Proof. This follows from the fact that a splitting of structure sheaves for a finite map of normal varieties may be checked on an open subset with complement of codimension at least two. In positive characteristic, this lemma has been observed in [Reference Schwede and SmithSS10, Proof of Proposition 6.3]. In mixed characteristic, the proof is analogous to [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMPSTWW20, Proposition 6.18].

Proof. In positive characteristic, see [Reference Ma, Schwede, Tucker, Waldron and WitaszekMSTWW22, Theorem A] or the proof of [Reference DasDas15, Theorem A]. In mixed characteristic, this is [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMPSTWW20, Corollary 7.9].

Proof. In positive characteristic, this follows by the same proof as [Reference Hacon and WitaszekHW19, Lemma 2.10]. It uses the equality between the different and the F-different. For this, note that [Reference DasDas15, Theorem B] assumes that the base field is algebraically closed, but the proof goes through for every F-finite base field.

In mixed characteristic, this is [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMPSTWW20, Corollary 7.5] (after completing or invoking [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMPSTWW20, Corollary 7.6]).

Finally, the normality of S around positive characteristic closed points follows from Proposition 2.8, and around characteristic 0 closed points by standard results ([Reference Kollár and MoriKM98, Theorem 5.50]).

Proof. This follows easily from the following commutative diagram, which exists as the different is equal to the F-different

Here, we use that $(X,S+B)$ is $\mathbb {Q}$ -factorial, strongly F-regular and of dimension at least three to guarantee that divisorial sheaves on X are $S_3$ (cf. [Reference Hacon and WitaszekHW19, Proposition 2.7]), which implies that the upper horizontal arrow is a surjection of sheaves (cf. [Reference Hacon and WitaszekHW19, Lemma 2.4]). This is turn gives the surjection on global sections for $e \gg 0$ by Serre’s vanishing.

Proof. The proof is the same as in [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMPSTWW20, Theorem 7.2]. We recall it for the convenience of the reader. We refer to [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMPSTWW20, Subsection 2.1] for a primer on Matlis duality.

Set $\mathcal {N} = {\mathcal {O}}_{X^+}(N)$ for $N = \pi ^*(K_X+S+B-L)$ , where $\pi \colon X^+ \to X$ is the natural map. Note that $\mathcal {N}$ is a line bundle as every ${\mathbb {Q}}$ -Cartier ${\mathbb {Q}}$ -divisor is rendered Cartier by some finite cover. Let $S^+$ be the chosen prime divisor on $X^+$ such that $\pi (S^+)=S$ , and consider the following diagram wherein the left square exists as B is effective:

Here, the right vertical dashed arrow exists by an easy diagram chase. Moreover, this arrow factorises through the $S_2$ -fification ${\mathcal {O}}_{S}(K_{S}-L|_S)$ of the upper term as $S^+$ is normal, and so ${\mathcal {O}}_{S^+} \otimes \mathcal {N}$ is $S_2$ . Now, apply local cohomology with $d = \dim X$ to get:

Here, the upper left vertical arrow is surjective as the cokernel of

$$\begin{align*}{\mathcal{O}}_X(K_X+S-L)\, /\, {\mathcal{O}}_X(K_X-L) \to {\mathcal{O}}_{S}(K_{S}-L|_S) \end{align*}$$

has codimension at least $1$ in S, and so the top local cohomology ( $H^{d-1} R\Gamma _{\mathfrak {m}} R\Gamma $ ) thereof vanishes. The bottom horizontal arrow is injective as

$$\begin{align*}H^{d-1} R\Gamma_{\mathfrak{m}} R\Gamma(X^+, {\mathcal{O}}_{X^+}(N)) = 0\\[-9pt] \end{align*}$$

by [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMPSTWW20, Corollary 3.7].

The upshot is that the image of $(*)$ injects into the image of $(**)$ . Note that

$$\begin{align*}{\mathcal{O}}_{S^+} \otimes \mathcal{N} \simeq {\mathcal{O}}_{S^+}((\pi|_{S^+})^*(K_S+B_S - L|_S))\\[-9pt] \end{align*}$$

by definition of the different (cf. the second part of [Reference Ma, Schwede, Tucker, Waldron and WitaszekMSTWW22, Subsection 2.1]), and so by taking Matlis duality we get that

$$\begin{align*}B^0_S(X,S+B;L)\to B^0(S,B_S; L|_S)\\[-9pt] \end{align*}$$

is surjective. Here, the left-hand side is Matlis dual to the image of $(**)$ by [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMPSTWW20, Definition 4.21 and Lemma 4.24], while the right-hand side is Matlis dual to the image of $(*)$ by [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMPSTWW20, Lemma 4.8].

Proof. The three-dimensional case of Theorem 2.12 is an immediate consequence of the proof of [Reference FujinoFuj07, Theorem 4.2.1] and the two-dimensional MMP [Reference TanakaTan18].

