PomeTheorem 20.1.28.C3EFB85C68F6 There is a natural bijection from $\mathcal{G}(\alpha )/\sim _{\mathcal{I}}$ to the set of big and nef b-divisors $\mathbb {D}$ over $X$ with $\mathbb {D}_ X=\alpha $.
Theorem 20.1.28C3EFB85C68F6
Proof. Given $T\in \mathcal{G}(\alpha )$, we associate the b-divisor $\mathbb {D}(T)$. It is big and nef due to theorem 20.1.15. This map clearly descends to $\mathcal{G}(\alpha )/\sim _{\mathcal{I}}$.
This map is surjective by theorem 20.1.24. Now we show that it is injective. Let $T,T'\in \mathcal{G}(\alpha )$. Assume that $\mathbb {D}(T)=\mathbb {D}(T')$, we want to show that $T\sim _{\mathcal{I}} T'$.
Let $E$ be a prime divisor over $X$, it suffices to show that
We may assume that $E$ is not a prime divisor on $X$, as otherwise both sides vanish.
Choose a sequence of blow-ups with smooth connected centers
\[ Y\coloneqq X_ k\rightarrow X_{k-1}\rightarrow \cdots \rightarrow X_0\coloneqq X \]
so that $E$ is a prime divisor on $Y$, exceptional with respect to $X_ k\rightarrow X_{k-1}$. Denote the composition by $\pi \colon Y\rightarrow X$. Thanks to proposition 20.1.2,
\[ \mathrm{H}^{1,1}(X_ k,\mathbb {R})=\mathrm{H}^{1,1}(X_{k-1},\mathbb {R})\oplus \mathbb {R}\{ E_ k\} , \]
where $E_ k=E$ is the exceptional divisor of $X_ k\rightarrow X_{k-1}$.
By induction,
\[ \mathrm{H}^{1,1}(Y,\mathbb {R})=\mathrm{H}^{1,1}(X,\mathbb {R})\oplus \bigoplus _{i=1}^{k} \mathbb {R} \{ E_ i\} , \]
where $E_ i$ is the exceptional divisor of $X_ i\rightarrow X_{i-1}$. Now by 18.1.38,
F7913C6F4559
20.1.30
\begin{equation} \operatorname{Reg}\pi ^*T=\pi ^*T-\sum _{i=1}^ k \nu (T,E_ i)[E_ i]. \end{equation}
In particular, the cohomology class of $\operatorname{Reg}\pi ^*T$ determines $\nu (T,E)$. Hence, (20.1.29) follows. $\square$
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