x1D706}_{i}H_{i}$ is ample where 
$B_{i}$ and 
$H_{i}$ denote the pushforwards of 
$B$ and 
$H$. Let 
$\unicode[STIX]{x1D706}_{i+1}:=\text{inf}\{t>0|K_{X_{i}}+B_{i}+tH_{i}~\text{is nef}\}$. It is easy to see that 
$0\leqslant \unicode[STIX]{x1D706}_{i+1}<\unicode[STIX]{x1D706}_{i}$ and 
$K_{X_{i}}+B_{i}+tH_{i}$ is ample for 
$\unicode[STIX]{x1D706}_{i}\geqslant t>\unicode[STIX]{x1D706}_{i+1}$. If 
$\unicode[STIX]{x1D706}_{i+1}=0$, then 
$K_{X_{i}}+B_{i}$ is nef and we have the required 
$K_{X}+B$ minimal model. Otherwise, by Theorem 2.1, there exists a 
$K_{X_{i}}+B_{i}+\unicode[STIX]{x1D706}_{i+1}H_{i}$-trivial and 
$K_{X_{i}}+B_{i}$ negative extremal ray 
$C_{i}$. Let 
$\unicode[STIX]{x1D708}_{i}:X_{i}\rightarrow X_{i+1}$ be the corresponding contraction. If 
$\dim X_{i+1}<2$, then we have the required 
$K_{X}+B$ Mori fiber space. Otherwise, 
$X_{i}\rightarrow X_{i+1}$ is a divisorial contraction. Since 
$H$ is general in 
$N^{1}(X/X^{\prime })$, 
$H_{i}$ is general in 
$N^{1}(X_{i}/X^{\prime })$ and hence 
$NE(X_{i})_{K_{X_{i}}+B_{i}+\unicode[STIX]{x1D706}_{i+1}H_{i}=0}=[C_{i}]$. It follows that 
$K_{X_{i}}+B_{i}+\unicode[STIX]{x1D706}_{i+1}H_{i}=\unicode[STIX]{x1D708}_{i}^{\ast }(K_{X_{i+1}}+B_{i+1}+\unicode[STIX]{x1D706}_{i+1}H_{i+1})$, where 
$K_{X_{i+1}}+B_{i+1}+\unicode[STIX]{x1D706}_{i+1}H_{i+1}$ is ample.◻
Proposition 2.4. Let 
$X$ be a projective surface over an algebraically closed field 
$k$. Assume that 
$(X,B)$ is a klt pair and 
$\unicode[STIX]{x1D708}:X^{\prime }\rightarrow X$ is a proper birational morphism such that 
$(X^{\prime },B^{\prime })$ is terminal where 
$K_{X^{\prime }}+B^{\prime }=\unicode[STIX]{x1D708}^{\ast }(K_{X}+B)$. Let 
$\unicode[STIX]{x1D6E9}=B^{\prime }-B^{\prime }\wedge N_{\unicode[STIX]{x1D70E}}(K_{X^{\prime }}+B^{\prime })$ and 
$\unicode[STIX]{x1D719}^{\prime }:X^{\prime }\rightarrow X_{M}^{\prime }$ the minimal model for 
$(X^{\prime },\unicode[STIX]{x1D6E9})$. If 
$\unicode[STIX]{x1D719}:X\rightarrow X_{M}$ is the minimal model for 
$(X,B)$, then the rational map 
$\unicode[STIX]{x1D707}:X_{M}^{\prime }\rightarrow X_{M}$ is a morphism and 
$K_{X_{M}^{\prime }}+\unicode[STIX]{x1D719}_{\ast }^{\prime }\unicode[STIX]{x1D6E9}=\unicode[STIX]{x1D707}^{\ast }(K_{X_{M}}+\unicode[STIX]{x1D719}_{\ast }B)$. If 
$\unicode[STIX]{x1D705}(K_{X}+B)=1$ and 
$B$ is big over 
$\operatorname{Proj}R(K_{X}+B)$, then 
$\unicode[STIX]{x1D6E9}$ is big over 
$\operatorname{Proj}R(K_{X^{\prime }}+\unicode[STIX]{x1D6E9})$.
Proof. Consider the morphism 
$\unicode[STIX]{x1D713}:X^{\prime }\rightarrow X_{M}$. Since 
$K_{X}+B=\unicode[STIX]{x1D719}^{\ast }(K_{X_{M}}+\unicode[STIX]{x1D719}_{\ast }B)+E$ where 
$E\geqslant 0$ is 
$\unicode[STIX]{x1D719}$-exceptional, then 
$K_{X^{\prime }}+B^{\prime }=\unicode[STIX]{x1D713}^{\ast }(K_{X_{M}}+\unicode[STIX]{x1D719}_{\ast }B)+\unicode[STIX]{x1D708}^{\ast }E$ where 
$K_{X_{M}}+\unicode[STIX]{x1D719}_{\ast }B$ is nef and 
$\unicode[STIX]{x1D708}^{\ast }E$ is effective and 
$\unicode[STIX]{x1D713}$ exceptional. It follows that 
$N_{\unicode[STIX]{x1D70E}}(K_{X^{\prime }}+B^{\prime })=\unicode[STIX]{x1D708}^{\ast }E$ and so 
$K_{X^{\prime }}+\unicode[STIX]{x1D6E9}=\unicode[STIX]{x1D713}^{\ast }(K_{X_{M}}+\unicode[STIX]{x1D719}_{\ast }B)+E^{\prime }$, where 
$0\leqslant E^{\prime }\leqslant \unicode[STIX]{x1D708}^{\ast }E$. In particular, 
$N_{\unicode[STIX]{x1D70E}}(K_{X^{\prime }}+\unicode[STIX]{x1D6E9})=E^{\prime }$ and so the divisors contracted by 
$\unicode[STIX]{x1D719}^{\prime }$ are precisely the divisors contained in 
$\text{Supp}(E^{\prime })$. Thus, 
$X^{\prime }\rightarrow X_{M}$ factors through 
$\unicode[STIX]{x1D719}^{\prime }$. We have 
$K_{X_{M}^{\prime }}+\unicode[STIX]{x1D719}_{\ast }^{\prime }\unicode[STIX]{x1D6E9}=\unicode[STIX]{x1D707}^{\ast }(K_{X_{M}}+\unicode[STIX]{x1D719}_{\ast }B)+\unicode[STIX]{x1D719}_{\ast }^{\prime }E^{\prime }$ where 
$\unicode[STIX]{x1D707}_{\ast }(\unicode[STIX]{x1D719}_{\ast }^{\prime }E^{\prime })\leqslant \unicode[STIX]{x1D719}_{\ast }E=0$ and hence 
$\unicode[STIX]{x1D719}_{\ast }^{\prime }E$ is 
$\unicode[STIX]{x1D707}$ exceptional. By the negativity lemma, it follows that 
$\unicode[STIX]{x1D719}_{\ast }^{\prime }E^{\prime }=0$.
