We start with the observation that we have

(7.1)

$$\Gamma(\mathscr{X}_{B},\mathscr{L}_{B}^{\otimes m})\neq 0$$

for some $m>0$. In fact we have

$$\Gamma(\mathscr{X}_{B},\mathscr{L}_{B}^{\otimes m})\otimes_{R}K\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\Gamma(X,L^{\otimes m})\neq 0$$

by flat base change and the ampleness of $L$ for some $m>0$.

We have a cartesian diagram

We observe that $\mathscr{X}_{B}$ is a smooth variety over the perfect field $k$ and write

$${\mathfrak{a}}_{B,m}=\mbox{Im}\bigl(H^{0}({{\mathscr{X}}}_{B},{\mathscr{L}}_{B}^{\otimes m})\otimes_{k}{\mathscr{L}}_{B}^{\otimes-m}\to{\mathcal{O}}_{{{\mathscr{X}}}_{B}}\bigr)$$

for the $m$-th base ideal of $\mathscr{L}_{B}$. Consider the ideal $g^{-1}({\mathfrak{a}}_{B,m})\cdot{\mathcal{O}}_{{\mathscr{X}}}$ in ${\mathcal{O}}_{{{\mathscr{X}}}}$ generated by $g^{-1}({\mathfrak{a}}_{B,m})$. We have $g^{-1}({\mathfrak{a}}_{B,m})\cdot{\mathcal{O}}_{{\mathscr{X}}}=g^{*}{\mathfrak{a}}_{B,m}$ as $g$ is flat. Sections of ${\mathfrak{a}}_{m}$ are locally of the form $s\cdot t^{-1}$ where $s\in\Gamma({{\mathscr{X}}},{\mathscr{L}}^{\otimes m})$ is a global section and $t$ is a local section of ${\mathscr{L}}^{\otimes m}$. Flat base change [Har77, Prop. III.9.3] gives

$$H^{0}(\mathscr{X},\mathscr{L}^{\otimes m})=H^{0}(\mathscr{X}_{B},\mathscr{L}_{B}^{\otimes m})\otimes_{R}K^{\circ}.$$

Hence the formation of base ideals is compatible with base change, i.e. we have

(7.2)

$${\mathfrak{a}}_{m}=g^{-1}({\mathfrak{a}}_{B,m})\cdot{\mathcal{O}}_{{\mathscr{X}}}=g^{*}{\mathfrak{a}}_{B,m}$$

for all $m\in\mathbb{N}_{>0}$.

The family ${\mathfrak{a}}_{B,\bullet}=({\mathfrak{a}}_{B,m})_{m>0}$ defines a graded sequence of ideals in the sense of Section 4. Let ${\mathfrak{b}}_{B,m}:=\tau({\mathfrak{a}}_{B,\bullet}^{m})$ denote the associated asymptotic test ideal of exponent $m$. Motivated by (7.2) we define

$${\mathfrak{b}}_{m}:=g^{-1}{\mathfrak{b}}_{B,m}\cdot{\mathcal{O}}_{{{\mathscr{X}}}}=g^{*}{\mathfrak{b}}_{B,m}$$

as the ideal in ${{\mathcal{O}}_{{\mathscr{X}}}}$ generated by ${\mathfrak{b}}_{B,m}.$ These ideals have the following properties:

1. (a)

   We have $\mathfrak{a}_{m}\subset\mathfrak{b}_{m}$ for all $m\in\mathbb{N}_{>0}$.

2. (b)

   We have $\mathfrak{b}_{ml}\subset\mathfrak{b}_{m}^{l}$ for all $l,m\in\mathbb{N}_{>0}$.

3. (c)

   There is $m_{0}\geq 0$ such that $\mathcal{A}^{\otimes m_{0}}\otimes\mathscr{L}^{\otimes m}\otimes\mathfrak{b}_{m}$ is globally generated for all $m>0$.

Properties (a) and (b) follow from the corresponding properties of $\mathfrak{a}_{B,m}$ and $\mathfrak{b}_{B,m}$ mentioned in (4.5), (4.6), and (4.7) if we observe (7.2).

