# Termination of a minimal model program

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?

## 1 Answer

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.