Proof. Since $X$ is normal, we have $f_{\ast }{\mathcal{O}}_{X^{\prime }}\simeq {\mathcal{O}}_{X}$ [Reference Hartshorne29, Proof of Corollary III.11.4]. By the projection formula, we then have

$$\begin{eqnarray}H^{0}(X,{\mathcal{O}}_{X}(D))\simeq H^{0}(X^{\prime },{\mathcal{O}}_{X^{\prime }}(f^{\ast }D)),\end{eqnarray}$$

and the lemma then follows by pulling back the evaluation map (1).◻

Construction 3.1. [Reference Hochster and Huneke34, (6.7) and (6.11)]

Let $(A,\mathfrak{m},k)$ be a Noetherian complete local ring of prime characteristic $p>0$. By the Cohen structure theorem, we may identify $k$ with a coefficient field $k\subseteq A$. Moreover, by Zorn’s lemma (see [Reference Matsumura48, p. 202]), we may choose a $p$-basis $\unicode[STIX]{x1D6EC}$ for $k$, which is a subset $\unicode[STIX]{x1D6EC}\subseteq k$ such that $k=k^{p}(\unicode[STIX]{x1D6EC})$, and such that for every finite subset $\unicode[STIX]{x1D6F4}\subseteq \unicode[STIX]{x1D6EC}$ with $s$ elements, we have $[k^{p}(\unicode[STIX]{x1D6F4}):k^{p}]=p^{s}$.

Now let $\unicode[STIX]{x1D6E4}\subseteq \unicode[STIX]{x1D6EC}$ be a cofinite subset, that is, a subset $\unicode[STIX]{x1D6E4}$ of $\unicode[STIX]{x1D6EC}$ such that $\unicode[STIX]{x1D6EC}\smallsetminus \unicode[STIX]{x1D6E4}$ is a finite set. For each integer $e\geqslant 0$, consider the subfield

$$\begin{eqnarray}k_{e}^{\unicode[STIX]{x1D6E4}}=k[\unicode[STIX]{x1D706}^{1/p^{e}}]_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6E4}}\subseteq k_{\text{perf}}\end{eqnarray}$$

of a perfect closure $k_{\text{perf}}$ of $k$. These form an ascending chain, and we then set

where $k_{e}^{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x27E6}A\unicode[STIX]{x27E7}$ is the completion of $k_{e}^{\unicode[STIX]{x1D6E4}}\otimes _{k}A$ at the extended ideal $\mathfrak{m}\cdot (k_{e}^{\unicode[STIX]{x1D6E4}}\otimes _{k}A)$. Note that if $A=k$ is a field, then $A^{\unicode[STIX]{x1D6E4}}=k^{\unicode[STIX]{x1D6E4}}$ is a field by construction.

Finally, let $X$ be a scheme essentially of finite type over $A$, and consider two cofinite subsets $\unicode[STIX]{x1D6E4}\subseteq \unicode[STIX]{x1D6EC}$ and $\unicode[STIX]{x1D6E4}^{\prime }\subseteq \unicode[STIX]{x1D6EC}$ such that $\unicode[STIX]{x1D6E4}\subseteq \unicode[STIX]{x1D6E4}^{\prime }$. We then have the following commutative diagram whose vertical faces are cartesian:

Proof. The ring $A^{\unicode[STIX]{x1D6E4}}$ is Noetherian and $F$-finite [Reference Hochster and Huneke34, (6.11)], hence $X^{\unicode[STIX]{x1D6E4}}$ is also by Example 2.6 and the fact that morphisms essentially of finite type are preserved under base change [Reference Nayak53, (2.2)]. The ring extensions $A\subseteq A^{\unicode[STIX]{x1D6E4}}$ and $A^{\unicode[STIX]{x1D6E4}}\subseteq A^{\unicode[STIX]{x1D6E4}^{\prime }}$ are purely inseparable and faithfully flat [Reference Hochster and Huneke34, (6.11)], hence induce faithfully flat universal homeomorphisms on spectra [Reference Grothendieck and Dieudonné24, Proposition 2.4.5(i)]. Thus, the morphisms $\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}}$ and $\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}\unicode[STIX]{x1D6E4}^{\prime }}$ are faithfully flat universal homeomorphisms by base change. Finally, the ring extension $A\subseteq A^{\unicode[STIX]{x1D6E4}}$ is flat with local complete intersection fibers [Reference Hashimoto30, Lemma 3.19], hence $\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}}$ is also by base change [Reference Avramov2, Corollary 4].◻

Proof. For (i), it suffices to note that $\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}}$ is faithfully flat with local complete intersection fibers by Lemma 3.2(ii).

For (ii), we first note that ($\unicode[STIX]{x1D6E4}3^{\prime }$) implies ($\unicode[STIX]{x1D6E4}3$), since there exists a cofinite subset $\unicode[STIX]{x1D6E4}_{1}\subseteq \unicode[STIX]{x1D6EC}$ such that the closed fiber is a field for every cofinite subset $\unicode[STIX]{x1D6E4}\subseteq \unicode[STIX]{x1D6E4}_{1}$ by [Reference Hochster and Huneke34, Lemma 6.13(b)]. From now on, we therefore assume that ${\mathcal{P}}$ satisfies ($\unicode[STIX]{x1D6E4}1$), ($\unicode[STIX]{x1D6E4}2$), and ($\unicode[STIX]{x1D6E4}3$).

For every cofinite subset $\unicode[STIX]{x1D6E4}\subseteq \unicode[STIX]{x1D6EC}$, the set ${\mathcal{P}}(X^{\unicode[STIX]{x1D6E4}})$ is open by ($\unicode[STIX]{x1D6E4}1$) since $X^{\unicode[STIX]{x1D6E4}}$ is Noetherian and $F$-finite by Lemma 3.2(i). Moreover, the morphisms $\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}}$ and $\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}\unicode[STIX]{x1D6E4}^{\prime }}$ are faithfully flat universal homeomorphisms for every cofinite subset $\unicode[STIX]{x1D6E4}^{\prime }\subseteq \unicode[STIX]{x1D6EC}$ such that $\unicode[STIX]{x1D6E4}\subseteq \unicode[STIX]{x1D6E4}^{\prime }$ by Lemmas 3.2(ii) and 3.2(iii), hence by ($\unicode[STIX]{x1D6E4}2$), we have the inclusions

(3)$$\begin{eqnarray}{\mathcal{P}}(X)\supseteq \unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}}({\mathcal{P}}(X^{\unicode[STIX]{x1D6E4}}))\supseteq \unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}^{\prime }}({\mathcal{P}}(X^{\unicode[STIX]{x1D6E4}^{\prime }}))\end{eqnarray}$$

in $X$, where $\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}}({\mathcal{P}}(X^{\unicode[STIX]{x1D6E4}}))$ and $\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}^{\prime }}({\mathcal{P}}(X^{\unicode[STIX]{x1D6E4}^{\prime }}))$ are open. Since $X$ is Noetherian, it satisfies the ascending chain condition on the open sets $\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}}({\mathcal{P}}(X^{\unicode[STIX]{x1D6E4}}))$, hence we can choose a cofinite subset $\unicode[STIX]{x1D6E4}_{0}\subseteq \unicode[STIX]{x1D6EC}$ such that $\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}_{0}}({\mathcal{P}}(X^{\unicode[STIX]{x1D6E4}_{0}}))$ is maximal with respect to inclusion.

