Proof. Let $f:Y\rightarrow X$ be a log resolution of the pairs $(X,\unicode[STIX]{x1D6E5}_{i}\geqslant 0)$ for all $i=1,2,\ldots ,m$ and

(3.1)$$\begin{eqnarray}K_{Y}+\unicode[STIX]{x1D6E4}_{i}=f^{\ast }(K_{X}+\unicode[STIX]{x1D6E5}_{i})+F_{i},\end{eqnarray}$$

where $\unicode[STIX]{x1D6E4}_{i}\geqslant 0$ and $F_{i}\geqslant 0$ do not share any common component and $f_{\ast }\unicode[STIX]{x1D6E4}_{i}=\unicode[STIX]{x1D6E5}_{i}$ and $f_{\ast }F_{i}=0$, for all $i=1,2,\ldots ,m$.

First we deal with the log canonical case. Define a divisor $\unicode[STIX]{x1D6E5}_{Y}$ on $Y$ as $\unicode[STIX]{x1D6E5}_{Y}:=(f_{\ast }^{-1}(\unicode[STIX]{x1D6E5}_{1}+\unicode[STIX]{x1D6E5}_{2}+\cdots +\unicode[STIX]{x1D6E5}_{m}))_{\text{red}}+E$, where $E$ is the reduced $f$-exceptional divisor with $\operatorname{Supp}E=\operatorname{Ex}(f)$. Then $(Y,\unicode[STIX]{x1D6E5}_{Y}\geqslant 0)$ is a LC pair. By [Reference Tanaka20, Theorem 1] and its proof it follows that there exists an effective $\mathbb{K}$-divisor $0\leqslant D^{\prime }{\sim}_{\mathbb{K}}D$ not containing any $Z_{j},1\leqslant j\leqslant l$ in its support such that $(Y,\unicode[STIX]{x1D6E5}_{Y}+f^{\ast }D^{\prime })$ is LC. From (3.1) we see that $\unicode[STIX]{x1D6E4}_{i}+f^{\ast }D^{\prime }\leqslant \unicode[STIX]{x1D6E5}_{Y}$ and thus $(Y,\unicode[STIX]{x1D6E4}_{i}+f^{\ast }D^{\prime })$ is LC, for all $i=1,2,\ldots ,m$. It then follows that $(X,\unicode[STIX]{x1D6E5}_{i}+D^{\prime })$ is LC for all $i=1,2,\ldots ,m$.

We now work with the KLT case. The divisor $mD$ is semi-ample $\mathbb{K}$-Cartier $\mathbb{K}$-divisor on $X$. Then by the log canonical case there exists an effective $\mathbb{K}$-divisor $0\leqslant D^{\prime \prime }{\sim}_{\mathbb{K}}mD$ not containing any $Z_{j},1\leqslant j\leqslant l$ in its support such that $(X,\unicode[STIX]{x1D6E5}_{i}+D^{\prime \prime })$ is LC for all $i=1,2,\ldots ,m$. Set $D^{\prime }:=(1/m)D^{\prime \prime }{\sim}_{\mathbb{K}}D$. We claim that $(X,\unicode[STIX]{x1D6E5}_{i}+D^{\prime })$ is KLT for all $i=1,2,\ldots ,m$. Since $(X,\unicode[STIX]{x1D6E5}_{i})$ is KLT, if $(X,\unicode[STIX]{x1D6E5}_{i}+D^{\prime })$ is not KLT then this will implies that $(X,\unicode[STIX]{x1D6E5}_{i}+D^{\prime \prime })$ is not LC (for $m\geqslant 2$), a contradiction. Therefore, $(X,\unicode[STIX]{x1D6E5}_{i}+D^{\prime })$ is KLT for all $i=1,2,\ldots ,m$.◻

Proof. Since $K_{Y}+\unicode[STIX]{x1D6E4}$ is KLT and $\unicode[STIX]{x1D6E4}$ is big, it follows from [Reference Birkar and Waldron6, Theorem 1.2] that $K_{Y}+\unicode[STIX]{x1D6E4}$ is semi-ample. Part $(2)$ and $(3)$ then follow from [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 3.6.6(3)].◻

Proof. Since $(X,\unicode[STIX]{x1D6E5})$ is DLT, there exists a log resolution $f:X^{\prime }\rightarrow X$ of $(X,\unicode[STIX]{x1D6E5})$ such that $a(E_{i},X,\unicode[STIX]{x1D6E5})>-1$ for every $f$-exceptional divisor $E_{i}$. We can write

$$\begin{eqnarray}K_{X^{\prime }}+\unicode[STIX]{x1D6E4}=f^{\ast }(K_{X}+\unicode[STIX]{x1D6E5})+E,\end{eqnarray}$$

where $\unicode[STIX]{x1D6E4}\geqslant 0$ and $E\geqslant 0$ do not share any common component, and $f_{\ast }\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6E5},f_{\ast }E=0$.

Let $F\geqslant 0$ be a reduced divisor with $\operatorname{Supp}F=\operatorname{Ex}(f)$. Then $(X^{\prime },\unicode[STIX]{x1D6E4}+\unicode[STIX]{x1D700}F)$ is DLT for $0<\unicode[STIX]{x1D700}\ll 1$. Furthermore, we have

$$\begin{eqnarray}K_{X^{\prime }}+\unicode[STIX]{x1D6E4}+\unicode[STIX]{x1D700}F{\sim}_{\mathbb{R},f}E+\unicode[STIX]{x1D700}F\geqslant 0.\end{eqnarray}$$

By [Reference Waldron21, Corollary 1.8] we can run a terminating $(K_{X^{\prime }}+\unicode[STIX]{x1D6E4}+\unicode[STIX]{x1D700}F)$-MMP over $X$. Let $\unicode[STIX]{x1D719}:X^{\prime }{\dashrightarrow}Y$ be the corresponding minimal model over $X$, that is, $K_{Y}+\unicode[STIX]{x1D6E5}_{Y}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D719}_{\ast }F$ is nef over $X$, where $K_{Y}+\unicode[STIX]{x1D6E5}_{Y}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D719}_{\ast }F=\unicode[STIX]{x1D719}_{\ast }(K_{X^{\prime }}+\unicode[STIX]{x1D6E4}+\unicode[STIX]{x1D700}F)$. It is easy to see from the Negativity lemma that $\unicode[STIX]{x1D719}$ contracts all $f$-exceptional divisors, that is, $\unicode[STIX]{x1D719}_{\ast }F=0$; in particular, if $\unicode[STIX]{x1D70B}:Y\rightarrow X$ is the structure morphism, then $\unicode[STIX]{x1D70B}$ is a small birational morphism, $Y$ is $\mathbb{Q}$-factorial, $(Y,\unicode[STIX]{x1D6E5}_{Y}\geqslant 0)$ is DLT and

$$\begin{eqnarray}\hspace{103.79993pt}K_{Y}+\unicode[STIX]{x1D6E5}_{Y}=\unicode[STIX]{x1D70B}^{\ast }(K_{X}+\unicode[STIX]{x1D6E5}).\hspace{103.79993pt}\square\end{eqnarray}$$

Sketch of the proof.

We only explain the part where Bertini’s theorem is used in the proof of [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 3.7.4], which is basically the second paragraph in [Reference Birkar, Cascini, Hacon and McKernan5, page 436]. All other arguments in the rest of the proof of [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 3.7.4] holds in our settings here without any change.

We basically show that we can choose ample divisors $A_{i}$ and $A^{\prime }$ as in the proof of [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 3.7.4] such that $(X,\unicode[STIX]{x1D6E5}+A^{\prime }-A)$ is LC, $(X,\unicode[STIX]{x1D6E5}+4/3A_{i}+A^{\prime }-A)$ is LC for all $1\leqslant i\leqslant l$ and for all $\unicode[STIX]{x1D6E5}\in {\mathcal{L}}_{S+A}(V)$, and $(X,A^{\prime }+\unicode[STIX]{x1D6E5}_{0})$ is DLT.

To that end, let $\unicode[STIX]{x1D6E4}_{1},\unicode[STIX]{x1D6E4}_{2},\ldots ,\unicode[STIX]{x1D6E4}_{m}$ be the vertices of the rational polytope ${\mathcal{L}}_{S+A}(V)$. Since $(X,\unicode[STIX]{x1D6E4}_{i})$ is LC for all $1\leqslant j\leqslant m$ and $(X,\unicode[STIX]{x1D6E5}_{0})$ is DLT, by Theorem 3.1 there exists a divisor $0\leqslant A^{\prime \prime }{\sim}_{\mathbb{Q}}A$ such that $(X,\unicode[STIX]{x1D6E4}_{j}+A^{\prime \prime })$ is LC for all $1\leqslant j\leqslant m$ and the support of $A^{\prime \prime }$ does not contain any LC center of $(X,\unicode[STIX]{x1D6E5}_{0})$. Set $A^{\prime }=\unicode[STIX]{x1D700}A^{\prime \prime }$ for a rational number $\unicode[STIX]{x1D700}\in (0,1/4]$. For $0<\unicode[STIX]{x1D700}\ll 1/4$ we see that $(X,\unicode[STIX]{x1D6E4}_{j}+A^{\prime })$ is LC for all $1\leqslant j\leqslant m$ and $(X,\unicode[STIX]{x1D6E5}_{0}+A^{\prime })$ is DLT. Furthermore, since $A_{i}$’s are general ample $\mathbb{Q}$-divisors and $0<\unicode[STIX]{x1D700}\ll 1/4$, it again follows from Theorem 3.1 that $(X,\unicode[STIX]{x1D6E4}_{j}+4/3A_{i}+A^{\prime })$ is LC for all $1\leqslant j\leqslant m$ and $1\leqslant i\leqslant l$. Now for any $\unicode[STIX]{x1D6E5}\in {\mathcal{L}}_{S+A}(V)$ we can write $\unicode[STIX]{x1D6E5}=\sum _{j=1}^{m}\unicode[STIX]{x1D706}_{j}\unicode[STIX]{x1D6E4}_{j}$ for some $\unicode[STIX]{x1D706}_{j}\geqslant 0$ such that $\sum _{j=1}^{m}\unicode[STIX]{x1D706}_{j}=1$. It is easy to see that convex sum of finitely many LC divisors are LC. It then follows that $(X,\unicode[STIX]{x1D6E5}+A^{\prime })$ and $(X,\unicode[STIX]{x1D6E5}+4/3A_{i}+A^{\prime })$ are both LC for all $1\leqslant i\leqslant l$ and for all $\unicode[STIX]{x1D6E5}\in {\mathcal{L}}_{S+A}(V)$. In particular, we finally have $(X,\unicode[STIX]{x1D6E5}+A^{\prime }-A)$ is LC, $(X,\unicode[STIX]{x1D6E5}+4/3A_{i}+A^{\prime }-A)$ is LC for all $1\leqslant i\leqslant l$ and for all $\unicode[STIX]{x1D6E5}\in {\mathcal{L}}_{S+A}(V)$, and $(X,A^{\prime }+\unicode[STIX]{x1D6E5}_{0})$ is DLT. ◻

Proof. This result corresponds to [Reference Birkar, Cascini, Hacon and McKernan5, Theorem 3.11.1].