Proof. Since divisorial contractions decrease the Picard rank, we may assume that the above sequence consists only of flips. Denote the flips by $\phi _i \colon X_i \dashrightarrow X_{i+1}$ , and denote the flipping and the flipped contractions by $f_i \colon X_i \to Z_i$ and $f_i^+ \colon X_{i+1} \to Z_i$ , respectively. Let S be an irreducible component of $\lfloor B \rfloor $ , let $S_i$ be its strict transform on $X_i$ and let $T_i := f_i(S_i)$ . Note that the normalisation of $S_i$ need not be ${\mathbb {Q}}$ -factorial.

By [Reference FujinoFuj07, Proposition 4.2.14], we may assume that $f^+_i|_{S_{i+1}} \colon S_{i+1} \to T_i$ is small. Moreover, if $f_i|_{S_i} \colon S_i \to T_i$ is divisorial, then the Picard rank as defined in [Reference Alexeev, Hacon and KawamataAHK07, Lemma 1.6] satisfies $\rho (S^{\nu }_i/U)>\rho (S^{\nu }_{i+1}/U)$ , where $S^{\nu }_i$ and $S^{\nu }_{i+1}$ are appropriate normalisations. Thus, again we may assume $f_i|_{S_i}$ is small, that is, $\phi _i|_{S_i} \colon S_i \dashrightarrow S_{i+1}$ is of type (SS). The sum of the dimensions of the flipping and the flipped locus of is at least three, so one of them must be two-dimensional (in fact, they must be of type $(2,1)$ , $(2,2)$ or $(1,2)$ [Reference Kawamata, Matsuda and MatsukiKMM87, Lemma 5.1.17] and [Reference Alexeev, Hacon and KawamataAHK07, Lemma 1.2]), and so they cannot be both contained in the non-klt locus.

Now, assume that $\dim U=1$ , $\kappa (K_X+B/U)\geq 0$ , and $\lfloor B \rfloor $ contains the support of the fibre $X_u$ for some point $u\in U$ . As observed above, after finitely many steps, $\lfloor B_i\rfloor $ contains no two-dimensional component of the flipping or flipped loci. Suppose that the flipping locus (and thus also the flipped locus) is contained in the fibre over u and hence in the non-klt locus. Since either the flipping or flipped locus is two dimensional, this can not happen infinitely often. Therefore, we may assume that for $i\gg 0$ the flipping and flipped loci dominate U.

Therefore, we can replace $(X,B)$ and $(X_i,B_i)$ by $(X_{\eta }, B_{\eta })$ and $((X_i)_{\eta }, (B_i)_{\eta })$ , respectively, so that these are three-dimensional ${\mathbb {Q}}$ -factorial pairs defined over a field of characteristic 0 or an F-finite field $k(\eta )$ for the generic point $\eta \in U$ . In particular, we can assume that log resolutions and terminalisations exist (see [Reference Das and WaldronDW19b] when $\eta $ is of positive characteristic and [Reference Birkar, Cascini, Hacon and McKernanBCHM10] when it is of characteristic 0).

First, we will show that $\lfloor B_i \rfloor $ is disjoint from the flipping, and so the flipped, locus. By what we have proven above, we may also assume that the intersection of the non-klt locus with the flipping and the flipped locus is at most zero dimensional. Now, suppose that a flipping curve C intersects $\lfloor B_i\rfloor $ . Then $C \cdot \lfloor B_i \rfloor> 0$ , and so the flipped locus is contained in $\lfloor B_{i+1} \rfloor $ which is impossible. Now, replacing $B_i$ by $\{B_i\}$ , we may assume that $(X_i,B_i)$ are klt, and so the MMP should terminate by Proposition 2.15 for $U = \operatorname {\mathrm {Spec}} k(\eta )$ .

However, the above replacement may render the assumption $\kappa (K_X+B) \geq 0$ invalid, so we need to be more careful. Precisely, write $K_X+B\sim _{{\mathbb {Q}}}M\geq 0$ , where the supports of the strict transforms $M_i$ on $X_i$ contain the flipping loci for all i. Since the flipping loci are disjoint from $\lfloor B_i \rfloor $ , the above $(K_X+B)$ -MMP (equivalently, a $(K_X+B+\epsilon M)$ -MMP), is also a $(K_X+\{B\} + \epsilon M)$ -MMP for $0 < \epsilon \ll 1$ . Since the flipping loci are contained in ${\mathrm {Supp}}(\{B\}+ \epsilon M)$ , this MMP terminates by Proposition 2.15.

Proof. This follows by the same argument as [Reference Alexeev, Hacon and KawamataAHK07, Theorem 2.15]. For the convenience of the reader, we recall the argument briefly. Define the following functions $w^{-}_{\alpha }, w^{+}_{\alpha }, W^{-}_{\alpha }, W^+_{\alpha } \colon (-\infty , 1) \to \mathbb {R}_{\geq 0}$ for $\alpha \in (0,1)$ by

$$ \begin{align*} w_{\alpha}^-(x) &= 1-x \text{ for } x \leq \alpha, \text{ and } w_{\alpha}^-(x) =0 \text{ otherwise, }\\ w_{\alpha}^+(x) &= 1-x \text{ for } x < \alpha, \text{ and } w_{\alpha}^+(x) =0 \text{ otherwise, }\\ W_{\alpha}^-(b) &= {\textstyle \sum_{k=1}^{\infty}} w_{\alpha}^-(k(1-b)), \\ W_{\alpha}^+(b) &= {\textstyle \sum_{k=1}^{\infty}} w_{\alpha}^+(k(1-b)). \end{align*} $$

