$D$ respectively.

If $0\leq d_i \leq 1$ for every $i$, then we say that $D$ is a [*[boundary]{}*]{} $\mathbb R$-divisor on $X$. We note that $\sim _{\mathbb Q}$ (resp. $\sim_{\mathbb R}$) denotes the [*[$\mathbb Q$-linear]{}*]{} (resp. [*[$\mathbb R$-linear]{}*]{}) [*[equivalence]{}*]{} of $\mathbb Q$-divisors (resp. $\mathbb R$-divisors).

In this paper, we will repeatedly use the following notation: $$D^{=1}=\sum _{d_i=1} D_i, \quad 
D^{>1}=\sum _{d_i>1} d_i D_i, \quad \text{and} \quad 
D^{<0}=\sum _{d_i<0}d_i D_i.$$

We close this section with the definition of exceptional divisors for proper surjective morphisms between normal varieties.

\[x-def2.6\] Let $f:X\to Y$ be a proper surjective morphism between normal varieties. Let $E$ be a Weil divisor on $X$. We say that $E$ is [*[$f$-exceptional]{}*]{} if $\mathrm{codim}_Yf({{\operatorname{Supp}}}E)\geq 2$. We note that $f$ is not always assumed to be birational.

Preliminary lemmas {#x-sec3}
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In this section, we collect some useful and important lemmas for the reader’s convenience. They are more or less well known to the experts.

Let us start with the following easy lemmas on $\mathbb R$-divisors. We will use them repeatedly in this paper.

\[x-lem3.1\] Let $A$ be a Cartier divisor on a normal variety $V$. Let $B$ be an $\mathbb R$-Cartier $\mathbb R$-divisor on $V$ such that $B=\sum _{i\in I} 