Tie-Breaking Trick for Log Canonical Pairs and F-pure pairs in Positive Characteristic

Question

Let $X$ be a projective 3-fold in characteristic $p>0$. Let $(X, D)$ be a klt pair, and $D'$ a $\mathbb{R}$-Cartier divisor such that $D'=A'+B'$, where $A\geq 0$ is an ample $\mathbb{Q}$-divisor and $B\geq 0$ is a $\mathbb{R}$-divisor. Further assume that $(X, D+B')$ is klt and $(X, D+D')$ is log canonical but not klt. Let $Z\subset X$ be a minimal log canonical center of $(X, D+D')$.

Then show that there exists an effective $\mathbb{R}$-divisor $D''\sim_{\mathbb{R}}D'$ such that $(X, D+D'')$ is log canonical and $Z$ is the only log canonical center, and there is only one exceptional divisor $E$ over $X$ such that the discrepancy $a(E; X, D+D'')=-1$.

Observer that most of what I asked in the statement above is a part of the standard tie-breaking trick, except the part where I want the new $D''$ to be ($\mathbb{R}$-) linearly equivalent to $D'$. This can be arranged easily in characteristic 0 using Bertini's theorem for base-point free linear system, which fails in positive characteristic. Let me explain below how does the proof of the above statement work in char 0 so that I can pinpoint the difficulty in char p>0.

By the standard Tie-breaking trick, for example, see Proposition 8.7.1, page 152 of the book "Flips for 3-folds and 4-folds" by Allesio Corti, there exist an ample $\mathbb{Q}$-divisor $A''\sim_{\mathbb{Q}} A'$, and $0<\varepsilon, \eta\ll 1$ such that $(X, D+B'+(1-\varepsilon)A'+\eta A'')$ is log canonical with $Z\subset X$ as the only log canonical center and also has a unique log canonical place over $Z$. This much is true over all characterisitc, including char p>0, as long as the existence of resolution of singularities is known.

Now let $f:Y\to X$ be a log resolution of $(X, D+B'+(1-\varepsilon)A'+\eta A'')$. Then there is only one exceptional divisor, say $E_0$ with discrepancy $-1$. Since $\varepsilon>0$ and $\eta>0$ are sufficiently small we may assume that $0<1-\varepsilon+\eta<1$. Choose $0<\delta<1$ such that $1-\varepsilon+\eta+\delta=1$.

Now in char 0 by Bertini's theorem for the base-point free linear system $|f^*A'|_{\mathbb{Q}}$, there exists an ample $\mathbb{Q}$-divisor $H'\sim_{\mathbb{R}} A'$ such that $f^*H'$ has smooth support, $f^*H'=f^{-1}_*H'$, and that $f^*H'$ intersect the exceptional divisors of $f$ and the strict transfrom of other divisors tranversally. Then $(X, D+D'')$, where $D''=B+(1-\varepsilon)A'+\eta A''+\delta H'\sim_{\mathbb{R}} B+A'=D'$, is log canonical with $Z$ as the only log canonical center and has a unique log canonical place over $Z$.

So in char p>0, without the Bertini's theorem for base-point free linear system I don't know how to complete the argument. I am wondering if there is any $F$-singularity version of Tie-breaking which could help here. More specifically, I am wondering if the following statement is true:

Let $(X, D)$ be a simple normal crossing pair, i.e., $X$ is smooth and $D$ has simple normal crossing support. Let $B\geq 0$ be an effective Cartier divisor such that the linear system $|B|$ is base-point free. Then there exists a $0\leq B'\sim_{\mathbb{Q}} B$ such that a closed subset $W\subset X$ is a sharply $F$-pure center of $(X, D+B')$ if and only if it is a sharply $F$-pure center of $(X, D)$.

Answer 1

We claim the following holds.

Proposition (cf. [Tan17, Prop. 3 and Idea of Thm. 1; Wan, Proof of Thm. 3.5, Case 2]). Let $X$ be a complete normal variety $X$ of dimension $d$ over an infinite perfect field of characteristic $p > 0$. Let $\Delta$ be an effective $\mathbf{Q}$-Weil divisor on $X$ such that the Cartier index of $K_X+\Delta$ is not divisible by $p$, and such that the pair $(X,\Delta)$ is sharply $F$-pure. Let $B$ be a free Cartier divisor. Then, there exists an effective $\mathbf{Q}$-Cartier divisor $B' \sim_{\mathbf{Q}} B$ such that

1. the Cartier index of $K_X+\Delta+B'$ is not divisible by $p$;
2. $(X,\Delta+B')$ is sharply $F$-pure; and
3. the $F$-pure centers of $(X,\Delta)$ and those of $(X,\Delta+B')$ coincide.