Since ${\mathcal{L}}_{A}(V)$ is compact, it is enough to prove the finiteness of $R^{\bot }$ locally in a neighborhood of a point $\unicode[STIX]{x1D6E5}\in {\mathcal{L}}_{A}(V)$. Now since there is a boundary divisor $\unicode[STIX]{x1D6E5}_{0}$ such that $(X,\unicode[STIX]{x1D6E5}_{0})$ is KLT and the image of a hyperplane under linear isomorphism of affine spaces is again a hyperplane, by Lemma 4.4 we may assume that $K_{X}+\unicode[STIX]{x1D6E5}$ is KLT. Fix $\unicode[STIX]{x1D700}>0$ such that if $\unicode[STIX]{x1D6E5}^{\prime }\in {\mathcal{L}}_{A}(V)$ and $\Vert \unicode[STIX]{x1D6E5}^{\prime }-\unicode[STIX]{x1D6E5}\Vert <\unicode[STIX]{x1D700}$, then $\unicode[STIX]{x1D6E5}^{\prime }-\unicode[STIX]{x1D6E5}+A/2$ is ample over $U$. Let $R$ be an extremal ray over $U$ such that $(K_{X}+\unicode[STIX]{x1D6E5}^{\prime })\cdot R=0$ for some $\unicode[STIX]{x1D6E5}^{\prime }\in {\mathcal{L}}_{A}(V)$ with $\Vert \unicode[STIX]{x1D6E5}^{\prime }-\unicode[STIX]{x1D6E5}\Vert <\unicode[STIX]{x1D700}$. Then we have

$$\begin{eqnarray}\displaystyle (K_{X}+\unicode[STIX]{x1D6E5}-A/2)\cdot R & = & \displaystyle (K_{X}+\unicode[STIX]{x1D6E5}^{\prime })\cdot R-(\unicode[STIX]{x1D6E5}^{\prime }-\unicode[STIX]{x1D6E5}+A/2)\cdot R\nonumber\\ \displaystyle & = & \displaystyle -(\unicode[STIX]{x1D6E5}^{\prime }-\unicode[STIX]{x1D6E5}+A/2)\cdot R<0.\nonumber\end{eqnarray}$$

Write $\unicode[STIX]{x1D6E5}=A+B$. Then $K_{X}+\unicode[STIX]{x1D6E5}-A/2=K_{X}+B+A/2$. By [Reference Waldron21, Theorem 1.7(3)] there are only finitely many extremal rays $R$ satisfying these properties.

Now ${\mathcal{N}}_{A,\unicode[STIX]{x1D70B}}(V)$ is clearly a closed subset of ${\mathcal{L}}_{A}(V)$. Let $\unicode[STIX]{x1D6E5}\in {\mathcal{L}}_{A}(V)$. If $K_{X}+\unicode[STIX]{x1D6E5}$ is not nef$/U$, then again by [Reference Waldron21, Theorem 1.7] there exists an extremal ray $R$ of $\overline{\operatorname{NE}}(X/U)$ generated by a rational curve $\unicode[STIX]{x1D6F4}$ such that $(K_{X}+\unicode[STIX]{x1D6E5})\cdot \unicode[STIX]{x1D6F4}<0$. In particular, ${\mathcal{N}}_{A,\unicode[STIX]{x1D70B}}(V)$ is contained in the half-spaces $R^{{\geqslant}0}=\{\unicode[STIX]{x1D6E4}\in {\mathcal{L}}_{A}(V):(K_{X}+\unicode[STIX]{x1D6E4})\cdot R\geqslant 0\}$ of the hyperplanes $R^{\bot }$. Then by the previous part, there exists finitely many extremal rays $R_{1},R_{2},\ldots ,R_{n}$ of $\overline{\operatorname{NE}}(X/U)$ such that ${\mathcal{N}}_{A,\unicode[STIX]{x1D70B}}(V)=\bigcap _{i=1}^{n}R_{i}^{{\geqslant}0}$. Since $R_{i}$’s are generated by irreducible curves, the hyperplanes $R_{i}^{\bot }$’s are all rational hyperplanes, in particular, ${\mathcal{N}}_{A}(V)$ is a rational polytope.◻

Proof. This result corresponds to [Reference Birkar, Cascini, Hacon and McKernan5, Corollary 3.11.2].

Replacing $V_{A}$ by the span of ${\mathcal{L}}_{A}(V)$ if necessary we may assume that ${\mathcal{L}}_{A}(V)$ spans $V_{A}$. By compactness, to prove that ${\mathcal{W}}_{\unicode[STIX]{x1D719},A,\unicode[STIX]{x1D70B}}(V)$ is a rational polytope, we may work locally about a divisor $\unicode[STIX]{x1D6E5}\in {\mathcal{W}}_{\unicode[STIX]{x1D719},A,\unicode[STIX]{x1D70B}}(V)$. By [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 3.7.4] we may assume that $K_{X}+\unicode[STIX]{x1D6E5}$ is KLT. Then $K_{Y}+\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D719}_{\ast }(K_{X}+\unicode[STIX]{x1D6E5})$ is KLT as well. Let $C=\unicode[STIX]{x1D719}_{\ast }A$ and $W=\unicode[STIX]{x1D719}_{\ast }(V)$. Then $C$ is a big$/U$$\mathbb{Q}$-divisor on $Y$. By [Reference Birkar, Cascini, Hacon and McKernan5, Lemmas 3.7.3 and 3.7.4] there exists a rational affine linear isomorphism $L:W\rightarrow W^{\prime }$ and an ample$/U$$\mathbb{Q}$-divisor $C^{\prime }$ such that $L(\unicode[STIX]{x1D6E4})$ is contained in the interior of ${\mathcal{L}}_{C^{\prime }}(W^{\prime })$ and $L(\unicode[STIX]{x1D6F9}){\sim}_{\mathbb{Q},U}\unicode[STIX]{x1D6F9}$ for every $\unicode[STIX]{x1D6F9}\in W$. Then by Theorem 4.6, ${\mathcal{N}}_{C^{\prime },\unicode[STIX]{x1D713}}(W^{\prime })$ is a nonempty rational polytope containing $L(\unicode[STIX]{x1D6E4})$, where $\unicode[STIX]{x1D713}:Y\rightarrow U$ is the structure morphism. Therefore, ${\mathcal{N}}_{C,\unicode[STIX]{x1D713}}(W)$ is a rational polytope locally around $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D6E5}$.

Consider the following resolution of $\unicode[STIX]{x1D719}:X{\dashrightarrow}Y$ which is also a log resolution of $(X,\unicode[STIX]{x1D6E5})$.

Then we have

$$\begin{eqnarray}\displaystyle & \displaystyle K_{W}+\unicode[STIX]{x1D6F9}=p^{\ast }(K_{X}+\unicode[STIX]{x1D6E5}) & \displaystyle \nonumber\\ \displaystyle & \displaystyle K_{W}+\unicode[STIX]{x1D6F7}=q^{\ast }(K_{Y}+\unicode[STIX]{x1D6E4}). & \displaystyle \nonumber\end{eqnarray}$$

Note that $\unicode[STIX]{x1D6E5}\,\in \,{\mathcal{W}}_{\unicode[STIX]{x1D719},A,\unicode[STIX]{x1D70B}}(V)$ if and only if $\unicode[STIX]{x1D6E4}\,=\,\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D6E5}\,\in \,{\mathcal{N}}_{C,\unicode[STIX]{x1D713}}(W)$ and $\unicode[STIX]{x1D6F9}\,-\,\unicode[STIX]{x1D6F7}\,\geqslant \,0$. Since the map $L:V\rightarrow W$ given by $\unicode[STIX]{x1D6E5}\rightarrow \unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D6E5}$ is rational and linear, in a neighborhood of $\unicode[STIX]{x1D6E5}$, ${\mathcal{W}}_{\unicode[STIX]{x1D719},A,\unicode[STIX]{x1D70B}}(V)$ is cut out from ${\mathcal{L}}_{A}(V)$ by finitely many half-spaces generated by affine rational hyperplanes. Therefore, by compactness, ${\mathcal{W}}_{\unicode[STIX]{x1D719},A,\unicode[STIX]{x1D70B}}(V)$ is a rational polytope.

Now for any $\unicode[STIX]{x1D6E5}\in {\mathcal{W}}_{\unicode[STIX]{x1D719},A,\unicode[STIX]{x1D70B}}(V)$ we have $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D6E5}=A^{\prime }+B^{\prime }$, where $A^{\prime }\geqslant 0$ is a big divisor on $Y$. Therefore, by perturbing $\unicode[STIX]{x1D6E4}$, from the base-point free theorem [Reference Birkar and Waldron6, Theorem 1.2] it follows that there is a contraction $g:Y\rightarrow Z$ satisfying the required conditions. The rational map $g\circ \unicode[STIX]{x1D719}:X{\dashrightarrow}Z$ is the ample model of $K_{X}+\unicode[STIX]{x1D6E5}$. Next we prove that there are only finitely many such contractions $g:Y\rightarrow Z$ corresponding to all $\unicode[STIX]{x1D6E5}\in {\mathcal{W}}_{\unicode[STIX]{x1D719},A,\unicode[STIX]{x1D70B}}(V)$.