For any subterminal log pair $(Y,B_Y)$ and $B_Y = \sum b_i B_i$ with $B_i$ irreducible, we define the difficulty ([Reference Alexeev, Hacon and KawamataAHK07, Definition 2.3])

$$\begin{align*}d^+_{\alpha}(Y,B_Y) = \sum_{b_i \leq 0} W^{+}_{\alpha}(b_i) \rho(\tilde B_i) + \sum_{v} w_{\alpha}^+(a_v) - \sum_{\tilde C_{i,j}} W^{+}_{\alpha}(b_i), \text{ where}\end{align*}$$

• • $\rho $ is as in [Reference Alexeev, Hacon and KawamataAHK07, Lemma 1.6] (in particular, stable under small birational maps),

• • v is taken over all valuations which are not echoes on Y (here, $a_v$ denotes the discrepancy of $(Y,B_Y\hspace{-1pt})$ at v) and

• • $\tilde C_{i,j}$ are codimension one irreducible components in $\tilde B_i$ of the inverse image of a codimension two integral subscheme $C \subseteq Y$ under the normalisation $\bigsqcup \tilde B_i \to \bigcup B_i$ . Here, we exclude those C which are centres of echoes.

We say that an irreducible $C \subseteq Y$ is a centre of an echo if it is of codimension two, contained in exactly one $B_i$ , but not in $\mathrm {Sing}\, B_i$ . Here, a valuation is called a (k-th) echo for such C if it corresponds to the last exceptional divisor of a sequence of k blowups at strict transforms of C (see [Reference Alexeev, Hacon and KawamataAHK07, Example 1.4]). The discrepancy of the k-th echo is $k(1-b_i)$ . The difficulty is well defined and finite because there are only finitely many valuations with discrepancy in $(-1,0]$ and also finitely many in $(0,1)$ if we exclude echoes ([Reference Alexeev, Hacon and KawamataAHK07, Lemma 1.5]; here, we use log resolutions).

We can similarly define $d^-_{\alpha }$ . The difficulties are stable under log pullbacks ([Reference Alexeev, Hacon and KawamataAHK07, Lemma 2.7]), and so we extend these definitions to arbitrary ${\mathbb {Q}}$ -factorial klt pairs by taking the difficulty of a terminalisation ([Reference Alexeev, Hacon and KawamataAHK07, Definition 2.8]). The difficulties are nonnegative for klt pairs ([Reference Alexeev, Hacon and KawamataAHK07, Lemma 2.9]) and decreasing for a sequence of flips $(X_n,B_n) \dashrightarrow (X_{n+1},B_{n+1})$ if we start with $n \gg 0$ ([Reference Alexeev, Hacon and KawamataAHK07, Theorem 2.12]).

We are ready to give the proof of the proposition. First, we show that there are only finitely many flips with flipping or flipped loci admitting a component C of codimension two and contained in $\mathrm {Sing}\, X_n$ (this is [Reference Alexeev, Hacon and KawamataAHK07, Lemma 2.14] with the use of Bertini replaced by localisation at C). By taking $n\gg 0$ , we can assume that the places with discrepancies smaller or equal to 0 are stable under flips: Let $E_1, \ldots , E_m$ be such places over $X_n$ with discrepancies $a_1, \ldots , a_m \leq 0$ . In particular, the minimal discrepancy of the two-dimensional singularity obtained by localisation at C must be equal to $a_i$ for some i. It is known by experts that minimal log discrepancies (mlds) for klt surface singularities satisfy the ascending chain condition, and since flips increase discrepancies, the statement follows. As this result is unpublished, we refer to Lemma 2.16 instead.

We are left to show that there are only finitely many flips with flipping or flipped loci admitting a component C of codimension two and contained in ${\mathrm {Supp}}\, B_n$ but not $\mathrm {Sing}\, X_n$ . The blowup along C produces a divisor with discrepancy $1 - \sum m_i b_i$ for $m_i \in \mathbb {Z}_{\geq 0}$ . There are only finitely many such discrepancies, and, thus, it is enough to prove the following claim: Given $\alpha \in (-1,1)$ , there cannot be infinitely many flips for which there exists a valuation v with $a_v = \alpha $ and centre contained in the flipping or the flipped locus ([Reference Alexeev, Hacon and KawamataAHK07, Theorem 2.13]). The case $\alpha \in (-1,0]$ follows by finiteness, while for $\alpha \in (0,1)$ we notice that $d^-_{\alpha }$ drops by at least $1-\alpha $ if the flipping locus admits such a valuation and so does $d^+_{\alpha }$ for the flipped locus. Since the difficulties are positive and finite, this can only happen finitely many times.

The last assertions follows by replacing B by $B + \epsilon M$ for $0 \leq M \sim _{{\mathbb {Q}}, U} K_X + B$ and $0 < \epsilon \ll 1$ since then every flipping curve $\Sigma $ satisfies $\Sigma \cdot M < 0$ , and so is contained in M.