We claim that ${\mathcal{P}}(X)=\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}_{0}}({\mathcal{P}}(X^{\unicode[STIX]{x1D6E4}_{0}}))$ for every cofinite subset $\unicode[STIX]{x1D6E4}\subseteq \unicode[STIX]{x1D6E4}_{0}$. By (3), it suffices to show the inclusion $\subseteq$. Suppose there exists $x\in {\mathcal{P}}(X)\smallsetminus \unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}_{0}}({\mathcal{P}}(X^{\unicode[STIX]{x1D6E4}_{0}}))$. By ($\unicode[STIX]{x1D6E4}3$), there exists a cofinite subset $\unicode[STIX]{x1D6E4}_{1}\subseteq \unicode[STIX]{x1D6EC}$ such that $(\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}})^{-1}(x)\in {\mathcal{P}}(X^{\unicode[STIX]{x1D6E4}})$ for every cofinite subset $\unicode[STIX]{x1D6E4}\subseteq \unicode[STIX]{x1D6E4}_{1}$. Choosing $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6E4}_{0}\cap \unicode[STIX]{x1D6E4}_{1}$, we have $x\in \unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}}({\mathcal{P}}(X^{\unicode[STIX]{x1D6E4}}))\smallsetminus \unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}_{0}}({\mathcal{P}}(X^{\unicode[STIX]{x1D6E4}_{0}}))$, contradicting the maximality of $\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}_{0}}({\mathcal{P}}(X^{\unicode[STIX]{x1D6E4}_{0}}))$.◻

Proof. For (i), it suffices to note that these properties satisfy the condition in Proposition 3.3(i) by [Reference Avramov2, Corollary 2] and [Reference Matsumura48, Theorem 23.4, Corollary to Theorem 23.3, and Theorem 23.9(iii)], respectively.

We now prove (ii). We first note that (ii) holds for regularity since ($\unicode[STIX]{x1D6E4}1$) holds by the excellence of $X^{\unicode[STIX]{x1D6E4}}$, and ($\unicode[STIX]{x1D6E4}2$) and ($\unicode[STIX]{x1D6E4}3^{\prime }$) hold by [Reference Matsumura48, Theorem 23.7]. Since $\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}}$ preserves the dimension of local rings, we therefore see that (ii) holds for $(R_{n})$. (ii) for normality and reducedness then follows from (i) since they are equivalent to $(R_{1})+(S_{2})$ and $(R_{0})+(S_{1})$, respectively.

To prove (ii) holds in the remaining cases, we check the conditions in Proposition 3.3(ii). For weak normality, ($\unicode[STIX]{x1D6E4}1$) holds by [Reference Bingener and Flenner3, Theorem 7.1.3], and ($\unicode[STIX]{x1D6E4}2$) holds by [Reference Manaresi47, Corollary II.2]. To show that ($\unicode[STIX]{x1D6E4}3$) holds, recall by [Reference Manaresi47, Theorem I.6] that a reduced ring $B$ is weakly normal if and only if

(4)

is an equalizer diagram, where $B^{\unicode[STIX]{x1D708}}$ is the normalization of $B$. Now suppose $B$ is weakly normal, and let $\unicode[STIX]{x1D6E4}_{1}\subseteq \unicode[STIX]{x1D6EC}$ be a cofinite subset such that $B^{\unicode[STIX]{x1D6E4}}$ is reduced, $(B^{\unicode[STIX]{x1D708}})^{\unicode[STIX]{x1D6E4}}$ is normal, and $((B^{\unicode[STIX]{x1D708}}\otimes _{B}B^{\unicode[STIX]{x1D708}})_{\text{red}})^{\unicode[STIX]{x1D6E4}}$ is reduced for every cofinite subset $\unicode[STIX]{x1D6E4}\subseteq \unicode[STIX]{x1D6E4}_{1}$; such a $\unicode[STIX]{x1D6E4}_{1}$ exists by the previous paragraph. We claim that $B^{\unicode[STIX]{x1D6E4}}$ is weakly normal for every $\unicode[STIX]{x1D6E4}\subseteq \unicode[STIX]{x1D6E4}_{1}$ cofinite in $\unicode[STIX]{x1D6EC}$. Since (

) is an equalizer diagram and $A\subseteq A^{\unicode[STIX]{x1D6E4}}$ is flat, the diagram

is an equalizer diagram. Moreover, since $B^{\unicode[STIX]{x1D6E4}}\subseteq (B^{\unicode[STIX]{x1D708}})^{\unicode[STIX]{x1D6E4}}$ is an integral extension of rings with the same total ring of fractions, and $(B^{\unicode[STIX]{x1D708}})^{\unicode[STIX]{x1D6E4}}$ is normal, we see that $(B^{\unicode[STIX]{x1D708}})^{\unicode[STIX]{x1D6E4}}=(B^{\unicode[STIX]{x1D6E4}})^{\unicode[STIX]{x1D708}}$. Finally, $((B^{\unicode[STIX]{x1D708}}\otimes _{B}B^{\unicode[STIX]{x1D708}})_{\text{red}})^{\unicode[STIX]{x1D6E4}}$ is reduced, hence we have the natural isomorphism

$$\begin{eqnarray}((B^{\unicode[STIX]{x1D708}}\otimes _{B}B^{\unicode[STIX]{x1D708}})_{\text{red}})^{\unicode[STIX]{x1D6E4}}\simeq ((B^{\unicode[STIX]{x1D6E4}})^{\unicode[STIX]{x1D708}}\otimes _{B^{\unicode[STIX]{x1D6E4}}}(B^{\unicode[STIX]{x1D6E4}})^{\unicode[STIX]{x1D708}})_{\text{red}}.\end{eqnarray}$$

Thus, since the analogue of (

) with $B$ replaced by $B^{\unicode[STIX]{x1D6E4}}$ is an equalizer diagram, we see that $B^{\unicode[STIX]{x1D6E4}}$ is weakly normal for every $\unicode[STIX]{x1D6E4}\subseteq \unicode[STIX]{x1D6E4}_{1}$ cofinite in $\unicode[STIX]{x1D6EC}$, hence ($\unicode[STIX]{x1D6E4}3$) holds for weak normality.

We now prove (ii) for strong $F$-regularity, $F$-purity, and $F$-rationality. First, ($\unicode[STIX]{x1D6E4}1$) holds for strong $F$-regularity by [Reference Hashimoto30, Lemma 3.29], and the same argument shows that ($\unicode[STIX]{x1D6E4}1$) holds for $F$-purity since the $F$-pure and $F$-split loci coincide for $F$-finite rings [Reference Hochster and Roberts36, Corollary 5.3]. Next, ($\unicode[STIX]{x1D6E4}1$) for $F$-rationality holds by [Reference Vélez66, Theorem 1.11] since the reduced locus is open and reduced $F$-finite rings are admissible in the sense of [Reference Vélez66, Definition 1.5] by Theorem 2.7. It then suffices to note that ($\unicode[STIX]{x1D6E4}2$) holds by [Reference Hashimoto30, Lemma 3.17], [Reference Hochster and Roberts36, Proposition 5.13], and [Reference Vélez66, (6) on p. 440], respectively, and ($\unicode[STIX]{x1D6E4}3$) holds by [Reference Hashimoto30, Corollary 3.31], [Reference Ma45, Proposition 5.4], and [Reference Vélez66, Lemma 2.3], respectively.