For two contractions $f:Y\rightarrow Z,f_{\ast }{\mathcal{O}}_{Y}={\mathcal{O}}_{Z}$ and $f^{\prime }:Y\rightarrow Z^{\prime },f_{\ast }^{\prime }{\mathcal{O}}_{Y}={\mathcal{O}}_{Z^{\prime }}$ over $U$, there exists an isomorphism $\unicode[STIX]{x1D702}:Z\rightarrow Z^{\prime }$ satisfying $f^{\prime }=\unicode[STIX]{x1D702}\circ f$ if and only if $f$ and $f^{\prime }$ contracts exactly same curves, that is, $f(C)=\operatorname{pt}$ if and only if $f^{\prime }(C)=\operatorname{pt}$ for irreducible curves $C\subseteq X$ (see [Reference Debarre9, Proposition 1.14 and Lemma 1.15]). Let $f:Y\rightarrow Z$ be a contraction over $U$ such that

$$\begin{eqnarray}K_{Y}+\unicode[STIX]{x1D6E4}=K_{Y}+\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D6E5}{\sim}_{\mathbb{R},U}f^{\ast }D,\end{eqnarray}$$

where $\unicode[STIX]{x1D6E5}\in {\mathcal{W}}_{\unicode[STIX]{x1D719},A,\unicode[STIX]{x1D70B}}(V)$ and $D$ is an ample$/U$$\mathbb{R}$-divisor on $Z$. $\unicode[STIX]{x1D6E4}$ belongs to the interior of a unique face $G$ of ${\mathcal{N}}_{C,\unicode[STIX]{x1D713}}(W)$. Let $\unicode[STIX]{x1D6E4}_{1},\unicode[STIX]{x1D6E4}_{2},\ldots ,\unicode[STIX]{x1D6E4}_{k}$ be the vertices of the $G$. Write $\unicode[STIX]{x1D6E4}=\sum _{j=1}^{k}\unicode[STIX]{x1D706}_{j}\unicode[STIX]{x1D6E4}_{j}$, where $\sum _{j=1}^{k}\unicode[STIX]{x1D706}_{j}=1$ and $\unicode[STIX]{x1D706}_{j}\geqslant 0$ for all $j=1,2,\ldots ,k$. Since $\unicode[STIX]{x1D6E4}$ is contained in the interior of $G$, for any given (fixed) index $i$, we can choose $\unicode[STIX]{x1D706}_{i}>0$ in $\unicode[STIX]{x1D6E4}=\sum _{j=1}^{k}\unicode[STIX]{x1D706}_{j}\unicode[STIX]{x1D6E4}_{j}$. Let $C\subseteq Y$ be a curve contracted by $f$, then $0=(K_{Y}+\unicode[STIX]{x1D6E4})\cdot C=\sum _{j=1}^{k}\unicode[STIX]{x1D706}_{j}(K_{Y}+\unicode[STIX]{x1D6E4}_{j})\cdot C\geqslant 0$. This implies that $\unicode[STIX]{x1D706}_{i}(K_{Y}+\unicode[STIX]{x1D6E4}_{i})\cdot C=0$, that is, $(K_{Y}+\unicode[STIX]{x1D6E4}_{i})\cdot C=0$, since $\unicode[STIX]{x1D706}_{i}\neq 0$. Therefore, if $C$ is contracted by $f$, then $(K_{Y}+\unicode[STIX]{x1D6E4}_{j})\cdot C=0$ for all $j=1,2,\ldots ,k$. Conversely, if $C$ is a curve on $Y$ such that $(K_{Y}+\unicode[STIX]{x1D6E4}_{j})\cdot C=0$ for all $j=1,2,\ldots ,k$, then clearly $(K_{Y}+\unicode[STIX]{x1D6E4})\cdot C=0$, and hence $C$ is contracted by $f$. Therefore, the curves contacted by $f$ are uniquely determined by $G$. Now $\unicode[STIX]{x1D6E5}$ is contained in the interior of a unique face $F$ of ${\mathcal{W}}_{\unicode[STIX]{x1D719},A,\unicode[STIX]{x1D70B}}(V)$ and $G$ is determined by $F$. But since ${\mathcal{W}}_{\unicode[STIX]{x1D719},A,\unicode[STIX]{x1D70B}}(V)$ is a rational polytope it has only finitely many faces.◻

Proof. This result corresponds to [Reference Birkar, Cascini, Hacon and McKernan5, Corollary 3.11.3].

By Theorem 4.6 there exists finitely many extremal rays $R_{1},R_{2},\ldots ,R_{k}$ of $\overline{\operatorname{NE}}(Y/U)$ such that if $K_{Y}+\unicode[STIX]{x1D6E4}=K_{Y}+\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D6E5}$ is not nef over $U$ for some $\unicode[STIX]{x1D6E5}\in {\mathcal{L}}_{A}(V)$, then it is negative on one of these rays. If $\unicode[STIX]{x1D6E4}_{0}=\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D6E5}_{0}$, then we may write

$$\begin{eqnarray}K_{Y}+\unicode[STIX]{x1D6E4}=K_{Y}+\unicode[STIX]{x1D6E4}_{0}+(\unicode[STIX]{x1D6E4}-\unicode[STIX]{x1D6E4}_{0}){\sim}_{\mathbb{R},U}g^{\ast }H+\unicode[STIX]{x1D719}_{\ast }(\unicode[STIX]{x1D6E5}-\unicode[STIX]{x1D6E5}_{0}),\end{eqnarray}$$

where $g:Y\rightarrow Z$ is the structure morphism.

Claim. If $\unicode[STIX]{x1D6E5}\in {\mathcal{L}}_{A}(V)$ is sufficiently close to $\unicode[STIX]{x1D6E5}_{0}$ and $(K_{Y}+\unicode[STIX]{x1D6E4})\cdot R_{i_{0}}=\unicode[STIX]{x1D719}_{\ast }(K_{X}+\unicode[STIX]{x1D6E5})\cdot R_{i_{0}}<0$ for some $i_{0}\in \{1,2,\ldots ,k\}$, then $(K_{Y}+\unicode[STIX]{x1D6E4}_{0})\cdot R_{i_{0}}=\unicode[STIX]{x1D719}_{\ast }(K_{X}+\unicode[STIX]{x1D6E5}_{0})\cdot R_{i_{0}}=0$.

Proof of the claim.

On the contrary assume that $(K_{Y}+\unicode[STIX]{x1D6E4}_{0})\cdot R_{i_{0}}>0$. Let $\unicode[STIX]{x1D6FC}=\frac{1}{2}(K_{Y}+\unicode[STIX]{x1D6E4}_{0})\cdot R_{i_{0}}>0$. Then $(K_{Y}+\unicode[STIX]{x1D6E4}_{0})\cdot R_{i_{0}}>\unicode[STIX]{x1D6FC}$. Let $\unicode[STIX]{x1D6E4}^{\prime }=\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D6E5}^{\prime }$ for some $\unicode[STIX]{x1D6E5}^{\prime }\in {\mathcal{L}}_{A}(V)$ such that $K_{Y}+\unicode[STIX]{x1D6E4}^{\prime }$ is LC. Then by [Reference Waldron21, Theorem 1.7] we have $(K_{Y}+\unicode[STIX]{x1D6E4}^{\prime })\cdot R_{i_{0}}\geqslant -6$. Choose $\unicode[STIX]{x1D6E5}\in {\mathcal{L}}_{A}(V)$ such that $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D6E5}$ lies on the line segment joining $\unicode[STIX]{x1D6E4}_{0}$ and $\unicode[STIX]{x1D6E4}^{\prime }$, that is, $\unicode[STIX]{x1D6E4}=r\unicode[STIX]{x1D6E4}_{0}+s\unicode[STIX]{x1D6E4}^{\prime }$ for some $r\geqslant 0$ and $s>0$ satisfying $r+s=1$. Then

$$\begin{eqnarray}\displaystyle (K_{Y}+\unicode[STIX]{x1D6E4})\cdot R_{i_{0}} & = & \displaystyle r(K_{Y}+\unicode[STIX]{x1D6E4}_{0})\cdot R_{i_{0}}\nonumber\\ \displaystyle & & \displaystyle +\,s(K_{Y}+\unicode[STIX]{x1D6E4}^{\prime })\cdot R_{i_{0}}>2r\unicode[STIX]{x1D6FC}-6s>0\quad \text{if }r>\frac{3s}{\unicode[STIX]{x1D6FC}}.\nonumber\end{eqnarray}$$

This is a contradiction. Therefore, if $\unicode[STIX]{x1D6E5}$ is sufficiently close to $\unicode[STIX]{x1D6E5}_{0}$ then $(K_{Y}+\unicode[STIX]{x1D6E4})\cdot R_{i_{0}}<0$ implies that $(K_{Y}+\unicode[STIX]{x1D6E4}_{0})\cdot R_{i_{0}}=0$.◻

In other words, there exists a neighborhood $P_{0}$ of $\unicode[STIX]{x1D6E5}_{0}$ in ${\mathcal{L}}_{A}(V)$ such that if $\unicode[STIX]{x1D6E5}\in P_{0}$ and $K_{Y}+\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D719}_{\ast }(K_{X}+\unicode[STIX]{x1D6E5})$ is not nef over $U$, then $(K_{Y}+\unicode[STIX]{x1D6E4})\cdot R_{i_{0}}<0$ for some $i_{0}\in \{1,2,\ldots ,k\}$ and $R_{i_{0}}$ is extremal over $Z$ (otherwise $(K_{Y}+\unicode[STIX]{x1D6E4}_{0})\cdot R_{i_{0}}>0$), and consequently $K_{Y}+\unicode[STIX]{x1D6E4}$ is not nef over $Z$. Contra-positively, if $\unicode[STIX]{x1D719}$ is a log minimal model of $K_{X}+\unicode[STIX]{x1D6E5}$ over $Z$, then it is a log minimal model over $U$. The other direction is obvious.◻

Proof of Proposition 4.9.

Replacing $V_{A}$ by the span of ${\mathcal{C}}$ if necessary we may assume that ${\mathcal{C}}$ spans $V_{A}$. We proceed by induction on the dimension of ${\mathcal{C}}$.

First assume that $\operatorname{dim}{\mathcal{C}}=0$. Then ${\mathcal{C}}=\{\unicode[STIX]{x1D6E5}_{0}\}$ for some $\unicode[STIX]{x1D6E5}_{0}\in {\mathcal{L}}_{A}(V)$. If $\unicode[STIX]{x1D6E5}_{0}\in {\mathcal{E}}_{A,\unicode[STIX]{x1D70B}}(V)$, then by [Reference Birkar3, Theorem 1.2] there exists a log minimal model $\unicode[STIX]{x1D719}:X{\dashrightarrow}Y/U$ for $K_{X}+\unicode[STIX]{x1D6E5}_{0}$. By induction assume that the statement is true for any such rational polytope ${\mathcal{C}}^{\prime }$ with $\operatorname{dim}{\mathcal{C}}^{\prime }<\operatorname{dim}{\mathcal{C}}$.