Property (c) is a consequence of the generalization of Mustaţă’s uniform generation property given in Theorem 4.6. Write ${\mathscr{L}}_{B}={\mathcal{O}}(D)$ for some divisor $D$ on ${{\mathscr{X}}}_{B}$ and choose a divisor $H$ on ${{\mathscr{X}}}_{B}$ such that ${\mathcal{O}}(H)$ is ample and globally generated. Fix $d>\dim{{\mathscr{X}}}_{B}$ and a canonical divisor $K_{{{\mathscr{X}}}_{B}/k}$ on the smooth $k$-variety ${{\mathscr{X}}}_{B}$. As $\mathcal{A}_{B}$ is ample we find some $m_{0}\in\mathbb{N}$ such that $\mathcal{A}_{B}^{\otimes m_{0}}\otimes\mathcal{O}(-K_{X/k}-dH)$ is globally generated. Given $m\in\mathbb{N}_{>0}$ we put $E:=mD$. Since $\mathscr{L}_{B}$ satisfies (7.1), for any $m\in\mathbb{N}_{>0}$ we may use $E:=mD$ and $\lambda:=m$ in Theorem 4.6 to see that the sheaf

$\displaystyle\mathcal{O}(K_{{{\mathscr{X}}}_{B}/k}+dH)\otimes\mathscr{L}_{B}^{\otimes m}\otimes\mathfrak{b}_{B,m}$

is globally generated. As a consequence, our choice of $m_{0}$ implies that $\mathcal{A}_{B}^{\otimes m_{0}}\otimes\mathscr{L}_{B}^{\otimes m}\otimes\mathfrak{b}_{B,m}$ is globally generated. Base change to $K^{\circ}$ proves (c).

Now we follow the proof of [BFJ16a, Thm. 8.5]. Step 1 of loc. cit. holds not only on quasi-monomial points of ${X^{{\mathrm{an}}}}$, but pointwise on the whole ${X^{{\mathrm{an}}}}$ using Proposition 2.10 and our different definition of $P_{\theta}(0)$. Then Step 2 of loc. cit. works in our setting using properties (a), (b), and (c) above. The only difference is that all inequalities hold immediately on ${X^{{\mathrm{an}}}}$ and not only on the quasi-monomial points of ${X^{{\mathrm{an}}}}$. ∎

It follows from our assumptions that $X$ is base change of the generic fiber $\mathscr{X}_{R,\eta}$ of $\mathscr{X}_{R}/R$ to $K$. Since $X$ is smooth, we conclude that $\mathscr{X}_{R,\eta}$ is smooth as well [EGAIV, Cor. 17.7.3]. By Proposition 2.9(vii), it is enough to prove the claim for any positive multiple of $\theta$. Using this and Lemma 7.5 below, we see that by passing to dominant models, we may assume that $\mathscr{X}_{R}$ is regular and that the restriction of $\mathscr{L}_{R}$ to $\mathscr{X}_{R,\eta}$ extends to an ample line bundle on $\mathscr{X}_{R}$. By Remark 7.2, these conditions are equivalent to Assumption 7.1 and hence the claim follows from Theorem 7.3. ∎

The proof proceeds in three steps. First, we use a result of Lütkebohmert about vertical blowing ups to show that $L^{\prime}$ may be assumed to extend to an ample line bundle $\mathscr{H}$ on $\mathscr{X}_{R}$. In a second step, we show that $\mathscr{X}_{R}$ may be also assumed to be semi-factorial by a theorem of Pépin. In a third step, we use resolution of singularities to construct our desired regular model $\mathscr{X}_{R}^{\prime}$.

Step 1: Replacing $L^{\prime}$ by a positive tensor power, we may assume that $L^{\prime}$ has an ample extension $\mathscr{H}_{R}$ to a projective $R$-model ${\mathscr{Y}}_{R}$. There is a blow up $\pi:\mathscr{Z}_{R}\to{\mathscr{Y}}_{R}$ in an ideal sheaf $\mathcal{J}$ supported in the special fiber of ${\mathscr{Y}}_{R}$ such that the identity on $X^{\prime}$ extends to a morphism $\mathscr{Z}_{R}\to\mathscr{X}_{R}$ [Lü93, Lemma 2.2]. Then $\pi^{-1}(\mathcal{J})=\mathcal{O}_{\mathscr{Z}_{R}/{\mathscr{Y}}_{R}}(1)$ and hence there is $\ell\in\mathbb{N}_{>0}$ such that $\pi^{*}(\mathscr{H}^{\otimes\ell})\otimes\mathcal{O}_{\mathscr{Z}_{R}/{\mathscr{Y}}_{R}}(1)$ is ample [Har77, Prop. II.7.10]. We conclude that by replacing $\mathscr{X}_{R}$ by $\mathscr{Z}_{R}$ and by passing to a positive tensor power of $L^{\prime}$, we may assume that $L^{\prime}$ has an ample extension $\mathscr{H}_{R}$ to $\mathscr{X}_{R}$. This completes the first step.

