Suppose $\pi: X \dashrightarrow Y$ is a birational map which is an isomorphism in codimension 1 (such as a flip). Also suppose both $X$ and $Y$ have reasonable (say log terminal) singularities. We know that divisors in $X$ and $Y$ correspond bijectively to each other. If $D$ is a nef divisor in $X$, is it true that $\overline{\pi(D)}$ is also nef in $Y$?
$\begingroup$
$\endgroup$
2
-
3$\begingroup$ No, $\overline{\pi(D)}$ will typically have a non-empty base locus where it is negative - try to see what happens in the Atiyah flop.. $\endgroup$– Ennio Mori coneCommented Apr 26, 2020 at 21:21
-
3$\begingroup$ More precisely, for a klt flip, it carries through iff $D\cdot \Sigma =0$ where $\Sigma$ is the flipping curve (else $D\cdot \Sigma >0$ and so $\overline{\pi(D)} \cdot \Sigma ^+<0$ where $\Sigma ^+$ is a flipped curve....if $D\cdot \Sigma=0$, use the base point free thm). This fact is important when running the mmp with scaling. $\endgroup$– HaconCommented Apr 27, 2020 at 3:20
Add a comment
|