Now we prove the statement assuming that there is a divisor $\unicode[STIX]{x1D6E5}_{0}\in {\mathcal{C}}$ such that $K_{X}+\unicode[STIX]{x1D6E5}_{0}{\sim}_{\mathbb{R},U}0$. Let $\unicode[STIX]{x1D6E5}\in {\mathcal{C}}$ be a divisor such that $\unicode[STIX]{x1D6E5}\neq \unicode[STIX]{x1D6E5}_{0}$. Then there exists a divisor $\unicode[STIX]{x1D6E5}^{\prime }$ on one of the faces of ${\mathcal{C}}$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D706}\unicode[STIX]{x1D6E5}^{\prime }+(1-\unicode[STIX]{x1D706})\unicode[STIX]{x1D6E5}_{0},\end{eqnarray}$$

for some $0<\unicode[STIX]{x1D706}\leqslant 1$.

We have

$$\begin{eqnarray}K_{X}+\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D706}(K_{X}+\unicode[STIX]{x1D6E5}^{\prime })+(1-\unicode[STIX]{x1D706})(K_{X}+\unicode[STIX]{x1D6E5}_{0}){\sim}_{\mathbb{R},U}\unicode[STIX]{x1D706}(K_{X}+\unicode[STIX]{x1D6E5}^{\prime }).\end{eqnarray}$$

Therefore, $\unicode[STIX]{x1D6E5}\in {\mathcal{E}}_{A,\unicode[STIX]{x1D70B}}(V)$ if and only if $\unicode[STIX]{x1D6E5}^{\prime }\in {\mathcal{E}}_{A,\unicode[STIX]{x1D70B}}(V)$, and by [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 3.6.9] $K_{X}+\unicode[STIX]{x1D6E5}$ and $K_{X}+\unicode[STIX]{x1D6E5}^{\prime }$ have same log minimal models over $U$. Since ${\mathcal{C}}$ is a rational polytope, it has finitely many faces each of which are rational polytope themselves; therefore, by induction we are done.

Now we prove the general case. Applying [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 3.7.4] we may assume that ${\mathcal{C}}$ is contained in the interior of ${\mathcal{L}}_{A}(V)$. Note that ${\mathcal{C}}\cap {\mathcal{E}}_{A,\unicode[STIX]{x1D70B}}(V)$ is compact (as ${\mathcal{L}}_{A}(V)$ is compact and ${\mathcal{C}}\cap {\mathcal{E}}_{A,\unicode[STIX]{x1D70B}}$ is closed). So it is sufficient to prove the statement locally in a neighborhood of a divisor $\unicode[STIX]{x1D6E5}_{0}\in {\mathcal{C}}\cap {\mathcal{E}}_{A,\unicode[STIX]{x1D70B}}(V)$.

Let $\unicode[STIX]{x1D719}:X{\dashrightarrow}Y/U$ be a log minimal model for $K_{X}+\unicode[STIX]{x1D6E5}_{0}$ and $\unicode[STIX]{x1D6E4}_{0}=\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D6E5}_{0}$. Let ${\mathcal{C}}_{0}\subseteq {\mathcal{L}}_{A}(V)$ be a neighborhood around $\unicode[STIX]{x1D6E5}$, which is also a rational polytope. Since $\unicode[STIX]{x1D719}$ is $(K_{X}+\unicode[STIX]{x1D6E5}_{0})$-negative, by shirking ${\mathcal{C}}_{0}$ (without changing its dimension) around $\unicode[STIX]{x1D6E5}_{0}$ we may assume that $a(F,K_{X}+\unicode[STIX]{x1D6E5})<a(F,K_{Y}+\unicode[STIX]{x1D6E5})$ for all $\unicode[STIX]{x1D6E5}\in {\mathcal{C}}_{0}$ and for all $\unicode[STIX]{x1D719}$-exceptional divisors $F$. Note that $K_{Y}+\unicode[STIX]{x1D6E4}_{0}$ is KLT and $Y$ is $\mathbb{Q}$-factorial. Since KLT is an open condition, all nearby divisors of $\unicode[STIX]{x1D6E4}_{0}$ in $Y$ are also KLT. Therefore, by shrinking ${\mathcal{C}}_{0}$ further around $\unicode[STIX]{x1D6E5}_{0}$ we may assume that $K_{Y}+\unicode[STIX]{x1D6E4}$ is KLT for all $\unicode[STIX]{x1D6E5}\in {\mathcal{C}}_{0}$, where $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D6E5}$.

Replacing ${\mathcal{C}}$ by ${\mathcal{C}}_{0}$ we may assume that the rational polytope ${\mathcal{C}}^{\prime }=\unicode[STIX]{x1D719}_{\ast }({\mathcal{C}})$ is contained in ${\mathcal{L}}_{\unicode[STIX]{x1D719}_{\ast }A}(W)$, where $W=\unicode[STIX]{x1D719}_{\ast }V$. Note that $\unicode[STIX]{x1D719}_{\ast }A$ is not an ample divisor, however it is a big $\mathbb{Q}$-divisor on $Y$. By [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 3.7.3] there exists a rational affine linear isomorphism $L:W\rightarrow V^{\prime }$ on $Y$ and a general ample$/U$$\mathbb{Q}$-divisor $A^{\prime }$ on $Y$ such that $L({\mathcal{C}}^{\prime })\subseteq {\mathcal{L}}_{A^{\prime }}(V^{\prime })$, $L(\unicode[STIX]{x1D6E4}){\sim}_{\mathbb{Q},U}\unicode[STIX]{x1D6E4}$ for all $\unicode[STIX]{x1D6E4}\in {\mathcal{C}}^{\prime }$ and $K_{Y}+\unicode[STIX]{x1D6E4}$ is KLT for any $\unicode[STIX]{x1D6E4}\in L({\mathcal{C}}^{\prime })$.

By [Reference Birkar, Cascini, Hacon and McKernan5, Lemmas 3.6.9 and 3.6.10], any log minimal model of $(Y,L(\unicode[STIX]{x1D6E4}))$ over $U$ is also a log minimal model of $(X,\unicode[STIX]{x1D6E5})$ over $U$ for every $\unicode[STIX]{x1D6E5}\in {\mathcal{C}}$. Thus replacing $X$ by $Y$ and ${\mathcal{C}}$ by $L({\mathcal{C}}^{\prime })$ we may assume that $X$ is $\mathbb{Q}$-factorial and $K_{X}+\unicode[STIX]{x1D6E5}_{0}$ is $\unicode[STIX]{x1D70B}$-nef. Since $\unicode[STIX]{x1D6E5}_{0}$ is a big divisor, by the base-point free theorem [Reference Birkar and Waldron6, Theorem 1.2] $K_{X}+\unicode[STIX]{x1D6E5}_{0}$ has an ample model $\unicode[STIX]{x1D713}:X\rightarrow Z$. In particular, $K_{X}+\unicode[STIX]{x1D6E5}_{0}{\sim}_{\mathbb{R},Z}0$. By the case we have already proved, there exist finitely many birational maps $\unicode[STIX]{x1D719}_{i}:X{\dashrightarrow}Y_{i}$ over $Z$, $1\leqslant i\leqslant k$, such that for any $\unicode[STIX]{x1D6E5}\in {\mathcal{C}}\cap {\mathcal{E}}_{A,\unicode[STIX]{x1D713}}(V)$, there is an index $i$ such that $\unicode[STIX]{x1D719}_{i}$ is a log minimal model of $K_{X}+\unicode[STIX]{x1D6E5}$ over $Z$. Since there are only finitely many indices $1\leqslant i\leqslant k$, by shrinking ${\mathcal{C}}$ (without changing its dimension) if necessary, it follows from Corollary 4.8 that if $\unicode[STIX]{x1D6E5}\in {\mathcal{C}}$, then $\unicode[STIX]{x1D719}_{i}$ is a log minimal model for $K_{X}+\unicode[STIX]{x1D6E5}$ over $Z$ if and only if it is a log minimal model for $K_{X}+\unicode[STIX]{x1D6E5}$ over $U$.

Let $\unicode[STIX]{x1D6E5}\in {\mathcal{C}}\cap {\mathcal{E}}_{A,\unicode[STIX]{x1D70B}}(V)$. Then $\unicode[STIX]{x1D6E5}\in {\mathcal{C}}\cap {\mathcal{E}}_{A,\unicode[STIX]{x1D713}}(V)$, and there exists an index $1\leqslant j\leqslant k$ such that $\unicode[STIX]{x1D719}_{j}$ is a log minimal model for $K_{X}+\unicode[STIX]{x1D6E5}$ over $Z$. But then $\unicode[STIX]{x1D719}_{j}$ is a log minimal model for $K_{X}+\unicode[STIX]{x1D6E5}$ over $U$.◻

Proof. This result corresponds to [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 7.2].

By [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 3.7.4] for every $\unicode[STIX]{x1D6E5}\in {\mathcal{C}}$ there exists a $\unicode[STIX]{x1D6E5}^{\prime }\geqslant 0$ such that $K_{X}+\unicode[STIX]{x1D6E5}^{\prime }$ is KLT and $K_{X}+\unicode[STIX]{x1D6E5}^{\prime }{\sim}_{\mathbb{R},U}K_{X}+\unicode[STIX]{x1D6E5}$. Then by [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 3.6.9] $\unicode[STIX]{x1D713}:X{\dashrightarrow}Z$ is a weak log canonical model of $K_{X}+\unicode[STIX]{x1D6E5}$ over $U$ if and only if $\unicode[STIX]{x1D713}^{\prime }$ is a weak log canonical model of $K_{X}+\unicode[STIX]{x1D6E5}^{\prime }$ of over $U$. Therefore, by [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 3.7.4] we may assume that $K_{X}+\unicode[STIX]{x1D6E5}$ is KLT for every $\unicode[STIX]{x1D6E5}\in {\mathcal{C}}$.

