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

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 $(X, D)$.

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].

• Thank you @Takumi, it is going to be very useful in my future research! – Omprokash Das May 17 '19 at 20:39