I am reading ["The dual complex of
singularities"](https://arxiv.org/abs/1212.1675) 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?



Termination of a minimal model program
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.