Let $G\geqslant 0$ be a divisor such that it contains the support of every divisor in $V$ and $f:Y\rightarrow X$ a log resolution of $(X,G)$. For a $\unicode[STIX]{x1D6E5}\in {\mathcal{L}}_{A}(V)$ we can write

$$\begin{eqnarray}K_{Y}+\unicode[STIX]{x1D6E4}=f^{\ast }(K_{X}+\unicode[STIX]{x1D6E5})+E,\end{eqnarray}$$

where $\unicode[STIX]{x1D6E4}\geqslant 0$ and $E\geqslant 0$ have no common components, $f_{\ast }\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6E5}$ and $f_{\ast }E=0$.

If $\unicode[STIX]{x1D713}:X{\dashrightarrow}Z$ is a weak log canonical model of $K_{X}+\unicode[STIX]{x1D6E5}$ over $U$, then $\unicode[STIX]{x1D713}\circ f:Y{\dashrightarrow}Z$ is a weak log canonical model of $K_{Y}+\unicode[STIX]{x1D6E4}$ over $U$. Let ${\mathcal{C}}^{\prime }$ be the image of ${\mathcal{C}}$ under the map $\unicode[STIX]{x1D6E5}\rightarrow \unicode[STIX]{x1D6E4}$. Then ${\mathcal{C}}^{\prime }$ is a rational polytope and $K_{Y}+\unicode[STIX]{x1D6E4}$ is KLT for all $\unicode[STIX]{x1D6E4}\in {\mathcal{C}}^{\prime }$. In particular, $\mathbf{B}_{+}(f^{\ast }A/U)$ does not contain any LC centers of $K_{Y}+\unicode[STIX]{x1D6E4}$ for any $\unicode[STIX]{x1D6E4}\in {\mathcal{C}}^{\prime }$. Let $W$ be the subspace of $\operatorname{WDiv}_{\mathbb{R}}(Y)$ spanned by the strict transforms of the components of $G$ and the exceptional divisors of $f$. Then by [Reference Birkar, Cascini, Hacon and McKernan5, Lemmas 3.7.3 and 3.6.9] we may assume that there exists a general ample$/U$$\mathbb{Q}$-divisor $A^{\prime }$ on $Y$ such that ${\mathcal{C}}^{\prime }\subseteq {\mathcal{L}}_{A^{\prime }}(W)$. Replacing $X$ by $Y$ and ${\mathcal{C}}$ by ${\mathcal{C}}^{\prime }$ we assume that $X$ is smooth.

Let $H_{1}\geqslant 0,H_{2}\geqslant 0,\ldots ,H_{q}\geqslant 0$ be general ample$/U$$\mathbb{Q}$-Cartier divisor on $X$ such that they generate $\operatorname{WDiv}_{\mathbb{R}}(X)$ modulo numerical equivalence over $U$. Let $H=H_{1}+H_{2}+\cdots +H_{q}$. By [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 3.7.4] we may assume that if $\unicode[STIX]{x1D6E5}\in {\mathcal{C}}$, then $\unicode[STIX]{x1D6E5}$ contains the support of $H$. Let $W$ be the affine subspace of $\operatorname{WDiv}_{\mathbb{R}}(X)$ spanned by $V$ and the support of $H$. Let ${\mathcal{C}}^{\prime }$ be a rational polytope in ${\mathcal{L}}_{A}(W)$ containing ${\mathcal{C}}$ in its interior such that $K_{X}+\unicode[STIX]{x1D6E5}$ is KLT for all $\unicode[STIX]{x1D6E5}\in {\mathcal{C}}^{\prime }$.

Then by Proposition 4.9 there are finitely many rational maps $\unicode[STIX]{x1D719}_{i}:X{\dashrightarrow}Y_{i},\;1\leqslant i\leqslant k$ over $U$, such that for any $\unicode[STIX]{x1D6E5}^{\prime }\in {\mathcal{C}}^{\prime }\cap {\mathcal{E}}_{A,\unicode[STIX]{x1D70B}}(W)$ there exists an index $1\leqslant j\leqslant k$ such that $\unicode[STIX]{x1D719}_{j}$ is a log minimal model of $K_{X}+\unicode[STIX]{x1D6E5}^{\prime }$ over $U$. By Corollary 4.7 for each index $1\leqslant i\leqslant k$ there are finitely many projective contractions $f_{i,m}:Y_{i}\rightarrow Z_{i,m}$ over $U$ such that if $\unicode[STIX]{x1D6E5}^{\prime }\in {\mathcal{W}}_{\unicode[STIX]{x1D719}_{i},A,\unicode[STIX]{x1D70B}}(W)$ and there is a contraction $f:Y_{i}\rightarrow Z$ over $U$, with

$$\begin{eqnarray}K_{Y_{i}}+\unicode[STIX]{x1D6E4}_{i}=K_{Y_{i}}+\unicode[STIX]{x1D719}_{i,\ast }\unicode[STIX]{x1D6E5}^{\prime }{\sim}_{\mathbb{ R},U}f^{\ast }D,\end{eqnarray}$$

for some ample$/U$$\mathbb{R}$-divisor $D$ on $Z$, then there is an index $(i,m)$ and an isomorphism $\unicode[STIX]{x1D709}:Z_{i,m}\rightarrow Z$ such that $f=\unicode[STIX]{x1D709}\circ f_{i,m}$. Let $\unicode[STIX]{x1D713}_{j}:X{\dashrightarrow}Z_{j}$, $1\leqslant j\leqslant l$ be the finitely many rational maps obtained by composing every $\unicode[STIX]{x1D719}_{i}$ with every $f_{i,j}$. Pick $\unicode[STIX]{x1D6E5}\in {\mathcal{C}}$ and let $\unicode[STIX]{x1D713}:X{\dashrightarrow}Z$ be a weak log canonical model of $K_{X}+\unicode[STIX]{x1D6E5}$ over $U$. Then $K_{Z}+\unicode[STIX]{x1D6E9}$ is KLT and nef over $U$, where $\unicode[STIX]{x1D6E9}=\unicode[STIX]{x1D713}_{\ast }\unicode[STIX]{x1D6E5}$. Since $K_{Z}+\unicode[STIX]{x1D6E9}$ is KLT, by [Reference Birkar3, Theorem 1.6] $Z$ has a $\mathbb{Q}$-factorization $\unicode[STIX]{x1D702}:Y^{\prime }\rightarrow Z$, where $\unicode[STIX]{x1D702}$ is a small birational morphism and $Y^{\prime }$ is $\mathbb{Q}$-factorial. Then by [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 3.6.12] we may find $\unicode[STIX]{x1D6E5}^{\prime }\in {\mathcal{C}}^{\prime }\cap {\mathcal{E}}_{A,\unicode[STIX]{x1D70B}}(W)$ such that $\unicode[STIX]{x1D713}$ is an ample model of $K_{X}+\unicode[STIX]{x1D6E5}^{\prime }$ over $U$. Pick an index $1\leqslant i\leqslant k$ such that $\unicode[STIX]{x1D719}_{i}$ is a log minimal model of $K_{X}+\unicode[STIX]{x1D6E5}^{\prime }$ over $U$. By [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 3.6.6(4)] there exists a contraction $f:Y_{i}\rightarrow Z$ such that

$$\begin{eqnarray}K_{Y_{i}}+\unicode[STIX]{x1D6E4}_{i}=f^{\ast }(K_{Z}+\unicode[STIX]{x1D6E9}^{\prime }),\end{eqnarray}$$

where $\unicode[STIX]{x1D6E4}_{i}=\unicode[STIX]{x1D719}_{i\ast }\unicode[STIX]{x1D6E5}^{\prime }$ and $\unicode[STIX]{x1D6E9}^{\prime }=\unicode[STIX]{x1D713}_{\ast }\unicode[STIX]{x1D6E5}^{\prime }$. As $K_{Z}+\unicode[STIX]{x1D6E9}^{\prime }$ is ample over $U$, it follows that there is an index $m$ and isomorphism $\unicode[STIX]{x1D709}:Z_{i,m}\rightarrow Z$ such that $f=\unicode[STIX]{x1D709}\circ f_{i,m}$. But then

$$\begin{eqnarray}\unicode[STIX]{x1D713}=f\circ \unicode[STIX]{x1D719}_{i}=\unicode[STIX]{x1D709}\circ f_{i,m}\circ \unicode[STIX]{x1D719}_{i}=\unicode[STIX]{x1D709}\circ \unicode[STIX]{x1D713}_{j},\end{eqnarray}$$

for some index $1\leqslant j\leqslant l$.◻

Proof. This result corresponds to [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 7.3].

Since ${\mathcal{L}}_{A}(V)$ is a rational polytope, the statement follows from Theorem 4.11.◻

Proof of Theorem 1.2.

First we prove $(1)$ and $(2)$. Since ample models are unique by [Reference Birkar, Cascini, Hacon and McKernan5, Lemma 3.6.6(1)], by Corollaries 4.12 and 4.7 it suffices to prove that if $\unicode[STIX]{x1D6E5}\in {\mathcal{E}}_{A,\unicode[STIX]{x1D70B}}(V)$, then $K_{X}+\unicode[STIX]{x1D6E5}$ has both a log minimal model over $U$ and an ample model over $U$.

By [Reference Birkar, Cascini, Hacon and McKernan5, Lemmas 3.7.5 and 3.6.9] we may assume that $K_{X}+\unicode[STIX]{x1D6E5}$ is KLT. Then [Reference Birkar3, Theorem 1.2] gives the existence of a log minimal model of $K_{X}+\unicode[STIX]{x1D6E5}$ over $U$, and the existence of the ample model follows from Lemma 4.1.