Proof. Let $f \colon S'\to S$ be the minimal resolution at a point $s \in S$ with k exceptional curves $C_i$ (which are $k(s)$ -schemes), and let $\Gamma $ be the corresponding graph. Write $K_{S'} + B_{S'} = f^*(K_S+B_S)$ , and let $a_i$ be the log discrepancy of $C_i$ in $(S',B_{S'})$ . We will use the results of [Reference KollárKol13]. Set $r_i:=\dim _{k(s)}H^0(C_i, \mathcal O _{C_i})$ . Then $C_i^2=-r_ic_i$ for $c_i \in \mathbb {Z}$ . Note that $C_i$ are conics over the field $H^0(C_i, \mathcal {O}_{C_i})$ and in particular $\deg \omega _{C_i}=-2r_i$ ([Reference KollárKol13, Reduction 3.30.2]). By adjunction,

$$\begin{align*}a_iC^2_i = (K_{S'} + B_{S'} + a_iC_i) \cdot C_i \geq -2r_i, \end{align*}$$

and so $-c_i \leq \frac {2}{a_i}$ is bounded. By classification ([Reference KollárKol13, 3.31 and 3.41]), $r_i \leq 4$ , and so $-C_i^2$ is bounded. Further, $C_i \cdot C_j$ is bounded as well for every $i\neq j$ (see [Reference KollárKol13, 3.41]).

Recall that $\Gamma $ is a tree with at most one fork and three legs ([Reference KollárKol13, 3.31]), there are no $K_X$ -negative curves and the convexity of discrepancies holds ([Reference KollárKol13, Proposition 2.37]), that is, given three consecutive curves $C_1$ , $C_2$ and $C_3$ on a leg, we have $2a_2\leq a_1+a_3$ . In fact, it follows easily from the same proof that if $2a_2= a_1+a_3$ , then $c_2=-2$ , $f^{-1}_*B\cdot C_2=0$ and $C_2 \cdot C_1 = C_2 \cdot C_3 = r_2$ .

By the negativity lemma, the k equations

$$\begin{align*}C^2_i = -2r_i + B_{S'}\cdot C_i \end{align*}$$

are linearly independent and uniquely determine the discrepancies. Given a maximal sequence of successive curves $C_1,\ldots , C_r$ on one leg with the same discrepancies $a_i$ and parameters $r_i$ , we remove the equations corresponding to $C_2, \ldots , C_{r-1}$ and replace all occurrences of $a_2,\ldots , a_r$ by $a_1$ . This operation does not change the set of solutions (indeed, this is equivalent to adding equations $a_1=a_2=\ldots =a_r$ which does not change the set of solutions but renders the equations corresponding to $C_2, \ldots , C_{r-1}$ trivial), and so, eventually, given the shape of $\Gamma $ ([Reference KollárKol13, 3.31]), we are left with bounded-in-m number of equations with bounded coefficients and in the same or lower number of variables. Therefore, the set of discrepancies is bounded in terms of m as well.

Proof. Let $\tilde {C}$ be the normalisation of C, and write $K_{\tilde C} + B_{\tilde C} = (K_S+C+B)|_{\tilde C}$ . Then $(\tilde C, B_{\tilde C})$ is klt with standard coefficients (see Lemma 2.2). Since $\tilde C \to f(C)$ is birational, $(\tilde C, B_{\tilde C})$ is relatively globally F-regular ([Reference Das and WaldronDW19a, Theorem 5.1], cf. [Reference Hacon and XuHX15, Theorem 3.1], [Reference Hacon and WitaszekHW22, Proposition 2.9]) or completely relatively globally $+$ -regular ([Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMPSTWW20, Theorem 7.14]), respectively. Lemma 2.9 implies that $(S,C+B)$ is relatively purely F-regular (resp. completely relatively purely $+$ -regular) over a neighbourhood of $f(C)$ .

Proof. The statement is local, and hence we may assume that Z is affine and we work in a neighbourhood of some closed point $z\in Z$ of positive residue characteristic in the image of the flipping locus. Replacing $\lfloor \Delta \rfloor -S-A$ by $(1-\frac 1m)(\lfloor \Delta \rfloor -S-A)$ for some $m\gg 0$ , we may assume that $\lfloor \Delta \rfloor =S+A$ . Let $K_{S^{\nu }}+A_{S^{\nu }}+B_{S^{\nu }}=(K_X+S+A+B)|_{S^{\nu }}$ with $A_{S^{\nu }} = A|_{S^{\nu }}$ and $S^{\nu }$ being the normalisation of S. Then $(S^{\nu },A_{S^{\nu }}+B_{S^{\nu }})$ is a plt threefold with standard coefficients. Since $A_{S^{\nu }}$ is relatively ample, it is not exceptional, and so it is birational over its image in Z. As $-(K_{S^{\nu }}+A_{S^{\nu }}+B_{S^{\nu }})$ is relatively ample, Lemma 4.2 implies that $(S^{\nu },A_{S^{\nu }}+B_{S^{\nu }})$ is purely F-regular (resp. completely purely $+$ -regular) over a neighbourhood of $f(A_{S^{\nu }})$ . By Lemma 2.9, $(X,S+(1-\epsilon )A+B)$ is purely F-regular (resp. completely purely $+$ -regular) over a neighbourhood of $f(A_S)$ for any $\epsilon>0$ . Since every flipping curve is contained in S and intersects A, $(X,S+(1-\epsilon )A+B)$ is purely F-regular (resp. completely purely $+$ -regular) over a neighbourhood of the image of the flipping locus. In particular, S is normal (Lemma 2.9).

