globally $F$-regular type variety and vanishing theorem

Question

In [Smi00], Smith proved the following vanishing theorem.

Theorem. (cf. Corollary 5.4 in [Smi00]) Let $X$ be a globally $F$-regular type variety over $\mathbb{C}$, and $D,D'$ Cartier divisors on $X$. Suppose for any $n\gg 0$, $H^i(X,\mathcal{O}_X(nD+D'))=0$ holds for any $i\ge 1$. Then we have $H^i(X,\mathcal{O}_X(D))=0$ for any $i\ge 1$.

I don't understand the proof of this that how to use the semi-continuity. The Grauert's theorem combined with the flat base change theorem ensures us that if $A$ is a finitely generated $\mathbb{Z}$-algebra, and a model $(X_A,D_A,D'_A)$ of $(X,D,D')$, for any $n\gg 0$, there is a non-empty open subscheme $U_n\subseteq \mathrm{Spec}\,A$ such that for any closed point $s\in \mathrm{Spec}\,A$, $H^i(X_s,\mathcal{O}_s(nD+D'))=0$ for any $i\ge 1$. If we want to use the semi-continuity and Theorem 4.2 in [Smi00] to prove Corollary 5.4, for me, it is necessary to show that $U:=\bigcap_n U_n\subseteq \mathrm{Spec}\,A$ contains a closed point. But I don't know whether $U$ contains a closed point. How to show it? Or is there another approach to use the semi-continuity?

If we use the notion of globally ultra-$F$-regular variety, we may show the theorem without using the semi-continuity. Indeed, any globally $F$-regular variety is globally ultra-$F$-regualar by Theorem 3.5 in [Sch05], and thus Theorem 6.5 in [Sch05] shows the theorem. It is the only way of the proof of the theorem in my knowledge.

[Sch05] H. Schoutens: Log-terminal singularities and vanishing theorems via non-standard tight closure, J. Algebraic Geom. 14 (2005), no. 2, 357--390.

[Smi00] K. Smith: Globally F-regular varieties: applications to vanishing theorems for quotients of Fano varieties, Michigan Math. J. 48 (2000), 553--572.

Answer 1

I agree that the argument in [Smi00] seems incomplete for the reason you stated. This answer is more of a comment, and quite basic at that.

Consider the following example: let $E$ be an elliptic curve over $\mathbf{Q}$ and $L$ a line bundle of degree zero on $E$ which is non-torsion. Then $H^i(X, L^n)=0$ for all $i$ and all $n\neq 0$. However, for any model over $\mathbf{Z}[1/N]$ of $(E, L)$, upon reduction modulo $p$ (with $(p, N)=1$) the line bundle $L_p$ is torsion of some order $r_p$, and in particular $H^1(X_p, L_p^{r_p n})\neq 0$ for all $n$.

This shows that for every $n_0 > 0$, the intersection $\bigcap_{n>n_0} U_n$ consists only of the generic point.

However, of course $E$ is not of $F$-regular type, so this is not a counterexample to your statement. It only shows that more care is needed.