Part $(3)$ follows as in the proof of Corollary 4.7. Indeed if $\unicode[STIX]{x1D6E5}_{1},\unicode[STIX]{x1D6E5}_{2},\ldots ,\unicode[STIX]{x1D6E5}_{k}$ are the vertices of ${\mathcal{W}}_{\unicode[STIX]{x1D719},A,\unicode[STIX]{x1D70B}}(V)$ for a birational contraction $\unicode[STIX]{x1D719}:X{\dashrightarrow}Y$, and $\unicode[STIX]{x1D6E5}$ and $\unicode[STIX]{x1D6E5}^{\prime }$ are two divisors lying in the interior of ${\mathcal{W}}_{\unicode[STIX]{x1D719},A,\unicode[STIX]{x1D70B}}(V)$, then for a given (fixed) $0\leqslant l\leqslant k$ we can write $\unicode[STIX]{x1D6E5}=\sum \unicode[STIX]{x1D707}_{j}\unicode[STIX]{x1D6E5}_{j}$ and $\unicode[STIX]{x1D6E5}^{\prime }=\sum \unicode[STIX]{x1D707}_{j}^{\prime }\unicode[STIX]{x1D6E5}_{j}$ for some $\unicode[STIX]{x1D707}_{l}>0$. Therefore, from the base-point free theorem [Reference Birkar and Waldron6, Theorem 1.2] it follows that a curve $C$ is contracted by $K_{Y}+\unicode[STIX]{x1D6E4}=K_{Y}+\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D6E5}$ if and only if it is contracted by $K_{Y}+\unicode[STIX]{x1D6E4}^{\prime }=K_{Y}+\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D6E5}^{\prime }$. In particular, the interior of ${\mathcal{W}}_{\unicode[STIX]{x1D719},A,\unicode[STIX]{x1D70B}}(V)$ is contained in a single ample model ${\mathcal{A}}_{\unicode[STIX]{x1D713},A,\unicode[STIX]{x1D70B}}(V)$ for some projective contraction $\unicode[STIX]{x1D713}:Y\rightarrow Z$. Therefore, ${\mathcal{W}}_{\unicode[STIX]{x1D719},A,\unicode[STIX]{x1D70B}}(V)\subseteq \bar{{\mathcal{A}}}_{\unicode[STIX]{x1D719},A,\unicode[STIX]{x1D70B}}(V)$.

Part $(4)$ follows combining Part $(1),(2)$ and $(3)$.◻

Proof of Corollary 1.3.

Let $V$ be the finite dimensional affine subspace of $\operatorname{WDiv}(X)_{\mathbb{R}}$ generated by the irreducible components of $\unicode[STIX]{x1D6E5}_{1},\unicode[STIX]{x1D6E5}_{2},\ldots ,\unicode[STIX]{x1D6E5}_{k}$. Then by Theorem 1.2 there exist finitely many rational maps $\unicode[STIX]{x1D719}_{i}:X{\dashrightarrow}Y_{i}$ over $U$, $1\leqslant i\leqslant q$, such that for every $\unicode[STIX]{x1D6E5}\in {\mathcal{E}}_{A,\unicode[STIX]{x1D70B}}(V)$ there is an index $1\leqslant j\leqslant q$ such that $\unicode[STIX]{x1D719}_{j}$ is a log minimal model of $K_{X}+\unicode[STIX]{x1D6E5}$ over $U$. Let ${\mathcal{C}}\subseteq {\mathcal{L}}_{A}(V)$ be the (rational) polytope spanned by $\unicode[STIX]{x1D6E5}_{1},\unicode[STIX]{x1D6E5}_{2},\ldots ,\unicode[STIX]{x1D6E5}_{k}$ and let

$$\begin{eqnarray}{\mathcal{C}}_{j}={\mathcal{W}}_{\unicode[STIX]{x1D719}_{j},A,\unicode[STIX]{x1D70B}}(V)\cap {\mathcal{C}}.\end{eqnarray}$$

Then ${\mathcal{C}}_{j}$ is a rational polytope. Note that the ring ${\mathcal{R}}(\unicode[STIX]{x1D70B},D^{\bullet })$ is finitely generated if and only if the rings corresponding to the (rational) polytope ${\mathcal{C}}_{j}$ are finitely generated for all $j=1,2,\ldots ,q$. Therefore, replacing $\unicode[STIX]{x1D6E5}_{1},\unicode[STIX]{x1D6E5}_{2},\ldots ,\unicode[STIX]{x1D6E5}_{k}$ by the vertices of ${\mathcal{C}}_{j}$ we may assume that ${\mathcal{C}}={\mathcal{C}}_{j}$ and $\unicode[STIX]{x1D719}:X{\dashrightarrow}Y$ is a log minimal model for all $\unicode[STIX]{x1D6E5}_{1},\unicode[STIX]{x1D6E5}_{2},\ldots ,\unicode[STIX]{x1D6E5}_{k}$. Let $\unicode[STIX]{x1D70B}^{\prime }:Y\rightarrow U$ be the induced morphism. Let $\unicode[STIX]{x1D6E4}_{i}=\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D6E5}_{i}$ for all $1\leqslant i\leqslant k$. Let $g:W\rightarrow X$ and $h:W\rightarrow Y$ be a resolution of the graph of $\unicode[STIX]{x1D719}:X{\dashrightarrow}Y$.

Then we have

$$\begin{eqnarray}g^{\ast }(K_{X}+\unicode[STIX]{x1D6E5}_{i})=h^{\ast }(K_{Y}+\unicode[STIX]{x1D6E4}_{i})+F_{i}\end{eqnarray}$$

for $1\leqslant i\leqslant k$.

Note that $F_{i}\geqslant 0$ is an effective $h$-exceptional divisor, since $\unicode[STIX]{x1D719}:X{\dashrightarrow}Y$ is a log minimal model$/U$ for $\unicode[STIX]{x1D6E5}_{i}$, $1\leqslant i\leqslant k$. Let $m>0$ be positive integer such that $G_{i}=m(K_{Y}+\unicode[STIX]{x1D6E4}_{i})$ and $D_{i}=m(K_{X}+\unicode[STIX]{x1D6E5}_{i})$ are both Cartier for all $1\leqslant i\leqslant k$. Then from the projection formula it follows that

$$\begin{eqnarray}{\mathcal{R}}(\unicode[STIX]{x1D70B},D^{\bullet })\cong {\mathcal{R}}(\unicode[STIX]{x1D70B}^{\prime },G^{\bullet }).\end{eqnarray}$$

Therefore, replacing $X$ by $Y$ we may assume that $X$ is $\mathbb{Q}$-factorial and $K_{X}+\unicode[STIX]{x1D6E5}_{i}$ is KLT and nef over $U$ for all $1\leqslant i\leqslant k$. Since $\unicode[STIX]{x1D6E5}_{i}$ is big for all $1\leqslant i\leqslant k$, by [Reference Birkar and Waldron6, Theorem 1.2] $K_{X}+\unicode[STIX]{x1D6E5}_{i}$ is semi-ample for all $1\leqslant i\leqslant k$. Therefore, it follows that the ring ${\mathcal{R}}(\unicode[STIX]{x1D70B},D^{\bullet })$ is finitely generated.◻

Proof. This is [Reference Lazarsfeld15, Theorem 11.4.21]. The main ingredients of the proof of [Reference Lazarsfeld16, Theorem 11.4.21] are the Fujita’s approximation theorem, Hodge type inequalities [Reference Lazarsfeld15, Corollary 1.6.3, Lemma 11.4.22], and the continuity of volume [Reference Lazarsfeld15, Theorem 2.2.44, Example 2.2.47]. The Fujita’s approximation theorem is known in positive characteristic due to [Reference Takagi19] and the other two results are also known to hold in positive characteristic (their proofs in [Reference Lazarsfeld15] work in arbitrary characteristic). As a result, the proof of [Reference Lazarsfeld16, Theorem 11.4.21] holds in arbitrary characteristic. ◻

Proof of the Theorem 1.4.

It is well known that $\overline{\operatorname{Eff}}(X)\subseteq \overline{\operatorname{SNM}}(X)^{\ast }$. By contradiction, assume that the inclusion $\overline{\operatorname{Eff}}(X)\subsetneq \overline{\operatorname{SNM}}(X)^{\ast }$ is strict. Then there exists a class $\unicode[STIX]{x1D709}\in N^{1}(X)_{\mathbb{R}}$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D709}\in \text{boundary }(\overline{\operatorname{Eff}}(X))\quad \text{and}\quad \unicode[STIX]{x1D709}\in \text{interior }(\overline{\operatorname{SNM}}(X)^{\ast }).\end{eqnarray}$$

Fix an ample class $h$ such that $h\pm 2\unicode[STIX]{x1D709}$ is ample. Since $\unicode[STIX]{x1D709}$ lies in the interior of $\overline{\operatorname{SNM}}(X)^{\ast }$, there exists $\unicode[STIX]{x1D700}>0$ such that $\unicode[STIX]{x1D709}-\unicode[STIX]{x1D700}h\in \overline{\operatorname{SNM}}(X)^{\ast }$. In particular,

(5.1)$$\begin{eqnarray}\frac{(\unicode[STIX]{x1D709}\cdot \unicode[STIX]{x1D6FE})}{(h\cdot \unicode[STIX]{x1D6FE})}\geqslant \unicode[STIX]{x1D700}\end{eqnarray}$$

for every strongly movable class $\unicode[STIX]{x1D6FE}$ on $X$. Now consider the class

$$\begin{eqnarray}\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FF}}=\unicode[STIX]{x1D709}+\unicode[STIX]{x1D6FF}h\quad \text{for }\unicode[STIX]{x1D6FF}>0.\end{eqnarray}$$

Since $\unicode[STIX]{x1D709}$ is pseudo-effective, $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FF}}$ is big for all $\unicode[STIX]{x1D6FF}>0$. For small $\unicode[STIX]{x1D6FF}>0$ consider a Fujita approximation (by Theorem 5.3)

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FF}}:X_{\unicode[STIX]{x1D6FF}}^{\prime }\rightarrow X,\qquad \unicode[STIX]{x1D707}^{\ast }(\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FF}})=a_{\unicode[STIX]{x1D6FF}}+e_{\unicode[STIX]{x1D6FF}}\end{eqnarray}$$

such that

(5.2)$$\begin{eqnarray}\operatorname{vol}_{X_{\unicode[STIX]{x1D6FF}}^{\prime }}(a_{\unicode[STIX]{x1D6FF}})\geqslant \operatorname{vol}_{X}(\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FF}})-\unicode[STIX]{x1D6FF}^{2n}.\end{eqnarray}$$

Since $\unicode[STIX]{x1D6FF}^{2n}$ is a polynomial of degree $2n$ in $\unicode[STIX]{x1D6FF}$ and $\operatorname{vol}_{X}(\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FF}})=(\unicode[STIX]{x1D709}+\unicode[STIX]{x1D6FF}h)^{n}$ is a polynomial of degree $n$ in $\unicode[STIX]{x1D6FF}$ and $(h^{n})>0$, for $\unicode[STIX]{x1D6FF}>0$ sufficiently small we may assume that $0<\unicode[STIX]{x1D6FF}^{2n}<\frac{1}{2}\operatorname{vol}_{X}(\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FF}})$. In particular,