By Shokurov’s reduction to pl-flips (see, e.g., [Reference CortiCor07, Lemma 2.3.6]), it suffices to show that the restricted algebra

$$\begin{align*}R_S(K_X+S+A+B):=\textrm{Im}\left( R(K_X+S+A+B)\to R(K_S+A_S+B_S)\right)\end{align*}$$

is finitely generated. By Proposition 3.1 in fact $R_S(K_X+S+A+B)=R(K_S+A_S+B_S)$ (in divisible enough degrees). Since $(S,(1-\epsilon )A_S+B_S)$ is a klt threefold for $0 < \epsilon \leq 1$ , the ring $R(K_S+A_S+B_S)$ is finitely generated (see [Reference Das and WaldronDW19b, Theorem 1.4 and 1.6] when S is purely positive characteristic and [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMPSTWW20, Theorem F and G] when S is of mixed characteristic).

If $\phi \colon X \dashrightarrow X^+$ is the corresponding flip, then since the flipping contraction $f \colon X \to Z$ is relatively F-regular (resp. completely relatively $+$ -regular) over a neighbourhood of the image of the flipping locus and $\phi $ is an isomorphism in codimension two, the flipped contraction $f^+ \colon X^+ \to Z$ is also relatively F-regular (resp. completely relatively $+$ -regular); see Lemma 2.7.

Proof. Perturbing the coefficients, we may assume that $(X,S+B)$ is plt. We may pick an ample over $U \ {\mathbb {Q}}$ -divisor H such that $L=K_X+S+B+H$ is nef over U, and $L^{\perp }$ is spanned by $\Sigma $ . Let A be an ample ${\mathbb {Q}}$ -divisor such that $(S+A)\cdot \Sigma =0$ , then for any $0<\epsilon \ll 1$ we may pick an ample ${\mathbb {Q}}$ -divisor $H_{\epsilon } \sim _{\mathbb {Q}} H+ \epsilon (S+A)$ such that $L_{\epsilon } =K_X+S+B+H_{\epsilon }$ is nef over U, $(L_{\epsilon })^{\perp }=\mathbb R [\Sigma ]$ . Let $\mathbb E(L_{\epsilon })$ be the exceptional locus of the nef line bundle $L_{\epsilon }$ given by the union of the integral subvarieties $V\subset X$ such that $(L_{\epsilon }|_V)$ is not relatively big (cf. [Reference Keelkee99, Definition 0.1] and [Reference WitaszekWit22, Lemma 2.2]). Then $\mathbb E(L_{\epsilon })\subset S$ . To verify the last inclusion, notice that if $V\subset X$ is a subvariety not contained in S, then $L_{\epsilon } |_V=(L+\epsilon (S+A))|_V$ is nef and big over U, and so $ L_{\epsilon }^{\dim V}\cdot V>0$ . Replacing L by $L_{\epsilon }$ , we may assume that $\mathbb E(L)\subset S$ .

By adjunction, $(S^{\nu },B_{S^{\nu }})$ is klt, where $K_{S^{\nu }}+B_{S^{\nu }} = (K_X+S+B)|_{S^{\nu }}$ , the map $S^{\nu } \to S$ is the normalisation, and $H_{S^{\nu }} = H|_{S^{\nu }}$ . By [Reference Hashizume, Nakamura and TanakaHNT20, Theorem 1.4(1)] and [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMPSTWW20, Theorem G], $L|_{S^{\nu }}$ is semiample over U (note that in the mixed characteristic case, by [Reference Hashizume, Nakamura and TanakaHNT20, Theorem 1.4(1)], we may assume that $\dim \pi (S) \geq 1$ , and so [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMPSTWW20, Theorem G] applies). Since S is normal up to universal morphism, $L|_S$ is semiample by [Reference Keelkee99, Lemma 1.4] and Theorem 2.3. Since $\mathbb {E}(L) \subseteq S$ , [Reference Keelkee99, Theorem 0.2] and [Reference WitaszekWit22, Theorem 1.2] imply that L is semiample over U, and thus it induces a projective morphism $f\colon X\to Z$ contracting $\Sigma $ and such that $f_*{\mathcal {O}} _X={\mathcal {O}} _Z$ and $\rho (X/Z)=1$ .