Finally, we prove (ii) for $F$-injectivity. First, ($\unicode[STIX]{x1D6E4}1$) and ($\unicode[STIX]{x1D6E4}2$) hold by Lemmas A.2 and A.3, respectively. The proof of [Reference Enescu and Hochster17, Lemma 2.9(b)] implies ($\unicode[STIX]{x1D6E4}3$), since the residue field of $B$ is a finite extension of $k$, hence socles of Artinian $B$-modules are finite-dimensional $k$-vector spaces.◻

Proof. Let $A\rightarrow \widehat{A}$ be the completion of $A$ at $\mathfrak{m}$, and let $\unicode[STIX]{x1D6EC}$ be a $p$-basis for $\widehat{A}/\mathfrak{m}\widehat{A}$ as in Construction 3.1. For every cofinite subset $\unicode[STIX]{x1D6E4}\subseteq \unicode[STIX]{x1D6EC}$, consider the commutative diagram

where the squares are cartesian. By Theorem 3.4, there exists a cofinite subset $\unicode[STIX]{x1D6E4}\subseteq \unicode[STIX]{x1D6EC}$ such that $\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}}$ is a homeomorphism identifying $F$-pure loci. Since $X\times _{A}\widehat{A}^{\unicode[STIX]{x1D6E4}}$ is $F$-finite, the $F$-pure locus in $X\times _{A}\widehat{A}$ is therefore open by the fact that ($\unicode[STIX]{x1D6E4}1$) holds for $F$-purity (see the proof of Theorem 3.4(ii)).

Now let $x\in X\times _{A}\widehat{A}$. Since $A\rightarrow \widehat{A}$ is a regular homomorphism, the morphism $\unicode[STIX]{x1D70B}$ is also regular by base change [Reference Grothendieck and Dieudonné24, Proposition 6.8.3(iii)]. Thus, ${\mathcal{O}}_{X\times _{A}\widehat{A},x}$ is $F$-pure if and only if ${\mathcal{O}}_{X,\unicode[STIX]{x1D70B}(x)}$ is $F$-pure by [Reference Hochster and Roberts36, Proposition 5.13] and [Reference Hashimoto30, Propositions 2.4(4) and 2.4(6)]. Denoting the $F$-pure locus in $X$ by $W$, we see that $\unicode[STIX]{x1D70B}^{-1}(W)$ is the $F$-pure locus in $X\times _{A}\widehat{A}$. Since $\unicode[STIX]{x1D70B}^{-1}(W)$ is open and $\unicode[STIX]{x1D70B}$ is quasicompact and faithfully flat by base change, the $F$-pure locus $W\subseteq X$ is open by [Reference Grothendieck and Dieudonné24, Corollaire 2.3.12].◻

Proof. For (i), first write $\unicode[STIX]{x1D6E5}=\sum a_{i}D_{i}$, where $a_{i}\in \mathbf{R}$ and $D_{i}$ are prime divisors. By Theorem 3.4 applied to the regular locus of $X$ and of every set of intersections of the $D_{i}$, we see that there exists a cofinite subset $\unicode[STIX]{x1D6E4}_{0}\subseteq \unicode[STIX]{x1D6EC}$ such that $X^{\unicode[STIX]{x1D6E4}}$ is a regular variety and $(\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}})^{\ast }\unicode[STIX]{x1D6E5}=\sum a_{i}(\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D6E4}})^{\ast }D_{i}$ has simple normal crossing support for every $\unicode[STIX]{x1D6E4}\subseteq \unicode[STIX]{x1D6E4}_{0}$ cofinite in $\unicode[STIX]{x1D6EC}$. (ii) then follows by applying (i) to a log resolution of $(X,\unicode[STIX]{x1D6E5})$ while simultaneously choosing $\unicode[STIX]{x1D6E4}_{0}$ such that $X^{\unicode[STIX]{x1D6E4}}$ is normal for every cofinite subset $\unicode[STIX]{x1D6E4}\subseteq \unicode[STIX]{x1D6E4}_{0}$.◻

Proof of reduction.

Note that the formation of $K_{X}$ is compatible with ground field extensions [Reference Hartshorne28, Corollary V.3.4(a)], and that $f$-nefness is preserved under base change since $f$-ampleness is. By flat base change and the fact that field extensions are faithfully flat, $f$-semi-ampleness can be checked after a ground field extension. Since $k(x^{1/p^{\infty }})$ contains the infinite perfect field $\mathbf{F}_{p}(x^{1/p^{\infty }})$ and applying the gamma construction to $k(x^{1/p^{\infty }})$ results in an $F$-finite field (Construction 3.1 and Lemma 3.2(i)), it therefore suffices to show that for some choice of $\unicode[STIX]{x1D6E4}$, the base change of $(X,\unicode[STIX]{x1D6E5})$ under the sequence of ground field extensions

$$\begin{eqnarray}k\subseteq k(x^{1/p^{\infty }})\subseteq (k(x^{1/p^{\infty }}))^{\unicode[STIX]{x1D6E4}}\end{eqnarray}$$

is a log canonical surface. Moreover, Corollary 3.7(ii) implies it suffices to prove that the base change of $(X,\unicode[STIX]{x1D6E5})$ to $k(x^{1/p^{\infty }})$ is a log canonical surface.

Fix a log resolution $\unicode[STIX]{x1D707}:Y\rightarrow X$ for $(X,\unicode[STIX]{x1D6E5})$, and write $K_{Y}-\unicode[STIX]{x1D707}^{\ast }(K_{X}+\unicode[STIX]{x1D6E5})=\sum _{i}a_{i}E_{i}$. Note that $k(x^{1/p^{\infty }})=\bigcup _{e}k(x^{1/p^{e}})$, and that each field $k(x^{1/p^{e}})$ is isomorphic to $k(x)$. Since integrality, normality, and regularity are preserved under limits of schemes with affine and flat transition morphisms [Reference Grothendieck and Dieudonné24, Corollaire 5.13.4 and Proposition 5.13.7], it suffices to show that $X\times _{k}k(x)$ is a normal variety, $Y\times _{k}k(x)$ is a regular variety, and each $E_{i}\times _{k}k(x)$ is a regular variety such that every intersection of the $E_{i}\times _{k}k(x)$’s is regular. This follows for $X\times _{k}k(x)$, since if $\bigcup _{j}U_{j}$ is an affine open covering of $X$, then $X\times _{k}k(x)$ is covered by affine open subsets that are localizations of the normal varieties $U_{j}\times _{k}\mathbf{A}_{k}^{1}$, which pairwise intersect. A similar argument works for $Y$, the $E_{i}$’s, and the intersections of the $E_{i}$’s.◻

Proof. The proof of [Reference Küronya41, Proposition 2.9(1)] works in our setting with the additional hypothesis that $D$ is a Cartier divisor. It therefore suffices to reduce to this case. If the statement holds for integral $D$, then it also holds for $D\in \operatorname{Div}_{\mathbf{Q}}(X)$ by homogeneity of $\widehat{h}^{i}$ (Proposition 4.2). Moreover, the subspace of $\operatorname{Div}_{\mathbf{R}}(X)$ spanned by the Cartier divisors appearing in $D$ is finite-dimensional, hence by approximating each coefficient in $D$ by rational numbers, Proposition 4.2 implies the statement for $D\in \operatorname{Div}_{\mathbf{R}}(X)$ by continuity.◻

Proof. By Remark 4.4, it suffices to consider the case when $D$ is integral. Let $f:Y\rightarrow X$ be a regular alteration of degree $d$ [Reference de Jong38, Theorem 4.1]. We then have

$$\begin{eqnarray}\widehat{h}^{i}(Y,f^{\ast }D)=\limsup _{m\rightarrow \infty }\frac{h^{n-i}(Y,{\mathcal{O}}_{Y}(K_{Y}-f^{\ast }(mD)))}{m^{n}/n!}=\widehat{h}^{n-i}(Y,-f^{\ast }D)\end{eqnarray}$$

by Serre duality and [Reference Burgos Gil, Gubler, Jell, Künnemann and Martin6, Lemma 3.2.1], respectively. By Proposition 4.3, the left-hand side is equal to $d\cdot \widehat{h}^{i}(X,D)$ and the right-hand side is equal to $d\cdot \widehat{h}^{n-i}(X,-D)$, hence the statement follows after dividing by $d$.◻

Proof. The statement is trivial if $H^{0}(V,{\mathcal{O}}_{V}(mD))=0$ for every integer $m>0$, since in this case $\mathfrak{b}(|mqD|)=0$ for all positive integers $m,q$. We therefore assume $H^{0}(V,{\mathcal{O}}_{V}(mD))\neq 0$ for some integer $m>0$. We prove the statement in positive characteristic; see Remark 4.7 for the characteristic zero case.