(5.3)$$\begin{eqnarray}\operatorname{vol}_{X_{\unicode[STIX]{x1D6FF}}^{\prime }}(a_{\unicode[STIX]{x1D6FF}})\geqslant \frac{1}{2}\cdot \operatorname{vol}_{X}(\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FF}})\geqslant \frac{\unicode[STIX]{x1D6FF}^{n}}{2}\cdot (h^{n}).\end{eqnarray}$$

Consider the strongly movable class

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D6FF}}=\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FF},\ast }(a_{\unicode[STIX]{x1D6FF}}^{n-1}).\end{eqnarray}$$

Then by the projection formula and [Reference Lazarsfeld15, Corollary 1.6.3(ii)], we get

(5.4)$$\begin{eqnarray}\displaystyle (h\cdot \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D6FF}})_{X} & = & \displaystyle (\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FF}}^{\ast }(h)\cdot a_{\unicode[STIX]{x1D6FF}}^{n-1})_{X_{\unicode[STIX]{x1D6FF}}^{\prime }}\nonumber\\ \displaystyle & {\geqslant} & \displaystyle (h^{n})_{X}^{1/n}\cdot (a_{\unicode[STIX]{x1D6FF}}^{n})_{X_{\unicode[STIX]{x1D6FF}}^{\prime }}^{(n-1)/n}.\end{eqnarray}$$

On the other hand,

(5.5)$$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D709}\cdot \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D6FF}})_{X} & {\leqslant} & \displaystyle (\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FF}}\cdot \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D6FF}})_{X}\nonumber\\ \displaystyle & = & \displaystyle (\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FF}}^{\ast }(\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FF}})\cdot a_{\unicode[STIX]{x1D6FF}}^{n-1})_{X_{\unicode[STIX]{x1D6FF}}^{\prime }}\nonumber\\ \displaystyle & = & \displaystyle (a_{\unicode[STIX]{x1D6FF}}^{n})_{X_{\unicode[STIX]{x1D6FF}}^{\prime }}+(e_{\unicode[STIX]{x1D6FF}}\cdot a_{\unicode[STIX]{x1D6FF}}^{n-1})_{X_{\unicode[STIX]{x1D6FF}}^{\prime }}.\end{eqnarray}$$

Now $h\pm \unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FF}}$ is ample provided that $\unicode[STIX]{x1D6FF}<\frac{1}{2}$. Therefore, by Theorem 5.4 and (5.2) we get,

(5.6)$$\begin{eqnarray}\displaystyle (e_{\unicode[STIX]{x1D6FF}}\cdot a_{\unicode[STIX]{x1D6FF}}^{n-1})_{X_{\unicode[STIX]{x1D6FF}}^{\prime }} & {\leqslant} & \displaystyle (20\cdot (h^{n})_{X}\cdot (\operatorname{vol}_{X}(\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FF}})-\operatorname{vol}_{X_{\unicode[STIX]{x1D6FF}}^{\prime }}(a_{\unicode[STIX]{x1D6FF}})))^{1/2}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle C_{1}\cdot \unicode[STIX]{x1D6FF}^{n},\end{eqnarray}$$

where $C_{1}=20\cdot (h^{n})_{X}>0$ is independent of $\unicode[STIX]{x1D6FF}$.

Now from (5.3), (5.4), (5.5) and (5.6) we get

(5.7)$$\begin{eqnarray}\displaystyle 0\leqslant \frac{(\unicode[STIX]{x1D709}\cdot \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D6FF}})}{(h\cdot \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D6FF}})} & {\leqslant} & \displaystyle \frac{(a_{\unicode[STIX]{x1D6FF}}^{n})_{X_{\unicode[STIX]{x1D6FF}}^{\prime }}+C_{1}\cdot \unicode[STIX]{x1D6FF}^{n}}{(h^{n})_{X}^{1/n}\cdot (a_{\unicode[STIX]{x1D6FF}}^{n})_{X_{\unicode[STIX]{x1D6FF}}^{\prime }}^{(n-1)/n}}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \frac{1}{(h^{n})_{X}^{1/n}}\cdot (a_{\unicode[STIX]{x1D6FF}}^{n})_{X_{\unicode[STIX]{x1D6FF}}^{\prime }}^{1/n}+\frac{C_{1}\cdot \unicode[STIX]{x1D6FF}^{n}}{(h^{n})_{X}^{1/n}}\cdot \frac{2^{(n-1)/n}}{\unicode[STIX]{x1D6FF}^{n-1}\cdot (h^{n})_{X}^{(n-1)/n}}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle C_{2}\cdot (a_{\unicode[STIX]{x1D6FF}}^{n})_{X_{\unicode[STIX]{x1D6FF}}^{\prime }}^{1/n}+C_{3}\cdot \unicode[STIX]{x1D6FF},\end{eqnarray}$$

where $C_{2}$ and $C_{3}$ are constants independent of $\unicode[STIX]{x1D6FF}$.

Now recall that $\unicode[STIX]{x1D709}$ lies on the boundary of the big cone. Therefore, by the continuity of volume [Reference Lazarsfeld15, Theorem 2.2.44]

$$\begin{eqnarray}\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\operatorname{vol}_{X}(\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FF}})=\operatorname{vol}_{X}(\unicode[STIX]{x1D709})=0,\end{eqnarray}$$

and hence also

$$\begin{eqnarray}\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\operatorname{vol}_{X_{\unicode[STIX]{x1D6FF}}^{\prime }}(a_{\unicode[STIX]{x1D6FF}})=\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}(a_{\unicode[STIX]{x1D6FF}}^{n})_{X_{\unicode[STIX]{x1D6FF}}^{\prime }}=0.\end{eqnarray}$$

Thus from (5.7) we see that $(\unicode[STIX]{x1D709}\cdot \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D6FF}})/(h\cdot \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D6FF}})\rightarrow 0$ as $\unicode[STIX]{x1D6FF}\rightarrow 0^{+}$. But this contradicts (5.1), and completes the proof.◻

Proof. From the definition of nef curves it is clear that $\overline{\operatorname{NF}}(X)=\overline{\operatorname{Eff}}(X)^{\ast }$, and $\overline{\operatorname{Eff}}(X)^{\ast }=\overline{\operatorname{SNM}}(X)$ by Theorem 1.4. Thus we only need to prove that $\overline{\operatorname{NM}}(X)=\overline{\operatorname{SNM}}(X)$. The inclusion $\overline{\operatorname{SNM}}(X)\subseteq \overline{\operatorname{NM}}(X)$ is clear. We prove the other inclusion. Let $\unicode[STIX]{x1D6FE}\in \overline{\operatorname{NM}}(X)$. Then there exists a movable curve $C$ such that $\unicode[STIX]{x1D6FE}=[C]$. By Theorem 1.4 it is enough to show that $D\cdot \unicode[STIX]{x1D6FE}\geqslant 0$ for all effective Cartier divisors. Since $C$ belongs to an algebraic family of curves $\{C_{t}\}\text{}_{t\in T}$ such that $\bigcup _{t\in T}C_{t}$ covers a dense subset of $X$, we can find a curve $C_{t_{1}}$ in this family such that $C_{t_{1}}\nsubseteq \operatorname{Supp}(D)$. Thus $D\cdot \unicode[STIX]{x1D6FE}=D\cdot C_{t_{1}}\geqslant 0$, that is $\unicode[STIX]{x1D6FE}\in \overline{\operatorname{Eff}}(X)^{\ast }=\overline{\operatorname{SNM}}(X)$.◻

Proof of Theorem 1.6.

If there exists an algebraic family of $K_{X}$-negative rational curves covering a dense subset of $X$, then from Theorem 1.4 and Corollary 5.5 it follows that $K_{X}$ it not pseudo-effective.

Now assume that $K_{X}$ is not pseudo-effective. Then by Theorem 1.4 and Corollary 5.5, there exist a movable class $\unicode[STIX]{x1D6FE}\in \overline{\operatorname{NM}}(X)$ and an algebraic family of irreducible curves $\{C_{t}\}\text{}_{t\in T}$ representing $\unicode[STIX]{x1D6FE}$ such that $K_{X}\cdot C_{t}<0$ for all $t\in T$ and $\bigcup _{t\in T}C_{t}\subseteq X$ is dense in $X$. We fix a very ample divisor $H$ in $X$. Then by [Reference Miyaoka18, Theorem 1.1] there exist rational curves $\{C_{s}^{\prime }\}\text{}_{s\in S}$ of bounded degree through every point of $\bigcup _{t\in T}C_{t}\subseteq X$ such that

(5.8)$$\begin{eqnarray}\begin{array}{@{}c@{}}\displaystyle H\cdot C_{s}^{\prime }\leqslant \frac{2\operatorname{dim}X\cdot (H\cdot \unicode[STIX]{x1D6FE})}{-K_{X}\cdot \unicode[STIX]{x1D6FE}}\qquad \text{and}\\ \displaystyle 0<-(K_{X}\cdot C_{s}^{\prime })\leqslant \operatorname{dim}X+1\quad \text{ for all }s\in S.\end{array}\end{eqnarray}$$

