Example 20.1.97AF528B0FB3B

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

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

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,

for all $i$, whereas

Thus the inequality in (20.1.8) is strict for this sequence.