Definition 20.1.5.178D74F44C30 A (Weil) b-divisor $\mathbb {D}$ over $X$ is an assignment $(\mathbb {D}_{\pi })_{\pi \colon Y\rightarrow X}$, where $\pi \colon Y\rightarrow X$ runs over all modifications of $X$ such that
81114DDA7CAF
$\mathbb {D}_{\pi }\in \mathrm{H}^{1,1}(Y,\mathbb {R})$;
EC0926110C34
The classes are compatible under push-forwards: If $\pi '\colon Z\rightarrow X$ and $\pi \colon Y\rightarrow X$ are both modifications of $X$ and $\pi '$ dominates $\pi $ through $g\colon Z\rightarrow Y$, then $g_*\mathbb {D}_{\pi '}=\mathbb {D}_{\pi }$.
We also write $\mathbb {D}_ Y=\mathbb {D}_{\pi }$ if there is no risk of confusion.
Given two Weil b-divisors $\mathbb {D}$ and $\mathbb {D}'$ over $X$, we say $\mathbb {D}\leq \mathbb {D}'$ if for each modification $\pi $ of $X$, we have $\mathbb {D}_{\pi }\leq \mathbb {D}'_{\pi }$. Recall that by definition, this means the class $\mathbb {D}'_{\pi }-\mathbb {D}_{\pi }$ is pseudo-effective.
A Weil b-divisor with $\mathbb {D}\geq 0$ is called pseudo-effective.
The class $\mathbb {D}_ X$ is called the root of $\mathbb {D}$. The set of Weil b-divisors over $X$ has the natural structure of a real vector space.
Definition 20.1.6.E6F6681E7170 The volume of a pseudo-effective Weil b-divisor $\mathbb {D}$ over $X$ is
\[ \operatorname{vol}\mathbb {D}\coloneqq \lim _{\pi \colon Y\rightarrow X} \operatorname{vol}\mathbb {D}_ Y. \]
The right-hand side is a decreasing net due to proposition 16.6.17, hence the limit always exists. The volume of a non-pseudo-effective Weil b-divisor is defined to be $0$.
We say $\mathbb {D}$ is big if $\operatorname{vol}\mathbb {D}>0$.
Lemma 20.1.7.F5B2C1F1E318 Let $(\mathbb {D}_ i)_{i\in I}$ be a net of pseudo-effective b-divisors converging to a Weil b-divisor $\mathbb {D}$. Then
3BFB2E87E034
20.1.8
\begin{equation} \varlimsup _{i\in I}\operatorname{vol}\mathbb {D}_ i\leq \operatorname{vol}\mathbb {D}. \end{equation}
If the net is decreasing, then
\[ \lim _{i\in I}\operatorname{vol}\mathbb {D}_ i=\operatorname{vol}\mathbb {D}. \]
Here we say $(\mathbb {D}_ i)_{i\in I}$ converges to $\mathbb {D}$ if for any modification $\pi \colon Y\rightarrow X$, we have $\mathbb {D}_{i,Y}\to \mathbb {D}_ Y$ with respect to the Euclidean topology.
Proof.
First observe that $\mathbb {D}$ is pseudo-effective, since $\mathbb {D}_ i\geq 0$ for all $i$.
Let $\pi \colon Y\rightarrow X$ be a modification. Then
\[ \operatorname{vol}\mathbb {D}_ Y=\lim _{i\in I} \operatorname{vol}\mathbb {D}_{i,Y}\geq \varlimsup _{i\in I} \operatorname{vol}\mathbb {D}_ i. \]
The inequality (20.1.8) follows. As for the decreasing case, it suffices to observe that both sides of (20.1.8) can be written as
\[ \inf _ i \inf _{\pi \colon Y\rightarrow X} \operatorname{vol}\mathbb {D}_{i,Y}. \]
This completes the proof.
$\square$
Example 20.1.9.7AF528B0FB3B In general, equality need not hold in (20.1.8). Let $X=\mathbb {P}^2$, and let $H=c_1(\mathcal{O}_{\mathbb {P}^2}(1))$. Choose a sequence of points $p_ i\in X$ converging to the generic point in the following sense: for every algebraic curve $C\subseteq X$, the set of indices $i$ with $p_ i\in C$ is finite.
Let
\[ \pi _ i\colon X_ i\rightarrow X \]
be the blow-up of $X$ at $p_ i$, with exceptional divisor $E_ i$. Let $\mathbb {D}_ i$ be the Cartier b-divisor realized on $X_ i$ by the nef class
\[ \pi _ i^*H-\{ E_ i\} , \]
and let $\mathbb {D}$ be the Cartier b-divisor realized on $X$ by $H$. Then $\mathbb {D}_ i\to \mathbb {D}$. Indeed, for any fixed modification $\rho \colon Y\rightarrow X$, the morphism $\rho $ is an isomorphism over $p_ i$ for all large enough $i$, hence the trace of $\mathbb {D}_ i$ on $Y$ is $\rho ^*H$ for all large enough $i$.
On the other hand,
\[ \operatorname{vol}\mathbb {D}_ i=\left(\pi _ i^*H-\{ E_ i\} \right)^2=0 \]
for all $i$, whereas
\[ \operatorname{vol}\mathbb {D}=H^2=1. \]
Thus the inequality in (20.1.8) is strict for this sequence.
Definition 20.1.10.908967DFB37E A Cartier b-divisor $\mathbb {D}$ over $X$ is a Weil b-divisor $\mathbb {D}$ over $X$ such that there exists a modification $\pi \colon Y\rightarrow X$ and a class $\alpha _ Y\in \mathrm{H}^{1,1}(Y,\mathbb {R})$ so that for each $\pi '\colon Z\rightarrow X$ dominating $\pi $, the class $\mathbb {D}_ Z$ is the pull-back of $\alpha _ Y$. Any such $(\pi ,\alpha _ Y)$ is called a realization of $\mathbb {D}$.
By abuse of language, we also say $(Y,\alpha _ Y)$ is a realization of $\mathbb {D}$. The realization is not unique in general.
Definition 20.1.11.FB2640F8F828 A Cartier b-divisor $\mathbb {D}$ over $X$ is nef if there exists a realization $(\pi \colon Y\rightarrow X, \alpha _ Y)$ of $\mathbb {D}$ such that $\alpha _ Y$ is nef.
Definition 20.1.12.890427BD65D5 A Weil b-divisor $\mathbb {D}$ over $X$ is nef if it is the limit of a net of nef Cartier b-divisors $(\mathbb {D}_ i)_ i$ over $X$.
In other words, for each modification $\pi \colon Y\rightarrow X$, we have $\mathbb {D}_{i,Y}\to \mathbb {D}_ Y$.
Note that thanks to proposition 5.19.17, each $\mathbb {D}_ Y$ is necessarily modified nef, but it is not nef in general. The notion of modified nef classes is defined in definition 5.19.15.
A priori, for a Cartier b-divisor, nefness could mean either the notion defined by definition 20.1.11 or the one defined by definition 20.1.12. We will show in corollary 20.1.35 that they are actually equivalent. Before that, by a nef Cartier b-divisor, we always mean nefness in the sense of definition 20.1.11.
Our definition definition 20.1.12 amounts to defining the set of Weil b-divisors as the closure of the set of Cartier b-divisors in $\varprojlim _{\pi }\mathrm{H}^{1,1}(Y,\mathbb {R})$ with respect to the projective limit topology. In particular, the limit of a converging net of nef b-divisors is still nef.
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.