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 ${X^{{\mathrm{an}}}}$. ∎