Note that since 
$H^{0}(m(K_{X}+B))\cong H^{0}(m(K_{X^{\prime }}+\unicode[STIX]{x1D6E9}))$ for all 
$m\geqslant 0$, it follows that 
$Z:=\operatorname{Proj}R(K_{X}+B)=\operatorname{Proj}R(K_{X^{\prime }}+\unicode[STIX]{x1D6E9})$. We have 
$\dim Z=\unicode[STIX]{x1D705}(K_{X}+B)=1$. The bigness of 
$B$ over 
$Z$ is equivalent to 
$B\cdot X_{z}>0$ for general 
$z\in Z$. But then
$$\begin{eqnarray}\unicode[STIX]{x1D6E9}\cdot X_{z}^{\prime }=\unicode[STIX]{x1D707}_{\ast }\unicode[STIX]{x1D719}_{\ast }^{\prime }\unicode[STIX]{x1D6E9}\cdot (X_{M})_{z}=\unicode[STIX]{x1D719}_{\ast }B\cdot (X_{M})_{z}=B\cdot X_{z}>0\end{eqnarray}$$ and so 
$\unicode[STIX]{x1D6E9}$ is big over 
$Z$.◻
Consider now 
$X$ a smooth projective scheme over an integral Noetherian scheme 
$S$ and let 
$f:X\rightarrow S$ be the structure morphism. We say that a pair 
$(X,B)$ is log smooth over 
$S$ if 
$X$ is smooth over 
$S$ and 
$B$ is an effective 
$\mathbb{R}$-divisor whose support is simple normal crossings over 
$S$ so that 
$X$ is étale over 
$\mathbb{A}_{S}^{n}$ and some choice of local coordinates on 
$\mathbb{A}_{S}^{n}$ pulls back to a parameter system 
$t_{1},\ldots ,t_{n}$ on 
$X$ and 
$\text{Supp}(B)=\{t_{1}\ldots t_{r}=0\}$ for some 
$0\leqslant r\leqslant n$. We refer the reader to [16, Section 01V4] for a discussion of smooth morphisms. In particular, each strata of the support of 
$B$ is smooth over 
$S$. We say that a log smooth pair 
$(X,B=\sum b_{i}B_{i})$ is klt iff 
$0\leqslant b_{i}<1$ and 
$(X,B=\sum b_{i}B_{i})$ is terminal iff 
$0\leqslant b_{i}<1$ and 
$b_{i}+b_{j}<1$ if 
$i\neq j$ and 
$B_{i}\cap B_{j}\neq \emptyset$.
In what follows, 
$R$ will denote a DVR with residue field 
$k$ and fraction field 
$K$. Let 
$X$ be an integral Noetherian scheme over 
$\text{Spec}(R)$ and 
$f:X\rightarrow \text{Spec}(R)$ the structure morphism, then we let 
$X_{K}=X\times _{\text{Spec}(R)}\text{Spec}(K)$ be the generic fiber and 
$X_{k}=X\times _{\text{Spec}(R)}\text{Spec}(k)$ be the special fiber. As usual, we say that two Cartier divisors on 
$X$ are numerically equivalent 
$L_{1}\equiv _{R}L_{2}$ iff 
$(L_{1}-L_{2})\cdot C=0$ for any curve 
$C$ contained in a fiber 
$X_{K}$ or 
$X_{k}$. Note that it suffices to check this on the special fiber 
$X_{k}$. We then let
$$\begin{eqnarray}\displaystyle & N^{1}(X/R)=(\{\text{Cartier divisors }L\text{ on }X\}/\equiv _{R})\otimes _{\mathbb{Z}}\mathbb{R}, & \displaystyle \nonumber\\ \displaystyle & N_{1}(X/R)=(\{\text{curves on }X_{k}\}/\equiv _{R})\otimes _{\mathbb{Z}}\mathbb{R}. & \displaystyle \nonumber\end{eqnarray}$$
$NE(X/R)\subset N_{1}(X/R)$ is the closed cone spanned by effective curves. Note that the natural map 
$N_{1}(X/R)\rightarrow N^{1}(X_{k})$ is injective and the dual map 
$N_{1}(X_{k})\rightarrow N_{1}(X/R)$ is surjective and so is the induced map 
$NE(X_{k})\rightarrow NE(X/R)$.
Lemma 2.5. Let 
$f:X\rightarrow \text{Spec}(R)$ be a smooth projective morphism from a smooth variety to a DVR and 
$L$ a line bundle on 
$X$, then
(1)
$L$ is ample if and only if $L_{k}:=L|_{X_{k}}$ is ample, and(2)
$L$ is nef if and only if $L_{k}:=L|_{X_{k}}$ is nef.
Proof. Clearly, if 
$L$ is ample or nef, then so is 
$L_{k}$. It is well known that ampleness is an open condition and so if 
$L_{k}$ is ample, then so is 
$L$. Finally, if 
$L_{k}$ is nef and 
$H$ is ample, then 
$L_{k}+tH_{k}$ is ample for any 
$t>0$ so that 
$L+tH$ is ample and hence 
$L$ is nef.◻
Lemma 2.6. Let 
$f:X\rightarrow \text{Spec}(R)$ be a flat projective morphism from a variety to a DVR and 
$(X,B)$ a log pair. Then 
$(X,B)$ is log smooth over 
$\text{Spec}(R)$ if and only if 
$(X_{k},B_{k})$ is log smooth.
Proof. See [16, Section 01V4]. ◻
Lemma 2.7. Let 
$(X,B)$ be a log pair which is log smooth over 
$\text{Spec}(R)$, where 
$R$ is a DVR. If 
$R\subset \tilde{R}$ is an inclusion of DVR’s, then 
$(X_{\tilde{R}},B_{\tilde{R}})$ is log smooth over 
$\text{Spec}(\tilde{R})$. If 
$(X,B)$ is terminal (resp. klt), then so is 
$(X_{\tilde{R}},B_{\tilde{R}})$.
Proof. Since smoothness is preserved by base change, it follows that 
$(X_{\tilde{R}},B_{\tilde{R}})$ is log smooth over 
$\text{Spec}(\tilde{R})$. The pair 
$(X,B)$ is klt (resp. terminal) if and only if the coefficients of 
$B$ are 
${<}1$ (resp. the coefficients of 
$B$ are 
${<}1$ and if two components intersect, then the sum of the coefficients is 
${<}1$). The lemma now follows since if there are two intersecting components of 
$B_{\tilde{R}}$, then there are two intersecting components of 
$B$ (with the same coefficients).◻
Theorem 2.8. (Katsura–Ueno [Reference Katsura and Ueno11])
Let 
$f:X\rightarrow \text{Spec}(R)$ be an algebraic space which is smooth, proper and two-dimensional over 
$\text{Spec}(R)$, where 
$R$ is a DVR with algebraically closed residue field 
$k$ and field of fractions 
$K$. If 
$X_{k}$ contains a 
$-1$ curve 
$e\subset X_{k}$, then there exists a DVR 
$\tilde{R}\supset R$ with residue field 
$k$ and fraction field 
$\tilde{K}$ and a surjective proper morphism 
$\unicode[STIX]{x1D70B}:X_{\tilde{R}}\rightarrow {\tilde{Y}}$ over 
$\text{Spec}(\tilde{R})$ where 
${\tilde{Y}}\rightarrow \text{Spec}(\tilde{R})$ is smooth, proper, and two-dimensional, 
$\unicode[STIX]{x1D70B}_{k}$ contracts the 
$-1$ curve 
$e\subset X_{k}$ and 
$\unicode[STIX]{x1D70B}_{K}:X_{\tilde{K}}\rightarrow {\tilde{Y}}_{\tilde{K}}$ is also a contraction of a 
$-1$ curve.
