Example 10.17.105E3AD3C523AC

Example 10.17.10.5E3AD3C523AC Let $\mu :X\to \mathbb P^3$ be the blowup along a line $L$, and set $a=\mu ^*\{ \omega _{\mathrm{FS}}\} $, where $\omega _{\mathrm{FS}}$ has total volume one; the class $a$ is nef and $\int _ Xa^3=1$. Since $N_{L/\mathbb P^3}\simeq \mathcal O(1)\oplus \mathcal O(1)$, the exceptional divisor is $E=\mathbb P(N_{L/\mathbb P^3})\simeq \mathbb P^1\times \mathbb P^1$, and on $E$ the map $\mu $ is the projection to $L$. Hence $\int _ Ea^2=0$, whereas a section $Y$ of $E\to L$ satisfies $\int _ Ya=1$. Every irreducible subvariety not contained in $E$ has positive intersection number, because $\mu ^*\omega _{\mathrm{FS}}$ is positive on its nonempty open part outside $E$. By corollary 10.16.10,

The union of the subvarieties with zero invariant is the whole null locus $E$, although the curve $Y\subset E$ has positive invariant. Positivity along one subvariety does not place it outside the null locus.