Timeline for K3 surface as an anticanonical section
Current License: CC BY-SA 3.0
17 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| May 22, 2015 at 14:29 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| May 22, 2015 at 14:21 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| May 21, 2015 at 20:16 | comment | added | user47305 | Let $S \subset \mathbb P^3$ be a smooth quartic and let $X \to \mathbb P^3$ be the blow up a bunch of disjoint curves in $S$ (we can find as many as we want, eg if $S$ is elliptic). The strict transform of $S$ on $X$ is anticanonical, but won't be ample if you blow up enough curves. | |
| May 21, 2015 at 20:14 | comment | added | J.C. Ottem | Can't you just take a 3-fold where $-K_X$ has a smooth section (e.g., is bpf) but not ample and let $S\in |-K_X|$? For example, blow-up $P^3$ along the 16 nodes of a Kummer surface. | |
| May 21, 2015 at 20:03 | comment | added | Francesco Polizzi | I added this question at the end of my answer. | |
| May 21, 2015 at 20:01 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| May 21, 2015 at 19:58 | comment | added | Daniel Litt | I thought about it for a while and couldn't come up with one! | |
| May 21, 2015 at 19:44 | comment | added | Francesco Polizzi | Is there any example of a smooth K3 surface $S$ appearing as a non-ample anticanonical divisor in a smooth $3$-fold? | |
| May 21, 2015 at 19:38 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| May 21, 2015 at 19:15 | comment | added | Daniel Litt | No worries, still a very nice answer! | |
| May 21, 2015 at 19:15 | comment | added | Francesco Polizzi | Sorry, you are right. I misread the question, and I have only proved that $S$ cannot be a hyperplane section of a threefold (in this case the threefold is clearly Fano). | |
| May 21, 2015 at 19:13 | comment | added | Daniel Litt | I must still be missing something. First of all, to conclude an equality on restriction to S implies anything on V, you need amplitude of S to use Lefschetz. But that's what we're trying to prove. Second of all, you only show K_V=-S|_S, again, why is S|_S ample at all? | |
| May 21, 2015 at 19:07 | comment | added | Francesco Polizzi | By adjunction formula $$(K_V+S)|_S = K_S = \mathcal{O}_S,$$ hence $K_V=-$hyperplane class. | |
| May 21, 2015 at 19:03 | comment | added | Daniel Litt | Maybe I'm missing something easy--why does the threefold have to be Fano? Obviously the anticanonical is effective by hypothesis, but why ample? (That said +1, this is a very nice answer.) | |
| May 21, 2015 at 19:03 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| May 21, 2015 at 18:56 | history | answered | Francesco Polizzi | CC BY-SA 3.0 |