3 Main result
In this section, we will prove Theorem 1.1. We begin by showing that a more general version of this result holds when 
$S=\text{Spec}(R)$ is the spectrum of a DVR and then we will deduce the general case.
Theorem 3.1. Let 
$(X,B)$ be a klt pair which is log smooth, projective of dimension 
$2$ over 
$S=\text{Spec}(R)$, where 
$R$ is a DVR with residue field 
$k$ and fraction field 
$K$. If 
$K_{X}+B$ is 
$\mathbb{Q}$-Cartier, then 
$\unicode[STIX]{x1D705}(K_{X_{k}}+B_{k})=\unicode[STIX]{x1D705}(K_{X_{K}}+B_{K})$ and if either 
$\unicode[STIX]{x1D705}(K_{X_{k}}+B_{k})\neq 1$ or 
$\unicode[STIX]{x1D705}(K_{X_{k}}+B_{k})=1$ and 
$B_{k}$ is big over 
$\operatorname{Proj}R(K_{X_{k}}+B_{k})$, then there exists an integer 
$m_{0}>0$ such that for any integer 
$m\in m_{0}\mathbb{N}$, we have
$$\begin{eqnarray}h^{0}(m(K_{X_{k}}+B_{k}))=h^{0}(m(K_{X_{K}}+B_{K})).\end{eqnarray}$$Proof. Consider an inclusion of DVR’s 
$R\subset \tilde{R}$. If 
$\tilde{k}$ and 
$\tilde{K}$ denote the residue field and the fraction field of 
$\tilde{R}$, then 
$h^{0}(m(K_{X_{k}}+B_{k}))=h^{0}(m(K_{X_{\tilde{k}}}+B_{\tilde{k}}))$ and 
$h^{0}(m(K_{X_{K}}+B_{K}))=h^{0}(m(K_{X_{\tilde{K}}}+B_{\tilde{K}}))$. Note also that if 
$\tilde{X}=X\times _{\text{Spec}(R)}\text{Spec}(\tilde{R})$ and 
$\tilde{B}=B\times _{\text{Spec}(R)}\text{Spec}(\tilde{R})$, then 
$(\tilde{X},\tilde{B})$ is log smooth over 
$\tilde{R}$ and 
$\tilde{X}_{\tilde{k}}\cong X_{k}\times _{\text{Spec}(k)}\text{Spec}(\tilde{k})$. Thus, we are free to replace 
$X\rightarrow R$ by 
$\tilde{X}\rightarrow \tilde{R}$. In particular, we may assume that 
$k$ is algebraically closed.
If 
$h^{0}(m(K_{X_{k}}+B_{k}))=0$, then, by semicontinuity, 
$h^{0}(m(K_{X_{K}}+B_{K}))=0$. Therefore, the theorem holds trivially in the case 
$\unicode[STIX]{x1D705}(K_{X_{k}}+B_{k})=-\infty$. Thus, we may assume that 
$\unicode[STIX]{x1D705}(K_{X_{k}}+B_{k})\geqslant 0$.
Claim 3.2. The theorem holds under the additional assumption that 
$(X_{k},B_{k})$ is terminal and no component of the support of 
$B_{k}$ is contained in 
$\mathbf{B}(K_{X_{k}}+B_{k})$.
Proof. Since 
$k$ is algebraically closed, then by the Cone Theorem (Theorem 2.1),
$$\begin{eqnarray}\overline{NE}(X_{k})=\overline{NE}(X_{k})_{K_{X_{k}}+B_{k}\geqslant 0}+\mathop{\sum }_{i\in I}\mathbb{R}_{{\geqslant}0}C_{i},\end{eqnarray}$$ where 
$I$ is countable, 
$(K_{X_{k}}+B_{k})\cdot C_{i}<0$ and 
$C_{i}$ is rational.
Suppose that one of the above curves 
$C_{i}$ is contained in the support of 
$B_{k}$, then since 
$\unicode[STIX]{x1D705}(K_{X_{k}}+B_{k})\geqslant 0$ and 
$(K_{X_{k}}+B_{k})\cdot C_{i}<0$, we have 
$C_{i}\subset \mathbf{B}(K_{X_{k}}+B_{k})$, which we have assumed is impossible.
Note then that 
$C_{i}$ is not contained in the support of 
$B_{k}$ and thus 
$C_{i}\cdot B_{k}\geqslant 0$ and so 
$K_{X_{k}}\cdot C_{i}<0$. It follows that if 
$C_{i}$ spans a 
$K_{X_{k}}+B_{k}$-negative extremal ray, then it also spans a 
$K_{X_{k}}$-negative extremal ray and so it can be contracted by a divisorial contraction of a 
$-1$ curve 
$X_{k}\rightarrow X_{k}^{\prime }$. In particular, 
$X_{k}^{\prime }$ is also a smooth surface. Thus, we may assume that 
$C_{i}$ is a 
$-1$ curve. By Theorem 2.8 (after extending 
$R$), we may assume that there is a morphism 
$X\rightarrow X^{\prime }$ of smooth surfaces over 
$R$ such that 
$X_{K}\rightarrow X_{K}^{\prime }$ also contracts a 
$-1$ curve.
We now run an MMP by contracting a sequence of 
$K_{X}+B$-negative curves as above. Let 
$\unicode[STIX]{x1D708}:X\rightarrow \bar{X}$ be the induced morphism of smooth surfaces over 
$\text{Spec}(R)$. We may assume that 
$X_{K}\rightarrow \bar{X}_{K}$ and 
$X_{k}\rightarrow \bar{X}_{k}$ are given by a finite sequence of contractions of 
$-1$ curves such that the exceptional locus of 
$X_{k}\rightarrow \bar{X}_{k}$ contains no components of 
$B_{k}$. Then 
$(\bar{X}_{k},\bar{B}_{k})$ is terminal and 
$K_{X_{k}}+B_{k}=\unicode[STIX]{x1D708}_{k}^{\ast }(K_{\bar{X}_{k},}+\bar{B}_{k})+F_{k}$, where 
$B_{k}=\unicode[STIX]{x1D708}_{k,\ast }^{-1}\bar{B}_{k}$ and 
$B_{k}\wedge F_{k}=0$. In particular, 
$\mathbf{B}(K_{X_{k}}+B_{k})=\mathbf{B}(\unicode[STIX]{x1D708}_{k}^{\ast }(K_{\bar{X}_{k}}+\bar{B}_{k}))+F_{k}$. Suppose that 
$C\subset \bar{X}_{k}$ is contained in 
$\mathbf{B}(K_{\bar{X}_{k}}+\bar{B}_{k})\cap \text{Supp}(\bar{B}_{k})$, then 
$\unicode[STIX]{x1D708}_{\ast }^{-1}C\subset \mathbf{B}(K_{X_{k}}+B_{k})\cap \text{Supp}(B_{k})$ which is impossible. Therefore, if 
$K_{\bar{X}_{k}}+\bar{B}_{k}$ is not nef, we can continue to contract 
$-1$ curves. Since each contraction reduces the Picard number of the central fiber 
$X_{k}$ by one, this procedure must terminate after finitely many steps. We may therefore assume that 
$K_{\bar{X}_{k}}+\bar{B}_{k}$ is semiample. In particular, 
$K_{\bar{X}_{k}}+\bar{B}_{k}$ is nef and hence so is 
$K_{\bar{X}}+\bar{B}$ (see Lemma 2.5).