By [Reference Kollár and Mori14, Corollary 1.19(3)] there are finitely many subclasses of these rational curves, say $\{C_{s}^{\prime }\}\text{}_{s\in S_{i}}$, $1\leqslant i\leqslant n,S_{i}\subseteq S\text{ and }\coprod _{i=1}^{n}S_{i}=S$, such that for each fixed $i$, any two curves in $\{C_{s}^{\prime }\}\text{}_{s\in S_{i}}$ are numerically equivalent. Now since $\bigcup _{s\in S}C_{s}^{\prime }\subseteq X$ is dense in $X$, it follows that one of these subclasses, say $\{C_{\unicode[STIX]{x1D706}}^{\prime }\}\text{}_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}$, $\unicode[STIX]{x1D6EC}=S_{i_{k}}$ for some $i_{k}\in \{1,2,\ldots ,n\}$ has the property that $\bigcup _{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}C_{\unicode[STIX]{x1D706}}^{\prime }\subseteq X$ is dense in $X$. Let $d:=H\cdot C_{\unicode[STIX]{x1D706}}^{\prime }$. Then the curves $\{C_{\unicode[STIX]{x1D706}}^{\prime }\}\text{}_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}$ belong to the family $\operatorname{Univ}_{d}(\mathbb{P}^{1},X)\rightarrow \operatorname{Hom}_{d}(\mathbb{P}^{1},X)$, where $\operatorname{Hom}_{d}(\mathbb{P}^{1},X)$ is the scheme of degree $d$ morphisms $\mathbb{P}^{1}\rightarrow X$. Let $V^{\prime }\subseteq \operatorname{Hom}_{d}(\mathbb{P}^{1},X)$ be the connected component of $\operatorname{Hom}_{d}(\mathbb{P}^{1},X)$ which contains all the points corresponding to the curves $\{C_{\unicode[STIX]{x1D706}}^{\prime }\}\text{}_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}$ (since all $C_{\unicode[STIX]{x1D706}}^{\prime }$’s are numerically equivalent, they are contained in a connected component). Let ${\mathcal{U}}^{\prime }=V^{\prime }\times _{\operatorname{Hom}_{d}(\mathbb{P}^{1},X)}\operatorname{Univ}_{d}(\mathbb{P}^{1},X)$. Then ${\mathcal{U}}^{\prime }\rightarrow V^{\prime }$ is a (flat) family of rational curves $\unicode[STIX]{x1D6E4}_{t}$ such that $K_{X}\cdot \unicode[STIX]{x1D6E4}_{t}<0$ and $\bigcup _{t\in V^{\prime }}\unicode[STIX]{x1D6E4}_{t}\subseteq X$ is dense in $X$. Finally, let $V$ be an irreducible component of $V^{\prime }$ such that $\bigcup _{t\in V}\unicode[STIX]{x1D6E4}_{t}\subseteq X$ is dense in $X$. Set ${\mathcal{U}}=V\times _{V^{\prime }}{\mathcal{U}}^{\prime }=V\times _{\operatorname{Hom}_{d}(\mathbb{P}^{1},X)}\operatorname{Univ}_{d}(\mathbb{P}^{1},X)$. Then ${\mathcal{U}}\rightarrow V$ is a (flat) algebraic family of rational curves satisfying the required conditions.◻

Proof of Corollary 1.7.

Following the notations as in the proof of Theorem 1.6 we see that the evaluation map $\operatorname{ev}:\mathbb{P}^{1}\times _{k}V\rightarrow X$ is a dominant morphism, where $V$ is an irreducible variety. Thus by [Reference Debarre9, Remark 4.2(2)] $X$ is uniruled.◻

Proof. Let $H\geqslant 0$ be an ample $\mathbb{R}$-divisor on $X$ such that $K_{X}+\unicode[STIX]{x1D6E5}+H$ is KLT and nef (using Theorem 3.1). Since $K_{X}+\unicode[STIX]{x1D6E5}$ is pseudo-effective, by [Reference Birkar and Waldron6, Theorem 1.6] $(K_{X}+\unicode[STIX]{x1D6E5})$-MMP with the scaling of $H$ terminates with a log minimal model $\unicode[STIX]{x1D719}^{\prime }:X{\dashrightarrow}X^{\prime }$ such that $K_{X^{\prime }}+\unicode[STIX]{x1D6E5}^{\prime }=\unicode[STIX]{x1D719}_{\ast }^{\prime }(K_{X}+\unicode[STIX]{x1D6E5})$ is nef. Since $\unicode[STIX]{x1D6E5}^{\prime }$ is a big divisor, by perturbing $\unicode[STIX]{x1D6E5}^{\prime }$, from the base-point free theorem [Reference Birkar and Waldron6, Theorem 1.2] it follows that $K_{X^{\prime }}+\unicode[STIX]{x1D6E5}^{\prime }$ is semi-ample. Therefore, there exists a projective morphism $f:X^{\prime }\rightarrow Y$ and an ample $\mathbb{R}$-divisor $L$ on $Y$ such that $K_{X^{\prime }}+\unicode[STIX]{x1D6E5}^{\prime }{\sim}_{\mathbb{R}}f^{\ast }L$ (see [Reference Fujino10, Lemma 4.13]). Write $\unicode[STIX]{x1D6E5}^{\prime }=\unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D6E5}\equiv A^{\prime }+B^{\prime }$, where $A^{\prime }$ is an ample $\mathbb{R}$-divisor. Then $(X^{\prime },\unicode[STIX]{x1D6E5}^{\prime }+\unicode[STIX]{x1D700}B^{\prime })$ is KLT for $0<\unicode[STIX]{x1D700}\ll 1$. In particular, $(X^{\prime },(1-\unicode[STIX]{x1D700})\unicode[STIX]{x1D6E5}^{\prime }+\unicode[STIX]{x1D700}B^{\prime })$ is KLT. Let $G^{\prime }=(1-\unicode[STIX]{x1D700})\unicode[STIX]{x1D6E5}^{\prime }+\unicode[STIX]{x1D700}B^{\prime }\geqslant 0$. Then $(X^{\prime },G^{\prime })$ is KLT and $K_{X^{\prime }}+G^{\prime }\equiv _{f}-\unicode[STIX]{x1D700}A^{\prime }$.◻

Proof. This result corresponds to [Reference Lehmann17, Theorem 3.2].

For a given $\unicode[STIX]{x1D6E4}\in {\mathcal{E}}_{A}$ the existence of $\unicode[STIX]{x1D719}_{i}$ and $g_{i}$ is clear from Lemma 6.3. Then from Theorem 1.2 and Remark 6.4 it follows that there are only finitely many such $\unicode[STIX]{x1D719}_{i}$ and $g_{i}$ for all $\unicode[STIX]{x1D6E4}\in {\mathcal{E}}_{A}$.◻

Proof. This result corresponds to [Reference Lehmann17, Proposition 3.3].

Since $B$ is a big $\mathbb{R}$-divisor, $B\equiv H+E$ for some ample $\mathbb{R}$-divisor $H\geqslant 0$ and an effective $\mathbb{R}$-divisor $E$. Then $K_{X}+\unicode[STIX]{x1D6E5}+B+\unicode[STIX]{x1D700}E$ is KLT for $0<\unicode[STIX]{x1D700}\ll 1$. In particular, $K_{X}+\unicode[STIX]{x1D6E5}+(1-\unicode[STIX]{x1D700})B+\unicode[STIX]{x1D700}E$ is KLT. Let $A\geqslant 0$ be an ample $\mathbb{Q}$-divisor such that $\unicode[STIX]{x1D700}H-A$ is ample and $K_{X}+\unicode[STIX]{x1D6E5}+A+(1-\unicode[STIX]{x1D700})B+\unicode[STIX]{x1D700}E$ is KLT, by Theorem 3.1. Let $\{H_{j}\geqslant 0\}\text{}_{j=1}^{m}$ be a finite set of ample $\mathbb{Q}$-divisors such that the convex hull of the classes $[H_{j}]$’s contains an open set around $[\unicode[STIX]{x1D700}H-A]$ in $N^{1}(X)$. Let $U^{\prime }$ be an open neighborhood of $[B-A]$ contained in the convex hull of $[B-\unicode[STIX]{x1D700}H+H_{j}]$’s. We apply [Reference Lehmann17, Lemma 3.1] to $K_{X}+\unicode[STIX]{x1D6E5}+A+(1-\unicode[STIX]{x1D700})B+\unicode[STIX]{x1D700}E$ and $H_{j}$’s to obtain finitely many ample $\mathbb{Q}$-divisors $0\leqslant W_{j}{\sim}_{\mathbb{Q}}H_{j}$. Let $V$ be the vector space of $\mathbb{R}$-divisors spanned by the irreducible components of $(1-\unicode[STIX]{x1D700})B+\unicode[STIX]{x1D700}E$ and of the $W_{j}$’s. Therefore, $V$ is a finite dimensional subspace of $\operatorname{WDiv}(X)_{\mathbb{R}}$ such that every class in $U^{\prime }$ has an effective representative $\unicode[STIX]{x1D6E4}\in V$ with $(X,\unicode[STIX]{x1D6E5}+A+\unicode[STIX]{x1D6E4})$ KLT.

Let $\unicode[STIX]{x1D6E4}\in V$ be a representative of a class in $U^{\prime }$ such that $D=K_{X}+\unicode[STIX]{x1D6E5}+A+\unicode[STIX]{x1D6E4}$ is KLT. We can run the $D$-MMP with the scaling of an ample divisor. If $D$ lies on the boundary of $\overline{\operatorname{Eff}}(X)$, then by Proposition 6.5 there exists a birational contraction $\unicode[STIX]{x1D719}_{i}:X{\dashrightarrow}X_{i}$ and a log Fano fibration $g_{i}:X_{i}\rightarrow Z_{i}$ such that $K_{X_{i}}+{\unicode[STIX]{x1D719}_{i}}_{\ast }(\unicode[STIX]{x1D6E5}+A+\unicode[STIX]{x1D6E4}){\sim}_{\mathbb{R}}f^{\ast }L_{i}$ for some ample $\mathbb{R}$-divisor $L_{i}$ on $Z_{i}$. Let $C_{i}^{\prime }$ be a curve on a general fiber of $g_{i}$. Note that the exceptional locus $\operatorname{Ex}(\unicode[STIX]{x1D719}_{i}^{-1})$ where $\unicode[STIX]{x1D719}_{i}^{-1}:X_{i}{\dashrightarrow}X$ is not an isomorphism intersects the general fiber of $g_{i}$ along at least codimension $2$ subsets (see [Reference Birkar and Waldron6, Lemma 2.4]). Therefore, by choosing $C_{i}^{\prime }$ sufficiently general we see that $\unicode[STIX]{x1D719}_{i}:X_{i}{\dashrightarrow}X$ is an isomorphism in a neighborhood of $C_{i}^{\prime }$ and $C_{i}^{\prime }$ belongs to a family of curves dominating $X_{i}$. Let $C_{i}$ be the image of $C_{i}^{\prime }$ under $\unicode[STIX]{x1D719}_{i}^{-1}$. Then $C_{i}$ is a movable curve on $X$ and $D\cdot C_{i}=(K_{X_{i}}+{\unicode[STIX]{x1D719}_{i}}_{\ast }(\unicode[STIX]{x1D6E5}+A+\unicode[STIX]{x1D6E4}))\cdot C_{i}^{\prime }=f^{\ast }L_{i}\cdot C_{i}^{\prime }=0$.