We fix some notation. Set $D^{\prime }=\unicode[STIX]{x1D707}^{\ast }D$ and set $E=(\unicode[STIX]{x1D707}^{-1}(Z))_{\text{red}}$. We fix a very ample Cartier divisor $H$ on $V^{\prime }$, and set $A=K_{V^{\prime }}+(\dim V^{\prime }+1)H$. For every subvariety $W\subseteq V^{\prime }$, a complete intersection curve is a curve formed by taking the intersection of $\dim W-1$ hyperplane sections in $|H|_{W}|$, and a general complete intersection curve is one formed by taking these hyperplane sections to be general in $|H|_{W}|$. For each positive integer $q$, we consider the asymptotic test ideal

$$\begin{eqnarray}\unicode[STIX]{x1D70F}(V^{\prime },\Vert qD^{\prime }\Vert )=\unicode[STIX]{x1D70F}(\Vert qD^{\prime }\Vert )\subseteq {\mathcal{O}}_{V^{\prime }}.\end{eqnarray}$$

By uniform global generation for test ideals [Reference Sato57, Proposition 4.1], the sheaf

(6)$$\begin{eqnarray}\unicode[STIX]{x1D70F}(\Vert qD^{\prime }\Vert )\otimes {\mathcal{O}}_{V^{\prime }}(qD^{\prime }+A)\end{eqnarray}$$

is globally generated for every integer $q>0$.

Step 1. There exists an integer $\ell _{0}>0$ such that for every integer $\ell >\ell _{0}$ and for every irreducible component $F$ of $E$ that dominates $(Z_{\unicode[STIX]{x1D6FC}})_{\text{red}}$ for some $\unicode[STIX]{x1D6FC}$, we have

$$\begin{eqnarray}\unicode[STIX]{x1D70F}(\Vert \ell D^{\prime }\Vert )\subseteq {\mathcal{O}}_{V^{\prime }}(-F).\end{eqnarray}$$

Let $C\subseteq F$ be a general complete intersection curve; note that $C$ is integral by Bertini’s theorem [Reference Flenner, O’Carroll and Vogel20, Theorem 3.4.10 and Corollary 3.4.14] and dominates $(Z_{\unicode[STIX]{x1D6FC}})_{\text{red}}$ for some $\unicode[STIX]{x1D6FC}$, hence $(D^{\prime }\cdot C)<0$ by the projection formula and (5). If for some integer $q>0$, the curve $C$ is not contained in the zero locus of $\unicode[STIX]{x1D70F}(\Vert qD^{\prime }\Vert )$, then the fact that the sheaf (6) is globally generated implies

$$\begin{eqnarray}((qD^{\prime }+A)\cdot C)\geqslant 0.\end{eqnarray}$$

Letting $\ell _{0F}=-(A\cdot C)/(D^{\prime }\cdot C)$, we see that the ideal $\unicode[STIX]{x1D70F}(\Vert \ell D^{\prime }\Vert )$ vanishes everywhere along $C$ for every integer $\ell >\ell _{0F}$. By varying $C$, the ideal $\unicode[STIX]{x1D70F}(\Vert \ell D^{\prime }\Vert )$ must vanish everywhere along $F$ for every integer $\ell >\ell _{0F}$, hence we can set $\ell _{0}=\max _{F}\{\ell _{0F}\}$.

Step 2. Let $E_{i}$ be an irreducible component of $E$ not dominating $Z_{\unicode[STIX]{x1D6FC}}$ for every $\unicode[STIX]{x1D6FC}$. Suppose $E_{j}$ is another irreducible component of $E$ such that $E_{i}\cap E_{j}\neq \emptyset$ and for which there exists an integer $\ell _{j}$ such that for every integer $\ell >\ell _{j}$, we have

$$\begin{eqnarray}\unicode[STIX]{x1D70F}(\Vert \ell D^{\prime }\Vert )\subseteq {\mathcal{O}}_{V^{\prime }}(-E_{j}).\end{eqnarray}$$

Then, there is an integer $\ell _{i}\geqslant \ell _{j}$ such that for every integer $\ell >\ell _{i}$, we have

$$\begin{eqnarray}\unicode[STIX]{x1D70F}(\Vert \ell D^{\prime }\Vert )\subseteq {\mathcal{O}}_{V^{\prime }}(-E_{i}).\end{eqnarray}$$

Let $C\subseteq E_{i}$ be a complete intersection curve. By the assumption that $E$ is a simple normal crossing divisor, there exists at least one closed point $P\in C\cap E_{j}$. For every $\ell >\ell _{j}$ and every $m>0$, we have the sequence of inclusions

(7)$$\begin{eqnarray}\displaystyle \begin{array}{@{}rcl@{}} & & (\unicode[STIX]{x1D70F}(\Vert m\ell D^{\prime }\Vert )\otimes {\mathcal{O}}_{V^{\prime }}(m\ell D^{\prime }+A))\cdot {\mathcal{O}}_{C}\\ & & \qquad \subseteq (\unicode[STIX]{x1D70F}(\Vert \ell D^{\prime }\Vert )^{m}\otimes {\mathcal{O}}_{V^{\prime }}(m\ell D^{\prime }+A))\cdot {\mathcal{O}}_{C}\\ & & \qquad \subseteq ({\mathcal{O}}_{V^{\prime }}(-mE_{j})\otimes {\mathcal{O}}_{V^{\prime }}(m\ell D^{\prime }+A))\cdot {\mathcal{O}}_{C}\\ & & \qquad \subseteq {\mathcal{O}}_{C}(A|_{C}-mP),\end{array} & & \displaystyle\end{eqnarray}$$

where the first two inclusions follow from subadditivity [Reference Hara and Yoshida27, Theorem 6.10(2)] and by assumption, respectively. The last inclusion holds since $C$ maps to a closed point in $V$, hence ${\mathcal{O}}_{C}(D^{\prime })={\mathcal{O}}_{C}$. By the global generation of the sheaf in (6) for $q=m\ell$, the inclusion (7) implies that for every integer $\ell >\ell _{j}$, if $\unicode[STIX]{x1D70F}(\Vert m\ell D^{\prime }\Vert )$ does not vanish everywhere along $C$, then $(A\cdot C)\geqslant m$. Choosing $\ell _{i}=\ell _{j}\cdot ((A\cdot C)+1)$, we see that $\unicode[STIX]{x1D70F}(\Vert \ell D^{\prime }\Vert )$ vanishes everywhere along $C$ for every integer $\ell >\ell _{i}$. By varying $C$, we have $\unicode[STIX]{x1D70F}(\Vert \ell D^{\prime }\Vert )\subseteq {\mathcal{O}}_{V^{\prime }}(-E_{i})$ for every integer $\ell >\ell _{i}$.

Step 3. There exists an integer $a>0$ such that

$$\begin{eqnarray}\mathfrak{b}(|maD^{\prime }|)\subseteq {\mathcal{O}}_{V^{\prime }}(-mE)\end{eqnarray}$$

for every integer $m>0$.