Step 2: By a result of Pépin [Pé13, Thm. 3.1], there is a a blowing-up morphism $\pi^{\prime}\colon\mathscr{Z}_{R}^{\prime}\to\mathscr{X}_{R}$ centered in the special fiber of $\mathscr{X}_{R}$ such that $\mathscr{Z}_{R}^{\prime}$ is semi-factorial. The latter means that every line bundle on the generic fiber $\mathscr{Z}_{R,\eta}^{\prime}$ of $\mathscr{Z}_{R}^{\prime}$ over $R$ extends to a line bundle on $\mathscr{Z}_{R}^{\prime}$. Similarly as in the first step, we may assume that a positive tensor power of $L^{\prime}$ extends to an ample line bundle on $\mathscr{Z}_{R}^{\prime}$. Replacing $\mathscr{X}_{R}$ by $\mathscr{Z}_{R}^{\prime}$ and $L^{\prime}$ by this positive tensor power, we get the second step.

Step 3: We may assume that $B$ is affine. Using $R=\mathcal{O}_{B,b}$ for some $b\in B^{(1)}$, it is clear that $\mathscr{X}_{R}$ extends to a projective integral scheme $\mathscr{X}_{B}$ over $B$. By using resolution of singularities over $k$ in dimension $d+n$, there is a regular integral scheme $\mathscr{X}_{B}^{\prime}$ and a projective morphism $\varphi_{B}\colon\mathscr{X}_{B}^{\prime}\to\mathscr{X}_{B}$ which is an isomorphism over the regular locus of $\mathscr{X}_{B}$. Since $X^{\prime}$ is contained in the regular locus of $\mathscr{X}_{B}$, we conclude that $\varphi_{B}$ maps the generic fiber $\mathscr{X}_{B,\eta}^{\prime}$ of $\mathscr{X}_{B}^{\prime}$ over $B$ isomorphically onto $X^{\prime}=\mathscr{X}_{B,\eta}$. As usual, we read this isomorphism as an identification. Then we get an induced projective morphism

$$\varphi_{R}:\mathscr{X}_{R}^{\prime}\coloneqq\mathscr{X}_{B}^{\prime}\times_{B}{\Spec(R)}\longrightarrow\mathscr{X}_{R}$$

extending the identity on $X^{\prime}$. The same argument as in the first step gives $m\in\mathbb{N}_{>0}$ such that

$$\mathscr{L}_{R}^{\prime}\coloneqq\varphi_{R}^{*}(\mathscr{H}_{R}^{\otimes m})\otimes\mathcal{O}_{\mathscr{X}_{R}^{\prime}/\mathscr{X}_{R}}(1)$$

is an ample line bundle on $\mathscr{X}_{R}^{\prime}$. Let $F$ be the restriction of $\mathcal{O}_{\mathscr{X}_{R}^{\prime}/\mathscr{X}_{R}}(1)$ to $\mathscr{X}_{R,\eta}^{\prime}=\mathscr{X}_{R,\eta}=X^{\prime}$. Then $\mathscr{L}_{R}^{\prime}$ is a model of $(L^{\prime})^{\otimes m}\otimes F$. To prove the lemma, we have to ensure that $F$ may be assumed to be $\mathcal{O}_{X^{\prime}}$. To do so, we use that $\mathscr{X}_{R}$ is semi-factorial to extend $F$ to a line bundle $\mathscr{F}_{B}$ on $\mathscr{X}_{R}$. Then we may replace $\mathcal{O}_{\mathscr{X}_{R}^{\prime}/\mathscr{X}_{R}}(1)$ by $\mathcal{O}_{\mathscr{X}_{R}^{\prime}/\mathscr{X}_{R}}(1)\otimes\varphi_{R}^{*}(\mathscr{F}^{-1})$ to deduce the claim. ∎