Timeline for Pseudo-effective divisor which is not nef in any birational model
Current License: CC BY-SA 3.0
9 events
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| Oct 30, 2017 at 21:23 | history | edited | Joaquín Moraga | CC BY-SA 3.0 |
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| Oct 30, 2017 at 21:21 | comment | added | Joaquín Moraga | @Mark Yes, you are right, that example does not give a counter-example, I will clarify this in the question itself. May you elaborate a little more about those similar examples with CY3's? | |
| Oct 30, 2017 at 14:59 | comment | added | user47305 | Are you sure about condition $3$? If your example is what I suspect, you can just push the divisor forward via the blow-down to $\mathbb P^3$ and the pushforward class is $\mathcal O(1)$. On the other hand, there are probably similar examples on CY3's for which $B_{-}(D)$ is not closed and there are no contractions possible. | |
| Oct 29, 2017 at 5:19 | comment | added | R. van Dobben de Bruyn | @ChenJiang: if $U$ is the locus where $\pi$ is defined, then $X\setminus U$ has codimension $\geq 2$. Thus, we get an isomorphism $\operatorname{Cl}(X) \stackrel\sim\to \operatorname{Cl}(U)$. Define $\pi_*$ as the pushforward along $U \to Y$, in the sense of codimension $1$ cycles (see e.g. Fulton's Intersection Theory, $\S1.4$). Since $X$ is smooth, you don't need to worry about Weil vs. Cartier divisors, but it seems that $\pi_* D$ is a priori only a $\mathbb Q$-Weil divisor on $Y$. | |
| Oct 28, 2017 at 23:48 | comment | added | Joaquín Moraga | divisorial push-forward | |
| Oct 28, 2017 at 23:43 | history | edited | Joaquín Moraga | CC BY-SA 3.0 |
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| Oct 28, 2017 at 23:40 | comment | added | Chen Jiang | What do you mean by $\pi_*$? | |
| Oct 28, 2017 at 23:23 | history | edited | Joaquín Moraga |
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| Oct 28, 2017 at 23:03 | history | asked | Joaquín Moraga | CC BY-SA 3.0 |