Suppose now that 
$\unicode[STIX]{x1D708}(K_{X_{k}}+B_{k})=2$. In this case, 
$K_{\bar{X}_{k}}+\bar{B}_{k}$ is nef and big and 
$m_{0}(K_{\bar{X}_{k}}+\bar{B}_{k})$ is Cartier for some 
$m_{0}>0$. We may write
$$\begin{eqnarray}km_{0}(K_{\bar{X}_{k}}+\bar{B}_{k})=K_{\bar{X}_{k}}+\lceil (m_{0}-1)(K_{\bar{X}_{k}}+\bar{B}_{k}))\rceil +(k-1)m_{0}(K_{\bar{X}_{k}}+\bar{B}_{k})\end{eqnarray}$$ so that by [Reference Tanaka18, 2.6] 
$h^{i}(m(K_{\bar{X}_{k}}+\bar{B}_{k}))=0$ for all sufficiently big integers 
$m\in m_{0}\mathbb{N}$ and all 
$i>0$. Replacing 
$m_{0}$ by an appropriate multiple, this condition holds for all 
$m\in m_{0}\mathbb{N}$. By semicontinuity, we also have 
$h^{i}(m(K_{\bar{X}_{K}}+\bar{B}_{K}))=0$ for all 
$m\in m_{0}\mathbb{N}$ and 
$i>0$. The result now follows from cohomology and base change.
Suppose that 
$\unicode[STIX]{x1D705}(K_{X_{k}}+B_{k})=0$. Then we have 
$K_{\bar{X}_{k}}+\bar{B}_{k}{\sim}_{\mathbb{Q}}0$. By Lemma 2.5, it follows that 
$\pm (K_{\bar{X}_{K}}+\bar{B}_{K})$ is nef and hence that 
$K_{\bar{X}_{K}}+\bar{B}_{K}\equiv 0$. By [Reference Tanaka17, 1.2], 
$K_{\bar{X}_{K}}+\bar{B}_{K}{\sim}_{\mathbb{Q}}0$. Thus, there exists an integer 
$m_{0}>0$ such that 
$m_{0}(K_{\bar{X}_{K}}+\bar{B}_{K})\sim 0$ and 
$m_{0}(K_{\bar{X}_{k}}+\bar{B}_{k})\sim 0$. Thus, 
$h^{0}(m(K_{X_{K}}+B_{K}))=h^{0}(m(K_{X_{k}}+B_{k}))$ for all 
$m\geqslant 0$ divisible by 
$m_{0}$.
Suppose that 
$\unicode[STIX]{x1D705}(K_{X_{k}}+B_{k})=1$. Since 
$K_{\bar{X}_{k}}+\bar{B}_{k}$ is nef, so is 
$K_{\bar{X}_{K}}+\bar{B}_{K}$. In particular, 
$\unicode[STIX]{x1D705}(K_{\bar{X}_{K}}+\bar{B}_{K})\geqslant 0$ and, thus, by semicontinuity, we have 
$\unicode[STIX]{x1D705}(K_{\bar{X}_{K}}+\bar{B}_{K})\in \{0,1\}$. Let 
$H$ be a sufficiently ample divisor on 
$\bar{X}$. Then 
$(K_{\bar{X}_{K}}+\bar{B}_{K})\cdot H_{K}=(K_{\bar{X}_{k}}+\bar{B}_{k})\cdot H_{k}>0$ so 
$K_{\bar{X}_{K}}+\bar{B}_{K}\not \equiv 0$. Therefore, 
$\unicode[STIX]{x1D705}(K_{\bar{X}_{K}}+\bar{B}_{K})=1$.
Finally, suppose that 
$\unicode[STIX]{x1D705}(K_{X_{k}}+B_{k})=1$ and 
$B_{k}$ is big over 
$\operatorname{Proj}R(K_{X_{k}}+B_{k})$. Note that 
$\bar{B}_{k}$ is also big over 
$\operatorname{Proj}R(K_{\bar{X}_{k}}+\bar{B}_{k})$ and hence 
$\bar{B}_{k}+K_{\bar{X}_{k}}+\bar{B}_{k}$ is big. Thus, we may write 
$\bar{B}_{k}+K_{\bar{X}_{k}}+\bar{B}_{k}{\sim}_{\mathbb{Q}}\bar{A}_{k}+\bar{E}_{k}$, where 
$\bar{A}_{k}$ is ample and 
$\bar{E}_{k}$ is effective. For any rational number 
$0<\unicode[STIX]{x1D716}\ll 1$, the pair 
$(\bar{X}_{k},\unicode[STIX]{x1D6E5}_{k}=(1-\unicode[STIX]{x1D716})\bar{B}_{k}+\unicode[STIX]{x1D716}\bar{E}_{k})$ is Kawamata log terminal and so the corresponding multiplier ideal sheaf is trivial 
${\mathcal{J}}(\unicode[STIX]{x1D6E5}_{k})={\mathcal{O}}_{\bar{X}_{k}}$. If 
$L=N=m(K_{\bar{X}_{k}}+\bar{B}_{k})$, then 
$N$ is nef and not numerically equivalent to zero while
$$\begin{eqnarray}L-(K_{\bar{X}_{k}}+\unicode[STIX]{x1D6E5}_{k}){\sim}_{\mathbb{Q}}(m-1-\unicode[STIX]{x1D716})(K_{\bar{X}_{k}}+\bar{B}_{k})+\unicode[STIX]{x1D716}\bar{A}_{k}\end{eqnarray}$$ is ample, and so by [Reference Tanaka18, 0.3] and [Reference Kollár and Mori10, 2.70], 
$H^{i}({\mathcal{O}}_{\bar{X}_{k}}(m(l+1)(K_{\bar{X}_{k}}+\bar{B}_{k})))=0$ for 
$i>0$ and 
$l\gg 0$. By semicontinuity, 
$H^{i}({\mathcal{O}}_{\bar{X}_{K}}(m(l+1)(K_{\bar{X}_{K}}+\bar{B}_{K})))=0$ for 
$i>0$ and 
$l\gg 0$ and hence 
$h^{0}({\mathcal{O}}_{\bar{X}_{k}}(m(l+1)(K_{\bar{X}_{k}}+\bar{B}_{k})))=h^{0}({\mathcal{O}}_{\bar{X}_{K}}(m(l+1)(K_{\bar{X}_{K}}+\bar{B}_{K})))$.◻
We will now consider the general case. Since 
$(X,B)$ is log smooth over 
$R$, there is a sequence of blowups along strata of 
$\mathbf{M}_{B}$ say 
$\unicode[STIX]{x1D708}:X^{\prime }\rightarrow X$ such that 
$K_{X^{\prime }}+B^{\prime }=\unicode[STIX]{x1D708}^{\ast }(K_{X}+B)$ is terminal and, in particular, 
$B^{\prime }\geqslant 0$ and 