Proof. If $K_Y+\Delta $ is $\pi $ -nef, then we are done. Otherwise, we run a $(K_Y+\Delta )$ -MMP over X. Let be a sequence of flips and contractions, and let $\Delta _l$ be the strict transforms of $\Delta $ on $Y_l$ . Since X is ${\mathbb {Q}}$ -factorial, there exists an effective exceptional divisor E on $Y_l$ such that $-E$ is ample over X. In particular, $\lfloor \Delta _l\rfloor $ contains all $\pi _l\colon Y_l\to X$ exceptional curves. We must show that we can continue the MMP. The termination will then follow by Theorem 2.14 as both the flipping and flipped locus must be contained in $\mathrm {Ex}(\pi )\subseteq \lfloor \Delta \rfloor $ . Note that if $K_{Y_l}+\Delta _l $ is nef over X, then $Y_l$ is the required minimal model. Therefore, we may assume that $K_{Y_l}+\Delta _l $ is not nef over X.

We start by establishing the cone theorem. Let $S_i$ be all the $\pi _l$ -exceptional divisors for $1 \leq i \leq r$ . Restricting to the normalisation $S_i^{\nu }$ of $S_i$ , we get that $(S_i^{\nu }, \Delta _{S_i^{\nu }})$ is dlt, where $K_{S_i^{\nu }}+\Delta _{S_i^{\nu }}=(K_{Y_l}+\Delta _l)| _{S_i^{\nu }}$ . The three-dimensional cone theorem ([Reference Das and WaldronDW19b, Theorem 1.1] and [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMPSTWW20, Theorem H]) states that

$$\begin{align*}\mathrm{NE}(S_i^{\nu}/X)=\mathrm{NE}(S_i^{\nu}/X)_{K_{S_i^{\nu}}+\Delta_{S_i^{\nu}} \geq 0}+\sum _{j\geq 1}\mathbb R _{\geq 0}[\overline{\Gamma}_{i,j}],\end{align*}$$

where $0> (K_{S_i^{\nu }}+\Delta _{S_i^{\nu }})\cdot \overline {\Gamma }_{i,j}= (K_{Y_l}+\Delta _l)\cdot \Gamma _{i,j}.$ Here $\{\overline {\Gamma }_{i,j}\}_{j\geq 1}$ is a countable collection of curves on $S_i^{\nu }$ and $\Gamma _{i,j}$ denote their images on $S_i \subseteq Y_l$ . Since $\textrm {Ex }(\pi _l ) \subset \textrm {Supp }(E)$ , there is a surjection $\sum _{i=1}^r \mathrm {NE}(S_i^{\nu }/X)\to \mathrm {NE}(Y_l/X)$ and hence

$$\begin{align*}\mathrm{NE}(Y_l/X)=\mathrm{NE}(Y_l/X)_{K_{Y_l}+\Delta_l \geq 0}+\sum _{1 \leq i \leq r, 1 \leq j}\mathbb R _{\geq 0}[\Gamma_{i,j}]. \end{align*}$$

Note that for any ample ${\mathbb {Q}}$ -divisor $H_l$ on $Y_l$ , the set $\{ \Gamma _{i,j} \mid (K_{Y_l}+\Delta _l+H_l)\cdot \Gamma _{i,j}\leq 0\}$ is finite, and so extremal rays do not accumulate in $\mathrm {NE}(Y_l/X)_{K_{Y_l}+\Delta _l < 0}$ .

Now, pick an extremal curve $C = \Gamma _{i,j}$ for some $i,j$ . Then, there is an effective $\pi _l$ -exceptional divisor S such that $S \cdot C < 0$ , and hence a contraction $f \colon Y_l \to Z$ of C exists by Proposition 4.5 applied to $(Y_l,\Delta _l)$ . If f is a divisorial contraction, we let $Y_{l+1}=Z$ . If f is a flipping contraction, then by [Reference Hacon and WitaszekHW21, Lemma 3.1] (this lemma is stated over a field but it holds in much wider generality such as over DVRs), there exists an f-ample irreducible divisor A which is $\pi $ -exceptional and hence contained in $\lfloor \Delta _l \rfloor $ . By Theorem 4.3, the flip $f^+\colon Y^+\to Z$ exists. We let $Y_{l+1}=Y^+$ , , and $K_{Y_{l+1}}+\Delta _{l+1}=\psi _*(K_{Y_{l}}+\Delta _{l})$ .

Proof of Theorem 4.1

Note that the proof of Theorem 4.6 implies the existence of all relevant divisorial and flipping contractions and the termination of the relevant flips. It suffices therefore to show the existence of flips. Suppose that $f\colon Y\to Z$ is a flipping contraction over X. We must show that the corresponding flip $Y^+\to Z$ exists. Let $\zeta (\Delta )$ be the number of components of $\Delta $ whose coefficient is not contained in the standard set $\{ 1-\frac 1 m \mid m\in \mathbb N \}\cup \{ 1\}$ . We will prove the result by induction on $\zeta (\Delta )$ .

