Termination of a minimal model program

Question

I am reading "The dual complex of singularities" by de Fernex, Kollár and Xu and in the proof of Corollary 24 I have encountered a bit of reasoning that confuses me.

Let $(X, \Delta)$ be a $\mathbb{Q}$-factorial pair and let $0 \in X$ be a point such that $X$ is Kawamata log terminal. Let $p: Y \to X$ be a projective birational morphism such that its restriction to $X \setminus \{0\}$ is an isomorphism. Let $E$ be the excepctional locus of $p$ and assume that the pair $(Y, E + p_*^{-1} \Delta)$ is divisorially log terminal.

One of the claims of the Corollary 24 is that if $(X, \Delta)$ is Kawamata log terminal then the dual intersection complex of $E$ (a certain cellular complex associated to the divisor) of contractible, though my question is not about this conclusion really.

The main construction of the proof is to run $(Y, E + p_*^{-1}\Delta)$-MMP over $X$ to get a sequence of pairs $(Y,E)=(Y_0, E_0), \ldots, (Y_i, E_i), \ldots$ (call the projection $\pi_i$). As far as I understand such an MMP terminates with a pair $(Y_r, E_r)$ such that $K_{Y_r} + E_r$ is $\pi_r$-nef, but the proof of Corollary 24 claims that if $(X, \Delta)$ is supposed to be Kawamata log terminal then the MMP actually terminates with $X$ (not sure what the boundary is supposed to be in this case).

I am only starting to get to grips with the minimal model program techniques and would really appreciate it if someone could explain how the singularities of $(X, \Delta)$ are connected with the assertion about what the pair the MMP terminates with is.

Making this into a concrete question: if $(Y_i, E_i)$ is a pair as above, $Y_i \neq X$ and $(X, \Delta)$ is klt how does one see that $K_{Y_i} + E_i$ is not $\pi_i$-nef and so the MMP can continue to run?

Answer 1

We'll show a more general statement. Suppose $(X,\Delta)$ has klt singularities and $f : Y \to X$ is a projective birational morphism with $Y$ normal and $\mathbb{Q}$-factorial. Suppose further that $f$ is not small so that $Ex(f)$ contains some divisor.

Claim: Then $K_Y + f_*^{-1}\Delta + E$ is not $f$-nef where $E$ is the reduced exceptional divisor.

Indeed by the definition of klt, we have

$$ K_Y + f_*^{-1}\Delta + E \equiv f^*(K_X + \Delta) + \sum a_i E_i $$ where $a_i > 0$ and the sum runs over prime $f$-exceptional divisors. Let us denote $-B \colon = \sum a_i E_i$. Then for any $f$-exceptional curve $C$, we have $$ (K_Y + f_*^{-1}\Delta + E).C = (-B).C. $$

Now we apply the following negativity lemma (see for example Lemma 3.39 in Kollár-Mori).

Negativity Lemma: Let $f : Y \to X$ be a proper birational morphism between normal varieties. Suppose $-B$ is an $f$-nef $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor. Then $B$ is effective if and only if $f_*B$ is effective.

In our case, $B$ is $f$-exceptional so $f_*B = 0$ is effective. Thus if $-B$ were nef, this would contradict that $a_i > 0$. This proves the claim.

Now if $X$ is $\mathbb{Q}$-factorial, then the exceptional locus of any projective birational $f : Y \to X$ contains a divisor. In fact the exceptional locus is pure codimension $1$ (see for example Corollary 2.63 in Kollár-Mori). Therefore in the setting of the question, if the MMP relative $X$ terminates, it has to terminate with $X$ itself.

Note that without the $\mathbb{Q}$-factoriality assumption on $X$, the statement is false in general because the exceptional locus of the resolution may not be pure codimension $1$. In this case, the MMP terminates in a log minimal model $(Z, \Delta_Z)$ with a small contraction $q : Z \to X$ where $q^*(K_X + \Delta) = K_Z + \Delta_Z$ is $q$-nef.