$(X^{\prime },B^{\prime })$ is log smooth. Since 
$R(K_{X_{k}^{\prime }}+B_{k}^{\prime })\cong R(K_{X_{k}}+B_{k})$ is finitely generated, 
$N_{\unicode[STIX]{x1D70E}}(K_{X_{k}^{\prime }}+B_{k}^{\prime })$ is a 
$\mathbb{Q}$-divisor and hence so is
$$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{k}:=B_{k}^{\prime }-(B_{k}^{\prime }\wedge N_{\unicode[STIX]{x1D70E}}(K_{X_{k}^{\prime }}+B_{k}^{\prime })).\end{eqnarray}$$ Note that 
$R(K_{X_{k}^{\prime }}+\unicode[STIX]{x1D6E9}_{k})\cong R(K_{X_{k}^{\prime }}+B_{k}^{\prime })$, 
$(X_{k}^{\prime },\unicode[STIX]{x1D6E9}_{k})$ is terminal and no component of 
$\unicode[STIX]{x1D6E9}_{k}$ is contained in 
$\mathbf{B}(K_{X_{k}^{\prime }}+\unicode[STIX]{x1D6E9}_{k})$ [[Reference Hacon, McKernan and Xu6, 2.8.3] and [Reference Hacon and Xu7, 2.4]]. Let 
$\unicode[STIX]{x1D6E9}$ be the unique 
$\mathbb{Q}$-divisor supported on 
$B^{\prime }$ such that 
$\unicode[STIX]{x1D6E9}|_{X_{k}^{\prime }}=\unicode[STIX]{x1D6E9}_{k}$. We remark that if 
$\unicode[STIX]{x1D705}(K_{X_{k}}+B_{k})=1$ and 
$B_{k}$ is big over 
$\operatorname{Proj}R(K_{X_{k}}+B_{k})$, then by Proposition 2.4, 
$\unicode[STIX]{x1D6E9}_{k}$ is big over 
$\operatorname{Proj}R(K_{X_{k}^{\prime }}+\unicode[STIX]{x1D6E9}_{k})$. By Claim 3.2, it follows that 
$\unicode[STIX]{x1D705}(K_{X_{K}^{\prime }}+\unicode[STIX]{x1D6E9}_{K})=\unicode[STIX]{x1D705}(K_{X_{k}^{\prime }}+\unicode[STIX]{x1D6E9}_{k})$ and there exists an integer 
$m_{0}>0$ such that
$$\begin{eqnarray}h^{0}(m(K_{X_{K}^{\prime }}+\unicode[STIX]{x1D6E9}_{K}))=h^{0}(m(K_{X_{k}^{\prime }}+\unicode[STIX]{x1D6E9}_{k}))\quad \forall m\in m_{0}\mathbb{N}.\end{eqnarray}$$By semicontinuity, we then have
$$\begin{eqnarray}\displaystyle h^{0}(m(K_{X_{k}}+B_{k})) & {\geqslant} & \displaystyle h^{0}(m(K_{X_{K}}+B_{K}))\geqslant h^{0}(m(K_{X_{K}^{\prime }}+B_{K}^{\prime }))\nonumber\\ \displaystyle & {\geqslant} & \displaystyle h^{0}(m(K_{X_{K}^{\prime }}+\unicode[STIX]{x1D6E9}_{K}))=h^{0}(m(K_{X_{k}^{\prime }}+\unicode[STIX]{x1D6E9}_{k}))=h^{0}(m(K_{X_{k}}+B_{k}))\nonumber\end{eqnarray}$$ and hence 
$h^{0}(m(K_{X_{k}}+B_{k}))=h^{0}(m(K_{X_{K}}+B_{K}))$. The equality 
$\unicode[STIX]{x1D705}(K_{X_{k}}+B_{k})=\unicode[STIX]{x1D705}(K_{X_{K}}+B_{K})$ follows similarly.◻
Corollary 3.3. Let 
$(X,B)$ be a klt pair which is log smooth, projective of dimension 
$2$ over a DVR 
$R$ with residue field 
$k$ of characteristic 
$p>0$ and fraction field 
$K$. If 
$K_{X}+B$ is 
$\mathbb{Q}$-Cartier and either 
$\unicode[STIX]{x1D705}(K_{X_{k}}+B_{k})\in \{0,2\}$ or 
$\unicode[STIX]{x1D705}(K_{X_{k}}+B_{k})=1$ and 
$B_{k}$ is big over 
$\operatorname{Proj}R(K_{X_{k}}+B_{k})$, then 
$R(K_{X}+B)$ is finitely generated.
Proof. By Theorem 2.3, 
$R(K_{X_{k}}+B_{k})$ is finitely generated and hence there is a positive integer 
$m$ such that
$$\begin{eqnarray}R(m(K_{X_{k}}+B_{k}))\end{eqnarray}$$ is generated in degree 1, that is, by 
$H^{0}(m(K_{X_{k}}+B_{k}))$. By Theorem 3.1, after replacing 
$m$ by a multiple, we may assume that 
$m(K_{X}+B)$ is Cartier and
$$\begin{eqnarray}H^{0}(m(K_{X}+B))\rightarrow H^{0}(m(K_{X_{k}}+B_{k}))\end{eqnarray}$$is surjective. Therefore, the induced map
$$\begin{eqnarray}S^{k}H^{0}(m(K_{X}+B))\rightarrow S^{k}H^{0}(m(K_{X_{k}}+B_{k}))\rightarrow H^{0}(mk(K_{X_{k}}+B_{k}))\end{eqnarray}$$is surjective. By Nakayama’s lemma,
$$\begin{eqnarray}S^{k}H^{0}(m(K_{X}+B))\rightarrow H^{0}(mk(K_{X}+B))\end{eqnarray}$$ is surjective and so 
$R(m(K_{X}+B))$ is finitely generated.◻
Theorem 3.4. Let 
$(X,B)$ be a klt pair which is log smooth, projective of dimension 
$2$ over a DVR 
$R$ with residue field 
$k$ and fraction field 
$K$. If 
$K_{X}+B$ is 
$\mathbb{Q}$-Cartier, then (after possibly extending 
$R$) we may run a 
$K_{X}+B$ MMP over 
$R$ which is given by a sequence of divisorial contractions and terminates with a 
$K_{X}+B$ minimal model 
$X\rightarrow \bar{X}$ over 
$R$ or a 
$K_{X}+B$ Mori fiber space over 
$R$.