Write

$$\begin{eqnarray}E=\mathop{\bigcup }_{j}\mathop{\bigcup }_{i\in I_{j}}E_{ij},\end{eqnarray}$$

where the $E_{ij}$ are the irreducible components of $E$, and the $\bigcup _{i\in I_{j}}E_{ij}$ are the connected components of $E$. Since $V$ is normal, each preimage $\unicode[STIX]{x1D707}^{-1}(Z_{\unicode[STIX]{x1D6FC}})$ is connected by Zariski’s main theorem [Reference Hartshorne29, Corollary III.11.4], hence each connected component $\bigcup _{i\in I_{j}}E_{ij}$ of $E$ contains an irreducible component $E_{i_{0}j}$ that dominates $(Z_{\unicode[STIX]{x1D6FC}})_{\text{red}}$ for some $\unicode[STIX]{x1D6FC}$. By Step 1, there exists an integer $\ell _{0}$ such that for every $j$, we have $\unicode[STIX]{x1D70F}(\Vert \ell D^{\prime }\Vert )\subseteq {\mathcal{O}}_{V^{\prime }}(-E_{i_{0}j})$ for every integer $\ell >\ell _{0}$. For each $j$, by applying Step 2 $(|I_{j}|-1)$ times to the $j$th connected component $\bigcup _{i\in I_{j}}E_{ij}$ of $E$, we can find $\ell _{j}$ such that $\unicode[STIX]{x1D70F}(\Vert \ell D^{\prime }\Vert )\subseteq {\mathcal{O}}_{V^{\prime }}(-E_{ij})$ for every $i\in I_{j}$ and for every integer $\ell >\ell _{j}$. Setting $a=\max _{j}\{\ell _{j}\}+1$, we have $\unicode[STIX]{x1D70F}(\Vert aD^{\prime }\Vert )\subseteq {\mathcal{O}}_{V^{\prime }}(-E)$. Thus, for every integer $m>0$, we have

$$\begin{eqnarray}\mathfrak{b}(|maD^{\prime }|)\subseteq \unicode[STIX]{x1D70F}(|maD^{\prime }|)\subseteq \unicode[STIX]{x1D70F}(\Vert maD^{\prime }\Vert )\subseteq \unicode[STIX]{x1D70F}(\Vert aD^{\prime }\Vert )^{m}\subseteq {\mathcal{O}}_{V^{\prime }}(-mE),\end{eqnarray}$$

where the first inclusion follows by the fact that $V^{\prime }$ is regular hence strongly $F$-regular [Reference Takagi and Watanabe62, Propositions 5.6(1) and 5.6(5)], the second inclusion is by definition of the asymptotic test ideal, and the third inclusion is by subadditivity [Reference Hara and Yoshida27, Theorem 6.10(2)].

Let $\unicode[STIX]{x1D70B}:V^{\prime \prime }\rightarrow V^{\prime }$ be the normalized blowup of the ideal $\unicode[STIX]{x1D707}^{-1}\mathfrak{a}\cdot {\mathcal{O}}_{V^{\prime }}$, and write $(\unicode[STIX]{x1D707}\circ \unicode[STIX]{x1D70B})^{-1}\mathfrak{a}\cdot {\mathcal{O}}_{V^{\prime \prime }}={\mathcal{O}}_{V^{\prime \prime }}(-E^{\prime \prime })$ for a Cartier divisor $E^{\prime \prime }$ on $V^{\prime \prime }$. Note that since ${\mathcal{O}}_{V^{\prime \prime }}(-(\unicode[STIX]{x1D70B}^{\ast }E)_{\text{red}})$ is the radical of ${\mathcal{O}}_{V^{\prime \prime }}(-E^{\prime \prime })$, there exists an integer $b>0$ such that ${\mathcal{O}}_{V^{\prime \prime }}(-b(\unicode[STIX]{x1D70B}^{\ast }E)_{\text{red}})\subseteq {\mathcal{O}}_{V^{\prime \prime }}(-E^{\prime \prime })$. We then have

$$\begin{eqnarray}\displaystyle \mathfrak{b}(|mab\,\unicode[STIX]{x1D70B}^{\ast }D^{\prime }|) & = & \displaystyle \unicode[STIX]{x1D70B}^{-1}\mathfrak{b}(|mabD^{\prime }|)\cdot {\mathcal{O}}_{V^{\prime \prime }}\nonumber\\ \displaystyle & \subseteq & \displaystyle {\mathcal{O}}_{V^{\prime \prime }}(-mb\,\unicode[STIX]{x1D70B}^{\ast }E)\nonumber\\ \displaystyle & \subseteq & \displaystyle {\mathcal{O}}_{V^{\prime \prime }}(-mb(\unicode[STIX]{x1D70B}^{\ast }E)_{\text{red}})\nonumber\\ \displaystyle & \subseteq & \displaystyle {\mathcal{O}}_{V^{\prime \prime }}(-mE^{\prime \prime })\nonumber\end{eqnarray}$$

by Step 3, where the first equality holds by Lemma 2.3. Setting $q=ab$ and pushing forward by $\unicode[STIX]{x1D707}\circ \unicode[STIX]{x1D70B}$, we have

$$\begin{eqnarray}\displaystyle \mathfrak{b}(|mqD|) & \subseteq & \displaystyle (\unicode[STIX]{x1D707}\circ \unicode[STIX]{x1D70B})_{\ast }\mathfrak{b}(|mq\,\unicode[STIX]{x1D70B}^{\ast }D^{\prime }|)\nonumber\\ \displaystyle & \subseteq & \displaystyle (\unicode[STIX]{x1D707}\circ \unicode[STIX]{x1D70B})_{\ast }{\mathcal{O}}_{V^{\prime \prime }}(-mE^{\prime \prime })=\overline{\mathfrak{a}^{m}},\nonumber\end{eqnarray}$$

where the first inclusion follows from Lemma 2.3, and where $\overline{\mathfrak{a}^{m}}$ is the integral closure of $\mathfrak{a}^{m}$ [Reference Lazarsfeld44, Remark 9.6.4]. Finally, given any ideal $\mathfrak{a}\subseteq {\mathcal{O}}_{V}$, there exists an integer $c$ such that $\overline{\mathfrak{a}^{\ell +1}}=\mathfrak{a}\cdot \overline{\mathfrak{a}^{\ell }}$ for all $\ell \geqslant c$ [Reference Lazarsfeld44, Proof of Proposition 9.6.6], hence $\overline{\mathfrak{a}^{m}}\subseteq \mathfrak{a}^{m-c}$ for all $m\geqslant c$.◻

Proof of $\Rightarrow$ in Theorem B.

Let $A$ be a very ample Cartier divisor. Then, for all $t$ such that $L-tA$ is ample, we have $\widehat{h}^{i}(X,L-tA)=0$ by Serre vanishing and by homogeneity and continuity; see Remark 4.4.◻

Proof. We first construct a sequence

$$\begin{eqnarray}k\subseteq k^{\prime }\subseteq K\end{eqnarray}$$

of two field extensions such that $X\times _{k}K$ is integral, where $k^{\prime }$ is uncountable and $K$ is $F$-finite in positive characteristic. If $k$ is already uncountable, then let $k^{\prime }=k$. Otherwise, consider a purely transcendental extension

$$\begin{eqnarray}k^{\prime }:=k(x_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in A},\end{eqnarray}$$

where $\{{x_{\unicode[STIX]{x1D6FC}}\}}_{\unicode[STIX]{x1D6FC}\in A}$ is an uncountable set of indeterminates; note that $k^{\prime }$ is uncountable by construction. To show that $X\times _{k}k^{\prime }$ is integral, let $\bigcup _{j}U_{j}$ be an affine open covering of $X$. Then, $X\times _{k}k^{\prime }$ is covered by affine open subsets that are localizations of the integral varieties $U_{j}\times _{k}\operatorname{Spec}(k[x_{\unicode[STIX]{x1D6FC}}]_{\unicode[STIX]{x1D6FC}\in A})$, which pairwise intersect, hence $X\times _{k}k^{\prime }$ is integral. We set $K=k^{\prime }$ in characteristic zero, and in positive characteristic, the gamma construction (Theorem A) shows that there is a field extension $k^{\prime }\subseteq K$ such that $K$ is $F$-finite and the scheme $X\times _{k}K$ is integral. Note that $K$ is uncountable since it contains the uncountable field $k^{\prime }$.

