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I am reading the paper "On the birational automorphisms of varieties of general type", and I have a question about the property of potentially birational divisor.

Definition (potentially birational divisor, c.f. Def. 2.3.3). Let $X$ be a normal projective variety, and $D$ D be a big $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor on $X$. If $x$ and $y$ are two very general points of $X$ then, possibly switching $x$ and $y$, we may find $0 \leq \Delta \sim_{\mathbb Q} (1 − \epsilon)D$, for some $0 < \epsilon < 1$, where $(X,\Delta)$ is not kawamata log terminal at $y$, $(X, \Delta)$ is log canonical at $x$ and $\{x\}$ is a non-kawamata log terminal centre, then we say that $D$ is potentially birational.

After this definition, the following important property of potentially birational divisor is proven:

Lemma (c.f. Lemma 2.3.4(1)) Let $X$ be a normal projective variety and $D$ be a big $\mathbb Q$-Cartier divisor on $X$. If $D$ is potentially birational, then $\phi_{K_X+\lceil{D}\rceil}$ (i.e. the rational map defined by the linear system $|\lfloor K_X+\lceil{D}\rceil \rfloor|$) is birational.

In order to proving this, the authors reduce to consider the case where $X$ is smooth, $D \sim_\mathbb{Q} A + B$ with $A$ is ample and $B \geq 0$. Moreover, one can assume $\Delta\sim_{\mathbb Q} (1 − \epsilon)D$ in the definition of potentially birational divisor satisfying the property that $(X, \Delta)$ is klt in a punctured neighborhood of $x$. Then, by Nadel vanishing, one obtains $$H^1(X, \mathcal{O}_X(K_X + \lceil D \rceil) \otimes \mathcal{J}(\Delta + \epsilon B + \lceil D \rceil -D)) = 0,$$ where $\mathcal{J}(\Delta + \epsilon B + \lceil D \rceil -D)$ is the multiplier ideal sheaf.

Then the authors claim that one can find "a section $\sigma \in H^0(X, \mathcal{O}_X(K_X + \lceil D \rceil))$ vanishing at $y$ but not at $x$". It is this part that I was not able to follow. Granted this fact, one can find sections to separated very general points and hence define a birational map.

My question: why there exists section $\sigma$ satisfying claimed property as the above paragraph? Or, how does potentially birationality used above?

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    $\begingroup$ The point is that $O_X/\mathcal J =\mathcal O _x\oplus \mathcal O _Z$ where $y\in Z$. By the above vanishing, considering the long exact sequence in cohomology, we can lift the section $1\oplus 0$ from $H^0(\mathcal O _x)\oplus H^0(\mathcal O _Z)$ to a section $\sigma \in H^0(K_X+\lceil D\rceil)$ such that $s(x)=1$ and $s(y))=0$. Does this answer your question or did you mean something else? $\endgroup$
    – Hacon
    Commented Nov 8, 2016 at 3:48
  • $\begingroup$ Dear Prof. Hacon, this is exactly what I am asking and I see your point! Thank you very much! But I still have a technical detail: I understand that $(X, \Delta)$ is klt in a punctured neighborhood of $x$ with $\{x\}$ as non-klt centre. But the multiplier ideal is $\mathcal{J}(\Delta+ \epsilon B + \lceil D\rceil -D)$. How could I know the non-klt centre for $(X, \Delta+ \epsilon B + \lceil D\rceil -D)$ is still $\{x\}$. If the non-klt centre for the above is strictly larger than $\{x\}$, then one may not have direct sum $\mathcal{O}_x \oplus \mathcal{O}_Z$. Where did I got wrong? $\endgroup$
    – Li Yutong
    Commented Nov 8, 2016 at 7:04
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    $\begingroup$ $x$ is general, so may assume that $x$ is not contained in $B$ nor $D$, so adding them does not affect singularities around $x$ $\endgroup$
    – Chen Jiang
    Commented Nov 8, 2016 at 12:12
  • $\begingroup$ @Hacon By the way, it seems that as long as the singularity at $x$ is not klt (i.e. it is not necessarily to be lc), one will have such lifting property by the same argument (with $\mathcal{O}_X/\mathcal J = \mathcal{O}_X/{m^l_x} \oplus \mathcal{O}_Z$). $\endgroup$
    – Li Yutong
    Commented Nov 8, 2016 at 14:34
  • $\begingroup$ That's right. We use the LC condition for adjunction purposes (in the cutting down the LC centers process), but once the co-support is o an isolated point, we no longer need this. $\endgroup$
    – Hacon
    Commented Nov 8, 2016 at 19:01

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