Note that if $\zeta (\Delta )=0$ , then by Theorem 4.6 the required flip exists. Therefore, we assume that $\zeta (\Delta )>0$ , and so we may write $\Delta =aS+B$ , where $a\not \in \{ 1-\frac 1 m \mid m\in \mathbb N \}\cup \{ 1\}$ . Note that S is not exceptional over X (as the exceptional divisors occur with coefficient one). Let $\nu \colon Y'\to Y$ be a log resolution which is an isomorphism at the generic points of strata of $\lfloor \Delta \rfloor $ , and let $B'=\nu ^{-1}_*B+\textrm {Ex}(\nu )$ , $S'=\nu ^{-1}_* S$ . Since $\zeta (S'+B')<\zeta (\Delta )$ , we may run the $(K_{Y'}+S'+B')$ -MMP over Z. We obtain a birational contraction such that $K_{Y''}+S''+B''=\phi _*(K_{Y'}+S'+B')$ is nef over Z and all components of $\lfloor S''+B''\rfloor $ are normal up to universal homeomorphism. We now run a $(K_{Y''}+aS''+B'')$ -MMP over Z with scaling of $(1-a)S''$ . Note that this is also a $(K_{Y''}+B'')$ -MMP and $\zeta (B'')<\zeta (\Delta )$ . Therefore, the required minimal model exists. Since the log resolution was an isomorphism at the generic points of strata of $\lfloor \Delta \rfloor $ , it is easy to see that this is the required flip (cf. [Reference Hacon and WitaszekHW22, Section 7.1]).

Proof. Let $\pi \colon Y \to X$ be a log resolution, and run the $(K_Y+\textrm {Ex}(\pi ))$ -MMP over X. By standard arguments, this MMP contracts $\textrm {Ex}(\pi )$ , and so its output $Y'\to X$ is a small birational morphism and hence an isomorphism as X is ${\mathbb {Q}}$ -factorial. We shall show that if $g \colon Y \dashrightarrow Y_n$ is a sequence of steps of the MMP, with the induced map $\pi _n \colon Y_n \to X$ , then $R^i\pi _*W{\mathcal {O}}_{Y,{\mathbb {Q}}} = R^i(\pi _n)_*W{\mathcal {O}}_{Y_n,{\mathbb {Q}}}$ . As a consequence, $R^i\pi _*W{\mathcal {O}}_{Y,{\mathbb {Q}}} = 0$ for $i>0$ , and so X has $W{\mathcal {O}}$ -rational singularities. Thus, inductively, after replacing Y by some steps of the MMP, we may assume that $(Y, \textrm {Ex}(f))$ is dlt. Let $f \colon Y \to Z$ be a Mori contraction over X with $\rho (Y/Z)=1$ .

Assume that f is divisorial. There exists an f-antiample irreducible divisor $S \subseteq \textrm {Ex}(f)$ with the induced map $f_S \colon S \to X$ . Set $K_{S^{\nu }}+\Delta _{S^{\nu }} = (K_Y + \textrm {Ex}(f))|_{S^{\nu }}$ , where $S^{\nu }$ is the normalisation of S with the induced map $f_{S^{\nu }} \colon S^{\nu } \to X$ . Then $(S^{\nu },\Delta _{S^{\nu }})$ is dlt by Lemma 2.2 and $-(K_{S^{\nu }}+\Delta _{S^{\nu }})$ is $f_{S^{\nu }}$ -ample. Up to perturbation, we can assume that $(S^{\nu }, \Delta _{S^{\nu }})$ is klt, and so, by [Reference Nakamura and TanakaNT20, Theorem 3.11],

$$\begin{align*}R^i(f_{S^{\nu}})_*W{\mathcal{O}}_{S^{\nu},{\mathbb{Q}}}=0 \end{align*}$$

for $i>0$ . Since S is normal up to universal homeomorphism (Lemma 2.1), [Reference Nakamura and TanakaNT20, Lemma 2.17] and [Reference Gongyo, Nakamura and HiromuGNH19, Lemma 2.20 and 2.21] together with the Leray spectral sequence imply that $R^i(f_{S})_*W{\mathcal {O}}_{S,{\mathbb {Q}}}=0$ for $i>0$ . Let $u \colon S \to Y$ be the inclusion, and let $I_S$ be the ideal sheaf defining S. Then, we get the following short exact sequence (cf. [Reference Gongyo, Nakamura and HiromuGNH19, Proof of Proposition 3.4])

$$\begin{align*}0 \to WI_{S,{\mathbb{Q}}} \to W{\mathcal{O}}_{Y,{\mathbb{Q}}} \to u_*W{\mathcal{O}}_{S,{\mathbb{Q}}} \to 0. \end{align*}$$

By [Reference Gongyo, Nakamura and HiromuGNH19, Proposition 2.23], $R^if_* WI_{S,{\mathbb {Q}}}=0$ for $i>0$ , and so

$$\begin{align*}R^if_*W{\mathcal{O}}_{Y,{\mathbb{Q}}}=R^if_*(u_*W{\mathcal{O}}_{S,{\mathbb{Q}}}) = R^i(f_S)_*W{\mathcal{O}}_{S,{\mathbb{Q}}} = 0, \end{align*}$$

where the second equality follows from [Reference Gongyo, Nakamura and HiromuGNH19, Lemma 2.20 and Lemma 2.21]. Moreover, $f_*W{\mathcal {O}}_{Y,{\mathbb {Q}}} = W{\mathcal {O}}_{Z,{\mathbb {Q}}}$ by [Reference Nakamura and TanakaNT20, Lemma 2.17], and so we can conclude the proof when f is divisorial by the Leray spectral sequence.