Now suppose $X$ is a projective variety over $k$, and let $L$ be an Cartier divisor satisfying ($\star$ ) for some pair $(A,\unicode[STIX]{x1D700})$. Let

$$\begin{eqnarray}\unicode[STIX]{x1D70B}:X\times _{k}K\longrightarrow X\end{eqnarray}$$

be the first projection map, which we note is faithfully flat by base change. Then, the pullback $\unicode[STIX]{x1D70B}^{\ast }A$ of $A$ is very ample, and to show that $L$ is ample, it suffices to show that $\unicode[STIX]{x1D70B}^{\ast }L$ is ample by flat base change and Serre’s criterion for ampleness. By the special case of Theorem B over the ground field $K$, it therefore suffices to show that $\unicode[STIX]{x1D70B}^{\ast }L$ satisfies ($\star$ ) for the pair $(\unicode[STIX]{x1D70B}^{\ast }A,\unicode[STIX]{x1D700})$.

We want to show that for every $i>0$ and for all $t\in [0,\unicode[STIX]{x1D700})$, we have

(8)$$\begin{eqnarray}\widehat{h}^{i}(X,L-tA)=\widehat{h}^{i}(X\times _{k}K,\unicode[STIX]{x1D70B}^{\ast }(L-tA))=0.\end{eqnarray}$$

For every $D\in \operatorname{Div}(X)$ and every $i\geqslant 0$, the number $h^{i}(X,{\mathcal{O}}_{X}(D))$ is invariant under ground field extensions by flat base change, hence $\widehat{h}^{i}(X,D)$ is also. By homogeneity and continuity (see Remark 4.4), the number $\widehat{h}^{i}(X,D)$ is also invariant under ground field extensions for $D\in \operatorname{Div}_{\mathbf{R}}(X)$, hence (8) holds.◻

Proof. Suppose $L$ is a Cartier divisor satisfying ($\star$ ) for a pair $(A,\unicode[STIX]{x1D700})$. Choose $\unicode[STIX]{x1D6FF}\in (0,\unicode[STIX]{x1D700})\cap \mathbf{Q}$ and let $m$ be a positive integer such that $m\unicode[STIX]{x1D6FF}$ is an integer. Then, the Cartier divisor $m(L-\unicode[STIX]{x1D6FF}A)$ is nef since

$$\begin{eqnarray}\displaystyle \widehat{h}^{i}(X,m(L-\unicode[STIX]{x1D6FF}A)-tA) & = & \displaystyle \widehat{h}^{i}(X,mL-(t+m\unicode[STIX]{x1D6FF})A)\nonumber\\ \displaystyle & = & \displaystyle m\cdot \widehat{h}^{i}\biggl(X,L-\biggl(\frac{t}{m}+\unicode[STIX]{x1D6FF}\biggr)A\biggr)=0\nonumber\end{eqnarray}$$

for all $t\in [0,m\unicode[STIX]{x1D700}-\unicode[STIX]{x1D6FF})$ by homogeneity (Proposition 4.2). Thus, the Cartier divisor $L=(L-\unicode[STIX]{x1D6FF}A)+\unicode[STIX]{x1D6FF}A$ is ample by [Reference Lazarsfeld43, Corollary 1.4.10].◻

Proof of $\Leftarrow$ in Theorem B.

We proceed by induction on $\dim X$. Suppose $\dim X=1$; we show the contrapositive. If $L$ is not nef, then $\deg L<0$ and $-L$ is ample. Thus, by asymptotic Serre duality (Proposition 4.5), we have $\widehat{h}^{1}(X,L)=\widehat{h}^{0}(X,-L)\neq 0$, hence ($\star$ ) does not hold for every choice of $(A,\unicode[STIX]{x1D700})$.

We now assume $\dim X\geqslant 2$. Suppose by way of contradiction that there is a non-nef Cartier divisor $L$ satisfying ($\star$ ). We first claim that there exists a finite morphism $\unicode[STIX]{x1D708}:\widetilde{X}\rightarrow X$ such that $\unicode[STIX]{x1D708}^{\ast }L$ satisfies ($\star$ ), and such that $\widetilde{X}$ satisfies the hypotheses of Proposition 4.6 for $D=\unicode[STIX]{x1D708}^{\ast }L$. Choose an integral curve $Z\subset X$ such that $L\cdot Z<0$, and let $\unicode[STIX]{x1D711}:X^{\prime }\rightarrow X$ be a regular alteration for the pair $(X,Z)$ [Reference de Jong38, Theorem 4.1], in which case $(\unicode[STIX]{x1D711}^{-1}(Z))_{\text{red}}$ is a simple normal crossing divisor. Consider the Stein factorization [Reference Hartshorne29, Corollary III.11.5]

(9)

for the morphism $\unicode[STIX]{x1D711}$, in which case $\widetilde{X}$ is a normal projective variety. Now let $\widetilde{Z}$ be the scheme-theoretic inverse image of $Z$ under $\unicode[STIX]{x1D708}$, and write

$$\begin{eqnarray}\widetilde{Z}=\mathop{\bigcup }_{\unicode[STIX]{x1D6FC}}\widetilde{Z}_{\unicode[STIX]{x1D6FC}},\end{eqnarray}$$

where $\widetilde{Z}_{\unicode[STIX]{x1D6FC}}$ are the irreducible components of $\widetilde{Z}$. Since $\unicode[STIX]{x1D708}$ is finite, every $\widetilde{Z}_{\unicode[STIX]{x1D6FC}}$ is one-dimensional and dominates $Z$, hence the projection formula and (5) imply $\unicode[STIX]{x1D708}^{\ast }L\cdot \widetilde{Z}_{\unicode[STIX]{x1D6FC}}<0$. Finally, $(\unicode[STIX]{x1D711}^{-1}(Z))_{\text{red}}=(\unicode[STIX]{x1D707}^{-1}(\widetilde{Z}))_{\text{red}}$ is a simple normal crossing divisor by the factorization (

).

We now show that $\unicode[STIX]{x1D708}^{\ast }L$ satisfies ($\star$ ). Since $\unicode[STIX]{x1D708}^{\ast }A$ is ample [Reference Lazarsfeld43, Proposition 1.2.13], we can choose a positive integer $a$ such that $a\,\unicode[STIX]{x1D708}^{\ast }A$ is very ample. Then, Proposition 4.3 implies

(10)$$\begin{eqnarray}\widehat{h}^{i}(\widetilde{X},\unicode[STIX]{x1D708}^{\ast }L-ta\,\unicode[STIX]{x1D708}^{\ast }A)=(\deg \unicode[STIX]{x1D708})\cdot \widehat{h}^{i}(X,L-taA)=0\end{eqnarray}$$

for all $i>0$ and for all $t\in [0,\unicode[STIX]{x1D700}/a)$. Replacing $A$ by $aA$, we assume that $\unicode[STIX]{x1D708}^{\ast }A$ is very ample.