The idea is to stratify $X$ into a disjoint union of minimal $F$-pure centers $W_i^k(X,\Delta)$. We can then apply the strategy in [Tan17] and [Wan] to show that we can find $B'$ such that $(W_i^k(X,\Delta),\Delta_{W_i^k(X,\Delta)}+B'\rvert_{W_i^k(X,\Delta)})$ is strongly $F$-regular for every $i$ and $k$, which implies the conditions (2) and (3) above by $F$-adjunction [Sch09]. I would like to make the disclaimer that the proof below is probably not optimal.

Proof. We first stratify the $F$-pure centers of $(X,\Delta)$. Let $W_i^*(X,\Delta)$ be the union of $F$-pure centers of $(X,\Delta)$ of dimension $\le i$, let $$W_i(X,\Delta) := W_i^*(X,\Delta) \smallsetminus W_{i-1}^*(X,\Delta),$$ and let $W_i(X,\Delta) = \bigcup_k W_i^k(X,\Delta)$ be a decomposition into irreducible components. Since the intersection of $F$-pure centers is a union of $F$-pure centers of smaller dimension [Sch10, Lem. 3.5 and Prop. 4.7], we see that $X$ is the disjoint union $$X = U \sqcup \bigsqcup_{i,k} W_i^k(X,\Delta),$$ where $(X,\Delta)$ is strongly $F$-regular on $U$. Moreover, denoting $$Y_i^k(X,\Delta) := X \smallsetminus \bigl(\overline{W_i^k(X,\Delta)} \smallsetminus W_i^k(X,\Delta)\bigr),$$ we see that $W_i^k(X,\Delta)$ is a minimal $F$-pure center of $(Y_i^k(X,\Delta),\Delta\rvert_{Y_i^k(X,\Delta)})$ for every $i$ and $k$. In particular, $W_i^k(X,\Delta)$ is normal for every $i$ and $k$ by [Sch10, Cor. 7.8].

Next, for each $W^k_i(X,\Delta)$, [Sch09, (i) and (iv) of Main Thm.] implies that there exists an effective $\mathbf{Q}$-Weil divisor $\Delta_{W^k_i(X,\Delta)}$ on $W^k_i(X,\Delta)$ such that $$(K_X+\Delta)\rvert_{W^k_i(X,\Delta)} = K_{W^k_i(X,\Delta)} + \Delta_{W^k_i(X,\Delta)},$$ and such that $(W^k_i(X,\Delta),\Delta_{W^k_i(X,\Delta)})$ is strongly $F$-regular.

We now prove it suffices to show the following:

Claim. There exists a nonempty open subset $V \subseteq \lvert B \rvert$ such that for all sufficiently large integers $e > 0$ for all $k$-rational $d$-tuples $$(D_1,D_2,\ldots,D_d) \in V^{\times_k d} \subseteq \lvert B \rvert^{\times_k d},$$ the pairs $$\biggl(U,\Delta\rvert_U+\frac{1}{p^e-1}(D_1+D_2+\cdots+D_d)\rvert_{U}\biggr)$$ and $$\biggl(W_i^k(X,\Delta),\Delta_{W_i^k(X,\Delta)}+\frac{1}{p^e-1}(D_1+D_2+\cdots+D_d)\rvert_{W_i^k(X,\Delta)}\biggr)$$ are strongly $F$-regular for every $i$ and $k$.