Proof. After extending 
$R$, we may assume that 
$k$ is algebraically closed. Suppose that 
$H$ is ample and let
$$\begin{eqnarray}\unicode[STIX]{x1D70F}=\text{inf}\{t\geqslant 0|\unicode[STIX]{x1D705}(K_{X_{k}}+B_{k}+tH_{k})\geqslant 0\}.\end{eqnarray}$$ Pick 
$1\gg \unicode[STIX]{x1D70F}^{\prime }-\unicode[STIX]{x1D70F}>0$, then by Theorem 2.3 and its proof, 
$\text{Proj}R(K_{X_{k}}+B_{k}+\unicode[STIX]{x1D70F}^{\prime }H_{k})$ is the minimal model of 
$(X_{k},B_{k}+tH_{k})$ for 
$\unicode[STIX]{x1D70F}^{\prime }\geqslant t\geqslant \unicode[STIX]{x1D70F}$. Let 
$\unicode[STIX]{x1D708}_{k}:X_{k}^{\prime }\rightarrow X_{k}$ be a terminalization of 
$(X_{k},B_{k})$ given by a sequence of blowups along strata of 
$\mathbf{M}_{B_{k}}$, 
$K_{X_{k}^{\prime }}+B_{k}^{\prime }=\unicode[STIX]{x1D708}_{k}^{\ast }(K_{X_{k}}+B_{k})$, 
$H_{k}^{\prime }=\unicode[STIX]{x1D708}_{k}^{\ast }H_{k}$ and
$$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{k}=B_{k}^{\prime }-B_{k}^{\prime }\wedge N_{\unicode[STIX]{x1D70E}}(K_{X_{k}^{\prime }}+B_{k}^{\prime }+\unicode[STIX]{x1D70F}^{\prime }H_{k}^{\prime }).\end{eqnarray}$$ If 
$X_{k}^{\prime }\rightarrow X_{1,k}^{\prime }\rightarrow \cdots \rightarrow X_{n,k}^{\prime }$ is a MMP for 
$K_{X_{k}^{\prime }}+\unicode[STIX]{x1D6E9}_{k}+\unicode[STIX]{x1D70F}^{\prime }H_{k}^{\prime }$, then 
$K_{X_{n,k}^{\prime }}+\unicode[STIX]{x1D6E9}_{n,k}+\unicode[STIX]{x1D70F}^{\prime }H_{n,k}^{\prime }$ is semiample and induces a morphism 
$\unicode[STIX]{x1D708}_{n,k}:X_{n,k}^{\prime }\rightarrow X_{n,k}:=\text{Proj}R(K_{X_{k}^{\prime }}+\unicode[STIX]{x1D6E9}_{k}+\unicode[STIX]{x1D70F}^{\prime }H_{k}^{\prime })$. By Proposition 2.4, 
$R(K_{X_{k}^{\prime }}+\unicode[STIX]{x1D6E9}_{k}+\unicode[STIX]{x1D70F}^{\prime }H_{k}^{\prime })\cong R(K_{X_{k}}+B_{k}+\unicode[STIX]{x1D70F}^{\prime }H_{k})$, and so the induced birational map 
$X_{k}\rightarrow X_{n,k}$ is in fact the morphism corresponding to the minimal model of 
$K_{X_{k}}+B_{k}+\unicode[STIX]{x1D70F}H_{k}$. If 
$\unicode[STIX]{x1D70F}=0$, then 
$X_{k}\rightarrow X_{n,k}$ is a 
$K_{X_{k}}+B_{k}$ minimal model and if 
$\unicode[STIX]{x1D70F}>0$, then 
$X_{n,k}\rightarrow Z_{k}=\operatorname{Proj}R(K_{X_{k}}+B_{k}+\unicode[STIX]{x1D70F}H_{k})$ is a 
$K_{X_{k}}+B_{k}$ Mori fiber space.
We claim that the exceptional divisors of 
$X_{k}^{\prime }\rightarrow X_{n,k}$ are either contained in the support of 
$\mathbf{M}_{B_{k}}$ or in 
$N_{\unicode[STIX]{x1D70E}}(K_{X_{k}^{\prime }}+\unicode[STIX]{x1D6E9}_{k}+\unicode[STIX]{x1D70F}^{\prime }H_{k}^{\prime })$. To see this, note that the support of 
$\mathbf{M}_{B_{k}}$ contains the 
$X_{k}^{\prime }\rightarrow X_{k}$ exceptional divisors and so it suffices to show that the exceptional divisors of 
$X_{k}\rightarrow X_{n,k}$ are contained in the support of 
$B_{k}^{\prime }$ and 
$N_{\unicode[STIX]{x1D70E}}(K_{X_{k}^{\prime }}+\unicode[STIX]{x1D6E9}_{k}+\unicode[STIX]{x1D70F}^{\prime }H_{k}^{\prime })$. The exceptional divisors of 
$X_{k}\rightarrow X_{n,k}$ are given by the support of 
$N_{\unicode[STIX]{x1D70E}}(K_{X_{k}}+B_{k}+\unicode[STIX]{x1D70F}^{\prime }H_{k})$. The strict transforms of divisors in 
$N_{\unicode[STIX]{x1D70E}}(K_{X_{k}}+B_{k}+\unicode[STIX]{x1D70F}^{\prime }H_{k})$ are divisors in 
$N_{\unicode[STIX]{x1D70E}}(K_{X_{k}^{\prime }}+B_{k}^{\prime }+\unicode[STIX]{x1D70F}^{\prime }H_{k}^{\prime })$ and hence in 
$N_{\unicode[STIX]{x1D70E}}(K_{X_{k}^{\prime }}+\unicode[STIX]{x1D6E9}_{k}+\unicode[STIX]{x1D70F}^{\prime }H_{k}^{\prime })$ plus some divisors supported on 
$B_{k}^{\prime }$. Thus, the claim holds.
By the proof of Theorem 3.1, there is a sequence of divisorial contractions of smooth varieties 
$X^{\prime }\rightarrow X_{1}^{\prime }\rightarrow \cdots \rightarrow X_{n}^{\prime }$ extending the MMP 
$X_{k}^{\prime }\rightarrow X_{1,k}^{\prime }\rightarrow \cdots \rightarrow X_{n,k}^{\prime }$ which induces contractions of 
$-1$ curves on 
$X_{i,k}$ and 
$X_{i,K}$. It follows that if 
$P_{k}$ is an exceptional prime divisor of 
$X_{k}^{\prime }\rightarrow X_{n,k}$, then there is a prime divisor 
$P\subset X^{\prime }$ such that 
$P_{k}=P|_{X_{k}^{\prime }}$. To see this, note that either 
$P_{k}$ is a component of 
$\mathbf{M}_{B_{k}}$ and hence we may take 
$P$ as the corresponding component of 
$\mathbf{M}_{B}$ or 
$P_{k}$ is a component of 
$N_{\unicode[STIX]{x1D70E}}(K_{X_{k}^{\prime }}+\unicode[STIX]{x1D6E9}_{k}+\unicode[STIX]{x1D70F}^{\prime }H_{k})$ and hence the exceptional divisor for some divisorial contraction 
$X_{i,k}^{\prime }\rightarrow X_{i+1,k}^{\prime }$. We can then pick 
$P$ to be the exceptional divisor of 
$X_{i}^{\prime }\rightarrow X_{i+1}^{\prime }$.