For the rest of the proof, our goal is to show that

(11)$$\begin{eqnarray}\widehat{h}^{1}(\widetilde{X},\unicode[STIX]{x1D708}^{\ast }L-\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D708}^{\ast }A)\neq 0\end{eqnarray}$$

for $0<\unicode[STIX]{x1D6FF}\ll 1$, contradicting (10). Let $F\in |\unicode[STIX]{x1D708}^{\ast }A|$ be a very general divisor. By Bertini’s theorem [Reference Flenner, O’Carroll and Vogel20, Theorem 3.4.10 and Corollary 3.4.14], we may assume that $F$ is a subvariety of $\widetilde{X}$, in which case by inductive hypothesis and Lemma 4.12, we have that $\unicode[STIX]{x1D708}^{\ast }L|_{F}$ is ample. Since ampleness is an open condition in families [Reference Grothendieck and Dieudonné25, Corollaire 9.6.4], there exists an integer $b>0$ such that $b\,\unicode[STIX]{x1D708}^{\ast }L$ is very ample along the generic divisor $F_{\unicode[STIX]{x1D702}}\in |\unicode[STIX]{x1D708}^{\ast }A|$. By possibly replacing $b$ with a multiple, we may also assume that $mb\,\unicode[STIX]{x1D708}^{\ast }L|_{F_{\unicode[STIX]{x1D702}}}$ has vanishing higher cohomology for every integer $m>0$. Since the ground field $k$ is uncountable, we can then choose a sequence of very general Cartier divisors $\{E_{\unicode[STIX]{x1D6FD}}\}_{\unicode[STIX]{x1D6FD}=1}^{\infty }\subseteq |\unicode[STIX]{x1D708}^{\ast }A|$ such that the following properties hold:

1. (a) $E_{\unicode[STIX]{x1D6FD}}$ is a subvariety of $\widetilde{X}$ for all $\unicode[STIX]{x1D6FD}$ (by Bertini’s theorem [Reference Flenner, O’Carroll and Vogel20, Theorem 3.4.10 and Corollary 3.4.14]);

2. (b) for all $\unicode[STIX]{x1D6FD}$, $b\,\unicode[STIX]{x1D708}^{\ast }L|_{E_{\unicode[STIX]{x1D6FD}}}$ is very ample and $mb\,\unicode[STIX]{x1D708}^{\ast }L|_{E_{\unicode[STIX]{x1D6FD}}}$ has vanishing higher cohomology for every integer $m>0$ (by the constructibility of very ampleness in families [Reference Grothendieck and Dieudonné25, Proposition 9.6.3] and by semicontinuity); and

3. (c) for every positive integer $r$ and for all nonnegative integers $j$ and $m$, the $k$-dimension of cohomology groups of the form

   (12)$$\begin{eqnarray}H^{j}(E_{\unicode[STIX]{x1D6FD}_{1}}\cap E_{\unicode[STIX]{x1D6FD}_{2}}\cap \cdots \cap E_{\unicode[STIX]{x1D6FD}_{r}},{\mathcal{O}}_{E_{\unicode[STIX]{x1D6FD}_{1}}\cap E_{\unicode[STIX]{x1D6FD}_{2}}\cap \cdots \cap E_{\unicode[STIX]{x1D6FD}_{r}}}(mL))\end{eqnarray}$$
   is independent of the $r$-tuple $(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2},\ldots ,\unicode[STIX]{x1D6FD}_{r})$ (by semicontinuity; see [Reference Küronya41, Proposition 5.5]).

We denote by $h^{j}({\mathcal{O}}_{E_{1}\cap E_{2}\cap \cdots \cap E_{r}}(mL))$ the dimensions of the cohomology groups (12). By homogeneity (Proposition 4.2), we can replace $L$ by $bL$ so that $\unicode[STIX]{x1D708}^{\ast }L|_{E_{\unicode[STIX]{x1D6FD}}}$ is very ample with vanishing higher cohomology for all $\unicode[STIX]{x1D6FD}$.

To show (11), we now follow the proof in [Reference de Fernex, Küronya and Lazarsfeld12, pp. 453–454] with appropriate modifications. Given positive integers $m$ and $r$, consider the complex

$$\begin{eqnarray}\displaystyle K_{m,r}^{\bullet } & := & \displaystyle \biggl(\bigotimes _{\unicode[STIX]{x1D6FD}=1}^{r}({\mathcal{O}}_{\widetilde{X}}\longrightarrow {\mathcal{O}}_{E_{\unicode[STIX]{x1D6FD}}})\biggr)\otimes {\mathcal{O}}_{\widetilde{X}}(m\,\unicode[STIX]{x1D708}^{\ast }L)\nonumber\\ \displaystyle & = & \displaystyle \left\{\begin{array}{@{}lcl@{}}{\mathcal{O}}_{\widetilde{X}}(m\,\unicode[STIX]{x1D708}^{\ast }L)\, & \longrightarrow \, & \displaystyle \bigoplus _{\unicode[STIX]{x1D6FD}=1}^{r}{\mathcal{O}}_{E_{\unicode[STIX]{x1D6FD}}}(m\,\unicode[STIX]{x1D708}^{\ast }L)\\ \, & \longrightarrow \, & \displaystyle \bigoplus _{1\leqslant \unicode[STIX]{x1D6FD}_{1}<\unicode[STIX]{x1D6FD}_{2}\leqslant r}{\mathcal{O}}_{E_{\unicode[STIX]{x1D6FD}_{1}}\cap E_{\unicode[STIX]{x1D6FD}_{2}}}(m\,\unicode[STIX]{x1D708}^{\ast }L)\longrightarrow \cdots \end{array}\right\}.\nonumber\end{eqnarray}$$

By [Reference Küronya41, Corollary 4.2], this complex is acyclic away from ${\mathcal{O}}_{\widetilde{X}}(m\,\unicode[STIX]{x1D708}^{\ast }L)$, hence is a resolution for ${\mathcal{O}}_{\widetilde{X}}(m\,\unicode[STIX]{x1D708}^{\ast }L-r\,\unicode[STIX]{x1D708}^{\ast }A)$. In particular, we have

$$\begin{eqnarray}H^{j}(\widetilde{X},{\mathcal{O}}_{\widetilde{X}}(m\,\unicode[STIX]{x1D708}^{\ast }L-r\,\unicode[STIX]{x1D708}^{\ast }A))=\mathbf{H}^{j}(\widetilde{X},K_{m,r}^{\bullet }).\end{eqnarray}$$

The right-hand side is computed by an $E_{1}$-spectral sequence whose first page is

hence there is a natural inclusion

(13)$$\begin{eqnarray}\frac{\ker (u_{m,r})}{\operatorname{im}(v_{m,r})}\subseteq H^{1}(\widetilde{X},{\mathcal{O}}_{\widetilde{X}}(m\,\unicode[STIX]{x1D708}^{\ast }L-r\,\unicode[STIX]{x1D708}^{\ast }A)).\end{eqnarray}$$

We want to bound the left-hand side of (13) from below. First, there exists a constant $C_{1}>0$ such that $h^{0}({\mathcal{O}}_{E_{1}\cap E_{2}}(m\,\unicode[STIX]{x1D708}^{\ast }L))\leqslant C_{1}\cdot m^{n-2}$ for all $m\gg 0$ [Reference Lazarsfeld43, Example 1.2.20]. Thus, we have

$$\begin{eqnarray}\operatorname{codim}\biggl(\ker (u_{m,r})\subseteq \bigoplus _{\unicode[STIX]{x1D6FD}=1}^{r}H^{0}(E_{\unicode[STIX]{x1D6FD}},{\mathcal{O}}_{E_{\unicode[STIX]{x1D6FD}}}(m\,\unicode[STIX]{x1D708}^{\ast }L))\biggr)\leqslant C_{2}\cdot r^{2}m^{n-2}\end{eqnarray}$$

for some $C_{2}$ and for all $m\gg 0$. Now by Proposition 4.6, there are positive integers $q$ and $c$ such that $\mathfrak{b}(|mq\,\unicode[STIX]{x1D708}^{\ast }L|)\subseteq \mathfrak{a}^{m-c}$ for all $m>c$, where $\mathfrak{a}$ is the ideal sheaf of $\widetilde{Z}$. By replacing $L$ by $qL$, we can assume that this inclusion holds for $q=1$. The morphism $v_{m,r}$ therefore fits into the following commutative diagram:

We claim that there exists a constant $C_{3}>0$ such that for all $m\gg 0$,

(14)$$\begin{eqnarray}\operatorname{codim}(H^{0}({\mathcal{O}}_{E_{\unicode[STIX]{x1D6FD}}}(m\,\unicode[STIX]{x1D708}^{\ast }L)\otimes \mathfrak{a}^{m-c})\subseteq H^{0}({\mathcal{O}}_{E_{\unicode[STIX]{x1D6FD}}}(m\,\unicode[STIX]{x1D708}^{\ast }L)))\geqslant C_{3}\cdot m^{n-1}.\end{eqnarray}$$

Granted this, we have

$$\begin{eqnarray}\dim \biggl(\frac{\ker (u_{m,r})}{\operatorname{im}(v_{m,r})}\biggr)\geqslant C_{4}\cdot (rm^{n-1}-r^{2}m^{n-2})\end{eqnarray}$$

for some constant $C_{4}>0$ and for all $m\gg 0$. Fixing a rational number $0<\unicode[STIX]{x1D6FF}\ll 1$ and setting $r=m\unicode[STIX]{x1D6FF}$ for an integer $m>0$ such that $m\unicode[STIX]{x1D6FF}$ is an integer, we then see that there exists a constant $C_{5}>0$ such that

$$\begin{eqnarray}h^{1}(\widetilde{X},{\mathcal{O}}_{\widetilde{X}}(m(\unicode[STIX]{x1D708}^{\ast }L-\unicode[STIX]{x1D6FF}\,\unicode[STIX]{x1D708}^{\ast }A)))\geqslant C_{5}\cdot \unicode[STIX]{x1D6FF}m^{n}\end{eqnarray}$$

for all $m\gg 0$, contradicting (10).

It remains to show (14). Since the vanishing locus of $\mathfrak{a}$ may have no $k$-rational points, we pass to the algebraic closure of $k$ to bound the codimension on the left-hand side of (14) from below. Let $\overline{E}_{\unicode[STIX]{x1D6FD}}:=E_{\unicode[STIX]{x1D6FD}}\times _{k}\overline{k}$, and denote by $\unicode[STIX]{x1D70B}:\overline{E}_{\unicode[STIX]{x1D6FD}}\rightarrow E_{\unicode[STIX]{x1D6FD}}$ the projection morphism. Note that

$$\begin{eqnarray}\displaystyle & & \displaystyle \operatorname{codim}(H^{0}({\mathcal{O}}_{E_{\unicode[STIX]{x1D6FD}}}(m\,\unicode[STIX]{x1D708}^{\ast }L)\otimes \mathfrak{a}^{m-c})\subseteq H^{0}({\mathcal{O}}_{E_{\unicode[STIX]{x1D6FD}}}(m\,\unicode[STIX]{x1D708}^{\ast }L)))\nonumber\\ \displaystyle & & \displaystyle \quad =\operatorname{codim}(H^{0}({\mathcal{O}}_{\overline{E}_{\unicode[STIX]{x1D6FD}}}(m\,\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D708}^{\ast }L)\otimes \unicode[STIX]{x1D70B}^{-1}\mathfrak{a}^{m-c}\cdot {\mathcal{O}}_{\overline{E}_{\unicode[STIX]{x1D6FD}}})\subseteq H^{0}({\mathcal{O}}_{\overline{E}_{\unicode[STIX]{x1D6FD}}}(m\,\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D708}^{\ast }L)))\nonumber\end{eqnarray}$$

by the flatness of $k\subseteq \overline{k}$. Since ${\mathcal{O}}_{\overline{E}_{\unicode[STIX]{x1D6FD}}}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D708}^{\ast }L)$ is very ample by base change, we can choose a closed point $x\in Z(\unicode[STIX]{x1D70B}^{-1}\mathfrak{a}\cdot {\mathcal{O}}_{\overline{E}_{\unicode[STIX]{x1D6FD}}})\cap \overline{E}_{\unicode[STIX]{x1D6FD}}$, in which case $m\,\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D708}^{\ast }L$ separates $(m-c)$-jets at $x$ by [Reference Ito37, Proof of Lemma 3.7]. Finally, the dimension of the space of $(m-c)$-jets at $x$ is at least that for a regular point of a variety of dimension $n$, hence

$$\begin{eqnarray}\displaystyle & & \displaystyle \operatorname{codim}(H^{0}({\mathcal{O}}_{\overline{E}_{\unicode[STIX]{x1D6FD}}}(m\,\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D708}^{\ast }L)\otimes \unicode[STIX]{x1D70B}^{-1}\mathfrak{a}^{m-c}\cdot {\mathcal{O}}_{\overline{E}_{\unicode[STIX]{x1D6FD}}})\subseteq H^{0}({\mathcal{O}}_{\overline{E}_{\unicode[STIX]{x1D6FD}}}(m\,\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D708}^{\ast }L)))\nonumber\\ \displaystyle & & \displaystyle \quad \geqslant \operatorname{codim}(H^{0}({\mathcal{O}}_{\overline{E}_{\unicode[STIX]{x1D6FD}}}(m\,\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D708}^{\ast }L)\otimes \mathfrak{m}_{x}^{m-c})\subseteq H^{0}({\mathcal{O}}_{\overline{E}_{\unicode[STIX]{x1D6FD}}}(m\,\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D708}^{\ast }L)))\nonumber\\ \displaystyle & & \displaystyle \quad \geqslant \binom{m-c+n}{n-1}\geqslant C_{3}\cdot m^{n-1}\nonumber\end{eqnarray}$$

for some constant $C_{3}>0$ and all $m\gg 0$, as required.◻

Proof. Let $\{{A_{n}\}}_{n\geqslant 1}$ be a sequence of ample $\mathbf{R}$-Cartier divisors such that $\lim _{n\rightarrow \infty }\Vert A_{n}\Vert =0$ and such that $D+A_{n}$ is a $\mathbf{Q}$-Cartier divisor for every $n$. By [Reference Ein, Lazarsfeld, Mustaţă, Nakamaye and Popa15, Proposition 1.19], we have

By flat base change, we have $\unicode[STIX]{x1D70B}^{-1}(\mathbf{B}(D+A_{n}))=\mathbf{B}(\unicode[STIX]{x1D70B}^{\ast }(D+A_{n}))$, hence

where the second equality follows from applying [Reference Ein, Lazarsfeld, Mustaţă, Nakamaye and Popa15, Proposition 1.19] again to the sequence $\{\unicode[STIX]{x1D70B}^{\ast }{A_{n}\}}_{n\geqslant 1}$ of ample $\mathbf{R}$-Cartier divisors on $X\times _{k}k^{\prime }$.◻

Proof. We first prove that  for every pseudoeffective $\mathbf{R}$-Cartier divisor $D$, following [Reference Boucksom, Broustet and Pacienza5, Lemma 2.6]. Let , and let $v$ be a divisorial valuation such that $x\in c_{X}(v)$. By definition of , there exists an ample $\mathbf{R}$-Cartier divisor $A$ such that $D+A$ is $\mathbf{Q}$-Cartier divisor for which $x\notin \mathbf{B}(D+A)$. Thus, there exists an effective $\mathbf{Q}$-Cartier divisor $E$ such that $E{\sim}_{\mathbf{Q}}D+A$ and such that $x\notin \operatorname{Supp}E$. We therefore have $v_{\text{num}}(D+A)\leqslant v(E)=0$.