Assuming the Claim, since $\lvert B \rvert$ is free, we can find $$D_1,D_2,\ldots,D_{p^e-1} \in V$$ such that the intersection $\bigcap_{j \in J} D_j$ is empty for every subset $J \subseteq \{1,2,\ldots,p^e-1\}$ of cardinality $d+1$. For every point $x \in X$, there is an open neighborhood $T \subseteq X$ of $x$ such that \begin{align*} &\biggl(T \cap U,\Delta\rvert_{T \cap U}+\frac{1}{p^e-1}(D_1+D_2+\cdots+D_{p^e-1})\rvert_{T \cap U}\biggr)\\ &\qquad\qquad= \biggl(T \cap U,\Delta\rvert_{T \cap U}+\frac{1}{p^e-1}\sum_{i \in I} D_i\rvert_{T \cap U}\biggr) \end{align*} for a set of indicies $I$ with cardinality at most $d$ by the intersection condition on the $D_j$, and similarly \begin{align*} &\biggl(T \cap W_i^k(X,\Delta),\Delta_{W_i^k(X,\Delta)}\rvert_{T \cap W_i^k(X,\Delta)}+\frac{1}{p^e-1}(D_1+D_2+\cdots+D_{p^e-1})\rvert_{T \cap W_i^k(X,\Delta)}\biggr)\\ &\qquad\qquad=\biggl(T \cap W_i^k(X,\Delta),\Delta_{W_i^k(X,\Delta)}\rvert_{T \cap W_i^k(X,\Delta)}+\frac{1}{p^e-1}\sum_{i \in I_i^k} D_i\rvert_{T \cap W_i^k(X,\Delta)}\biggr) \end{align*} for sets of indices $I_i^k$ with cardinality at most $d$. These pairs are strongly $F$-regular by the choice of $V$ in the Claim. Finally, we note that setting $$B' = \frac{1}{p^e-1}(D_1+D_2+\cdots+D_{p^e-1}),$$ condition (1) holds by construction, and conditions (2) and (3) hold by [Sch09, (v) of Main Thm.].

It remains to show the Claim. Consider the morphisms $$W_i^k(X,\Delta) \times_k \lvert B \rvert^{\times_k d} \longrightarrow X \times_k \lvert B \rvert^{\times_k d} \overset{\mathrm{pr}_2}{\longrightarrow} \lvert B \rvert^{\times_k d}\tag{1}\label{eq:projectionmaps}$$ for every $i$ and $k$, where the first morphism is induced by the inclusion morphism $W_i^k(X,\Delta) \hookrightarrow X$. Choose an effective divisor $D' \in \lvert B \rvert$ containing no $F$-pure centers of $(X,\Delta)$ and an integer $e_0 > 0$ such that $$\biggl(U,\Delta\rvert_U + \frac{d}{p^e-1}D'\rvert_U\biggr)$$ and $$\biggl(W_i^k(X,\Delta),\Delta_{W_i^k(X,\Delta)} + \frac{d}{p^e-1}D'\rvert_{W_i^k(X,\Delta)}\biggr)$$ are strongly $F$-regular for all $i$ and $k$ and for every integer $e \ge e_0$. By applying [PSZ18, Cor. 4.19] to the morphisms in \eqref{eq:projectionmaps}, we see that there exists an open neighborhood $V \subseteq \lvert B \rvert$ of $D'$ such that the pairs $$\biggl(U,\Delta\rvert_U + \frac{1}{p^e-1}(D_1+D_2+\cdots+D_d)\rvert_U\biggr)$$ and $$\biggl(W_i^k(X,\Delta),\Delta_{W_i^k(X,\Delta)} + \frac{1}{p^e-1}(D_1+D_2+\cdots+D_d)\rvert_{W_i^k(X,\Delta)}\biggr)$$ are strongly $F$-regular for every $k$-rational $d$-tuple $$(D_1,D_2,\ldots,D_d) \in V^{\times_k d},$$ every $i$ and $k$, and every integer $e \ge e_0$. $\blacksquare$

Remark. I would imagine that one can run a similar argument with the weaker assumption that the ground field $k$ contains an infinite perfect field of characteristic $p > 0$, using ideas from [Tan17, Prop. 2] instead of [PSZ18, Cor. 4.19].

References

[PSZ18] Zsolt Patakfalvi, Karl Schwede, and Wenliang Zhang. "$F$-singularities in families." Algebr. Geom. 5 (2018), no. 3, 264–327. DOI: 10.14231/AG-2018-009. MR: 3800355.

[Sch09] Karl Schwede. "$F$-adjunction." Algebra Number Theory, 3 (2009), no. 8, 907–950. DOI: 10.2140/ant.2009.3.907. MR: 2587408.

[Sch10] Karl Schwede. "Centers of $F$-purity." Math. Z., 265 (2010), no. 3, 687–714. DOI: 10.1007/s00209-009-0536-5. MR: 2644316.

[Tan17] Hiromu Tanaka. "Semiample perturbations for log canonical varieties over an $F$-finite field containing an infinite perfect field." Internat. J. Math. 28 (2017), no. 5, 1750030, 13 pp. DOI: 10.1142/S0129167X17500306. MR: 3655076.

[Wan] Yuan Wang. "On the characterization of abelian varieties for log pairs in zero and positive characteristic." Jan. 22, 2018. arXiv:1610.05630v2 [math.AG].