Therefore, all 
$X_{k}^{\prime }\rightarrow X_{n,k}$ exceptional divisors extend to divisors on 
$X^{\prime }$ and hence 
$N^{1}(X^{\prime })\rightarrow N^{1}(X_{k}^{\prime }/X_{n,k})$ is surjective and so 
$N^{1}(X)\rightarrow N^{1}(X_{k}/X_{n,k})$ is also surjective.
We now replace 
$H$ by a sufficiently ample 
$\mathbb{Q}$-divisor on 
$X$ which is general in 
$N^{1}(X)$. Since 
$H_{k}$ is general in 
$N^{1}(X_{k}/X_{n,k})$, by Theorem 2.3, running the minimal model program with scaling of 
$H_{k}$, we obtain a sequence of rational numbers 
$\unicode[STIX]{x1D706}_{1}>\unicode[STIX]{x1D706}_{2}>\cdots >\unicode[STIX]{x1D706}_{n}=\unicode[STIX]{x1D70F}$ and divisorial contractions 
$X_{i,k}\rightarrow X_{i+1,k}$ such that 
$X_{i,k}=\operatorname{Proj}(R(K_{X_{k}}+B_{k}+tH_{k}))$ for 
$\unicode[STIX]{x1D706}_{i}\geqslant t>\unicode[STIX]{x1D706}_{i+1}$ where we let 
$X_{k}=X_{0,k}$ and 
$\unicode[STIX]{x1D706}_{0}=1$. By Corollary 3.3, 
$R(K_{X}+B+\unicode[STIX]{x1D706}_{i}H)$ is finitely generated over 
$R$. Let 
$X{\dashrightarrow}X_{i}=\operatorname{Proj}_{R}(R(K_{X}+B+\unicode[STIX]{x1D706}_{i}H))$ be the induced rational map. We claim that
(1)
$X_{i}$ is normal and $\mathbb{Q}$-factorial, $(X_{i},B_{i})$ is klt,(2)
$(X_{i},B_{i})_{k}=(X_{i,k},B_{i,k})$,(3)
$K_{X_{i}}+B_{i}+tH_{i}$ is ample for $\unicode[STIX]{x1D706}_{i}\geqslant t>\unicode[STIX]{x1D706}_{i+1}$ and(4)
$K_{X_{i}}+B_{i}+\unicode[STIX]{x1D706}_{i+1}H_{i}$ is semiample and induces a divisorial contraction $X_{i}\rightarrow X_{i+1}$.
We will prove this by induction. Clearly, the statements 
$(1-3)_{i=0}$ hold and 
$(4)_{i=-1}$ is vacuous. We will prove that 
$(1-3)_{i}$ and 
$(4)_{i-1}$ hold imply that 
$(1-3)_{i+1}$ and 
$(4)_{i}$ hold.
Since 
$R(K_{X}+B+\unicode[STIX]{x1D706}_{i+1}H)\cong R(K_{X_{i}}+B_{i}+\unicode[STIX]{x1D706}_{i+1}H_{i})$ and 
$K_{X_{i,k}}+B_{i,k}+\unicode[STIX]{x1D706}_{i+1}H_{i,k}$ is semiample, by Theorem 3.1, it follows that 
$K_{X_{i}}+B_{i}+\unicode[STIX]{x1D706}_{i+1}H_{i}$ is semiample (over 
$R$) and hence 
$|m(K_{X_{i}}+B_{i}+\unicode[STIX]{x1D706}_{i+1}H_{i})|$ defines a morphism 
$\unicode[STIX]{x1D707}_{i}:X_{i}\rightarrow X_{i+1}$ for 
$m>0$ sufficiently divisible which extends the morphism 
$\unicode[STIX]{x1D707}_{i,k}:X_{i,k}\rightarrow X_{i+1,k}$. Since 
$\unicode[STIX]{x1D707}_{i,k}$ is the divisorial contraction of a prime divisor 
$P_{k}$ which extends to a prime divisor 
$P$ on 
$X_{i}$, it follows that 
$X_{i}\rightarrow X_{i+1}$ is a divisorial contraction and so 
$(4)_{i}$ holds.
To show 
$(1)_{i+1}$, first observe that since 
$X_{i+1,k}$ is normal, so is 
$X_{i+1}$. By what we have seen above, 
$K_{X_{i+1}}+B_{i+1}+\unicode[STIX]{x1D706}_{i+1}H_{i+1}$ is 
$\mathbb{Q}$-Cartier and 
$\unicode[STIX]{x1D707}_{i}^{\ast }(K_{X_{i+1}}+B_{i+1}+\unicode[STIX]{x1D706}_{i+1}H_{i+1})=K_{X_{i}}+B_{i}+\unicode[STIX]{x1D706}_{i+1}H_{i}$. Since 
$(X_{i},B_{i}+\unicode[STIX]{x1D706}_{i+1}H_{i})$ is klt, it follows that 
$(X_{i+1},B_{i+1}+\unicode[STIX]{x1D706}_{i+1}H_{i+1})$ is klt. Therefore, to show that 
$(X_{i+1},B_{i+1})$ is klt, it suffices to show that 
$X_{i+1}$ is 
$\mathbb{Q}$-factorial.