Suppose that f is a flipping contraction, and let $\pi _Z \colon Z \to X$ be the induced map. Let $\phi \colon Y \dashrightarrow Y^+$ be the flip, let $f^+ \colon Y^+ \to Z$ be the flipped contraction and let $\pi ^+ \colon Y^+ \to X$ be the induced map. As before, by [Reference Hacon and WitaszekHW21, Lemma 3.1] and Theorem 4.3, both Y and $Y^+$ are relatively F-regular over Z. In particular, $Rf_* {\mathcal {O}}_Y = {\mathcal {O}}_Z$ and $Rf^+_* {\mathcal {O}}_{Y^+} = {\mathcal {O}}_Z$ (see [Reference Schwede and SmithSS10, Theorem 6.8] and the proof of [Reference Hacon and WitaszekHW22, Proposition 3.4]). By [Reference Nakamura and TanakaNT20, Lemma 2.17] and [Reference Gongyo, Nakamura and HiromuGNH19, Lemma 2.19 and Lemma 2.21], $f_* W{\mathcal {O}}_{Y,{\mathbb {Q}}} = f^+_* W{\mathcal {O}}_{Y^+,{\mathbb {Q}}} = W{\mathcal {O}}_{Z,{\mathbb {Q}}}$ and $R^if_* W{\mathcal {O}}_{Y,{\mathbb {Q}}} = R^if^+_* W{\mathcal {O}}_{Y^+,{\mathbb {Q}}} = 0$ for $i>0$ . By the Leray spectral sequence:

$$\begin{align*}R^i\pi_*W{\mathcal{O}}_{Y,{\mathbb{Q}}} = R^i(\pi_Z)_*W{\mathcal{O}}_{Z,{\mathbb{Q}}} = R^i\pi_*^+W{\mathcal{O}}_{Y^+,{\mathbb{Q}}}.\\[-38pt] \end{align*}$$

Proof of Corollary 1.3

Let $\pi \colon Y \to X$ be a resolution of singularities such that Y is rationally chain connected. Since X has $W\mathcal {O}_X$ -rational singularities by the above corollary, [Reference Gongyo, Nakamura and HiromuGNH19, Remark 3.1] and [Reference Berthelot, Bloch and EsnaultBBE07, Proposition 6.9] imply that $|X(\mathbb {F}_q)| \equiv |Y(\mathbb {F}_q)| \pmod q$ .

We can conclude the proof as $|Y(\mathbb {F}_q)| \equiv 1\ \pmod q$ by [Reference EsnaultEsn03, Theorem 1.1] and the argument explained between this theorem and [Reference EsnaultEsn03, Corollary 1.3]. Specifically, this follows from the Lefschetz trace formula for crystalline cohomology

$$\begin{align*}|Y(\mathbb{F}_q)| = \sum_i (-1)^i\mathrm{Trace}(F \mid H^i_{\textrm{crys}}(X/K)) \end{align*}$$

given that Y is rationally chain connected, and so all the slopes of Frobenius are $\geq 1$ by [Reference EsnaultEsn03, Theorem 1.1].

Proof. Let $\pi \colon Y \to X$ be a log resolution of $(X, \Delta )$ . Then, a dlt modification is a minimal model of $(Y,\pi ^{-1}_*\Delta + \textrm {Ex}(\pi ))$ over X (see Theorem 4.1).

Proof. By considering a log resolution of $(X,S+B)$ , it is easy to see that if $(X,S+B)$ is plt, then $(\tilde S,B_{\tilde S})$ is klt. Thus, we can assume that $(\tilde S,B_{\tilde S})$ is klt and aim to show that $(X,S+B)$ is plt near S.

Let $\pi \colon Y \to X$ be a dlt modification of $(X,S+B)$ (see Corollary 4.8), and write $K_Y + S_Y + B_Y = \pi ^*(K_X+S+B)$ . By definition (of a dlt modification) for any $\pi $ -exceptional irreducible divisor E, we have that $E \subseteq \lfloor B_Y \rfloor $ . Write

$$\begin{align*}(\pi|_{S_Y})^*(K_{\tilde S} + B_{\tilde S}) = (K_Y+S_Y+B_Y)|_{\tilde S_Y} = K_{\tilde S_Y} + B_{\tilde S_Y}, \end{align*}$$

where $\tilde S_Y \to S_Y$ is the normalisation of $S_Y$ , and $B_{\tilde S_Y}$ is the different. Let E be a $\pi $ -exceptional divisor intersecting $S_Y$ . Since $E \subseteq \lfloor B_Y \rfloor $ and $(Y,S_Y+\textrm {Ex}(\pi ))$ is dlt, we must have that $E \cap S_Y \subseteq \lfloor B_{\tilde S_Y} \rfloor $ . Since $E \cap S_Y \neq \emptyset $ , this contradicts $(\tilde S, B_{\tilde S})$ being klt.

Therefore, we may assume that $E\cap S_Y=\emptyset $ so that $Y=X$ near S, and hence $(X,S+B)$ is dlt on a neighbourhood of S. Since S is irreducible, $(X,S+B)$ is in fact plt.