Let 
$D_{i+1}$ be a divisor on 
$X_{i+1}$, we wish to show that 
$D_{i+1}$ is 
$\mathbb{Q}$-Cartier. We may assume that the support of 
$D_{i+1}$ does not contain 
$X_{i+1,k}$. Let 
$D_{k}$ be the pull back of 
$D_{i+1,k}$ to 
$X_{k}$. Fix 
$0<\unicode[STIX]{x1D716}\ll 1$. Since 
$N^{1}(X)\rightarrow N^{1}(X_{k}/X_{n,k})$ is surjective, we may pick a 
$\mathbb{Q}$-divisor 
$G$ on 
$X$ such that 
$G_{k}{\sim}_{\mathbb{Q}}\unicode[STIX]{x1D706}_{i+1}H_{k}+\unicode[STIX]{x1D716}D_{k}$. Since 
$0<\unicode[STIX]{x1D716}\ll 1$, it follows that 
$G_{k}$ is ample and 
$X_{k}\rightarrow X_{i,k}$ is a sequence of 
$K_{X_{k}}+B_{k}+G_{k}$ negative divisorial contractions. It then follows that 
$G$ is ample (over 
$R$) and 
$X\rightarrow X_{i}$ is a sequence of 
$K_{X}+B+G$ negative divisorial contractions. Note that by assumption, 
$K_{X_{i,k}}+B_{i,k}+G_{i,k}=\unicode[STIX]{x1D707}_{i,k}^{\ast }(K_{X_{i+1,k}}+B_{i+1,k}+G_{i+1,k})$. Here,
$$\begin{eqnarray}K_{X_{i+1,k}}+B_{i+1,k}+G_{i+1,k}{\sim}_{\mathbb{Q}}K_{X_{i+1,k}}+B_{i+1,k}+\unicode[STIX]{x1D706}_{i+1}H_{i+1,k}+\unicode[STIX]{x1D716}D_{i+1,k}\end{eqnarray}$$ is ample. Since 
$R(K_{X_{k}}+B_{k}+G_{k})\cong R(K_{X_{i+1,k}}+B_{i+1,k}+G_{i+1,k})$, by Theorem 3.1,
$$\begin{eqnarray}H^{0}(m(K_{X_{i+1}}+B_{i+1}+G_{i+1}))\rightarrow H^{0}(m(K_{X_{i+1,k}}+B_{i+1,k}+G_{i+1,k}))\end{eqnarray}$$ is surjective for 
$m>0$ sufficiently divisible. Since 
$K_{X_{i+1,k}}+B_{i+1,k}+G_{i+1,k}$ is ample (and, in particular, 
$\mathbb{Q}$-Cartier), we may assume that for any 
$x\in X_{i+1,k}$, there exists a global section 
$s_{i+1,k}\in H^{0}(m(K_{X_{i+1,k}}+B_{i+1,k}+G_{i+1,k}))$ which generates the line bundle 
${\mathcal{O}}_{X_{i+1,k}}(m(K_{X_{i+1,k}}+B_{i+1,k}+G_{i+1,k}))$ locally at 
$x$. Let 
$s_{i+1}\in H^{0}(m(K_{X_{i+1}}+B_{i+1}+G_{i+1}))$ be a lift of 
$s_{i+1,k}$ so that 
$s_{i+1}|_{X_{i+1,k}}=s_{i+1,k}$. It follows that 
${\mathcal{O}}_{X_{i+1}}(m(K_{X_{i+1}}+B_{i+1}+G_{i+1}))$ is generated by 
$s_{i+1}$ locally at 
$x$, and hence it is Cartier on a neighborhood of 
$x\in X$. Thus, 
$K_{X_{i+1}}+B_{i+1}+G_{i+1}$ is 
$\mathbb{Q}$-Cartier, and hence so is 
$D_{i+1}=\frac{1}{\unicode[STIX]{x1D716}}(G_{i+1}-H_{i+1})$. This concludes the proof that 
$(1)_{i+1}$ holds.
$(2)_{i+1}$ follows immediately from what we have observed above. To see 
$(3)_{i+1}$, note that 
$K_{X_{i+1,k}}+B_{i+1,k}+tH_{i+1,k}$ is ample for 
$\unicode[STIX]{x1D706}_{i+1}\leqslant t<\unicode[STIX]{x1D706}_{i+2}$ and apply Lemma 2.5.
If 
$\unicode[STIX]{x1D70F}=0$, then after finitely many steps, we have obtained a minimal model of 
$(X,B)$ over 
$\text{Spec}(R)$. Otherwise, there is a Mori fiber space 
$X_{n,k}\rightarrow Z_{k}$. By Theorem 3.1 and Corollary 3.3, 
$X_{n,k}\rightarrow Z_{k}$ extends to a morphism 
$X_{n}\rightarrow Z$ which is 
$K_{X}+B$ negative.◻
Proof of Theorem 1.1.
The independence of 
$\unicode[STIX]{x1D705}(K_{X_{s}}+B_{s})$ for 
$s\in S$ is an immediate consequence of Theorem 3.1; however, the statement regarding the log plurigenera 
$h^{0}(m(K_{X_{s}}+B_{s}))$ is more subtle as the integer 
$m_{0}$ given in Theorem 3.1 (with 
$R={\mathcal{O}}_{s,S}$) may depend on the point 
$s\in S$. Note, however, that it easily follows that the volumes 
$\text{vol}(K_{X_{s}}+B_{s})$ are independent of 
$s\in S$.
Assume now that 
$\text{vol}(K_{X_{s}}+B_{s})>0$. By [Reference Alexeev1, Theorem 7.7] (see also [Reference Hacon and Kovács5]), the corresponding canonical models 
$(X_{s}^{lc},B_{s}^{lc})$ belong to a bounded family and, in particular, there is an integer 
$m>0$ and finitely many degree-2 polynomials 
$P_{1},\ldots ,P_{l}\in \mathbb{Q}[x]$ such that for all 
$s\in S$, 
$m(K_{X_{s}^{lc}}+B_{s}^{lc})$ is Cartier, 
$R(m(K_{X_{s}^{lc}}+B_{s}^{lc}))$ is generated in degree 1 and for every 
$k>0$,
$$\begin{eqnarray}h^{0}(mk(K_{X_{s}^{lc}}+B_{s}^{lc}))=\unicode[STIX]{x1D712}(mk(K_{X_{s}^{lc}}+B_{s}^{lc}))=P_{j}(k)\end{eqnarray}$$ for some 
$1\leqslant j\leqslant l$. Let 
$\unicode[STIX]{x1D702}\in S$ be the generic point. Since
$$\begin{eqnarray}h^{0}(mk(K_{X_{s}^{lc}}+B_{s}^{lc}))=h^{0}(mk(K_{X_{s}}+B_{s}))=h^{0}(mk(K_{X_{\unicode[STIX]{x1D702}}}+B_{\unicode[STIX]{x1D702}}))\end{eqnarray}$$ for all 
$k>0$ sufficiently divisible, it follows that we may assume that 
$P_{1}=P_{2}=\cdots =P_{l}$ and so 
$h^{0}(mk(K_{X_{s}}+B_{s}))$ is constant for all 
$k>0$. But then, for any 
$k>0$, 
$f_{\ast }{\mathcal{O}}_{X}(mk(K_{X}+B))$ is locally free and 
$f_{\ast }{\mathcal{O}}_{X}(mk(K_{X}+B))\rightarrow H^{0}(mk(K_{X_{s}}+B_{s}))$ is surjective for any 
$s\in S$, where 
$f:X\rightarrow S$ is the given morphism. Since 
$S^{k}H^{0}(m(K_{X_{s}}+B_{s}))\rightarrow H^{0}(mk(K_{X_{s}}+B_{s}))$ is surjective for any 
$k>0$, it follows from Nakayama’s lemma that
$$\begin{eqnarray}S^{k}f_{\ast }{\mathcal{O}}_{X}(m(K_{X}+B))\rightarrow f_{\ast }{\mathcal{O}}_{X}(mk(K_{X}+B))\end{eqnarray}$$ is surjective for every 
$k>0$ and so 
$R(m(K_{X}+B))$ is finitely generated over 
$S$. The canonical model of 
$(X,B)$ over 
$S$ is then given by
$$\begin{eqnarray}\operatorname{Proj}_{{\mathcal{O}}_{S}}\left(\bigoplus _{k\geqslant 0}f_{\ast }{\mathcal{O}}_{X}(mk(K_{X}+B))\right).\square\end{eqnarray}$$
