Timeline for non-isomorphic birationally equivalent calabi-yau varieties
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Dec 30, 2016 at 20:51 | comment | added | Jason Starr | @MarkGross. The comment above of Mark (not really "Mark") is that it is difficult to make sense of "general" when working over a specified finite field, as the OP suggests. The example I give below works over every field, so long as that field has characteristic different from $2$. | |
| Dec 30, 2016 at 20:32 | comment | added | Mark Gross | Why not just take a quintic threefold in projective four-space containing a projective plane, $L$. If the quintic is general, then there will be 16 ODPs along the plane. There are two small resolutions, one of which is obtained by blowing up the plane. This should work for sufficiently large $p$. It is easy to see these two models are not isomorphic. | |
| Dec 30, 2016 at 14:48 | vote | accept | user102981 | ||
| Dec 30, 2016 at 14:48 | |||||
| Dec 30, 2016 at 13:42 | answer | added | Jason Starr | timeline score: 2 | |
| Dec 30, 2016 at 2:47 | comment | added | user47305 | (However, I am not so sure the Namikawa examples I mentioned can be carried out over $\mathbb F_p$: it requires some nondegeneracy of certain elliptic fibrations that might not be possible to arrange over a finite field.) | |
| Dec 30, 2016 at 2:45 | comment | added | user47305 | Your first answer is actually right: there are famous examples of Namikawa. But you want the paper "On the Birational Structure of Certain Calabi-Yau Threefolds" instead. The example there was (as I understand it) one impetus for the Kawamata-Morrison cone conjecture. (This gives an example of birational CY3's which are connected by flops, as indeed any such examples must be, but he shows that there are only finitely many isomorphism types within the birational class.) | |
| Dec 29, 2016 at 16:56 | comment | added | Jason Starr | @Mark. You are correct: the image of the flop is just an algebraic space. Okay, the third time is the charm. For a K3 surface $S$ of degree $2g-2$ in $\mathbb{P}^g$, with $g\geq 3$, the Hilbert scheme $\text{Hilb}^g_{S/\mathbb{F}_p}$ and the compactified relative Picard scheme parameterizing degree $g$, torsion-free, rank $1$ coherent sheaves on hyperplane sections of $S$ are birational, but usually not isomorphic. Each of these smooth, projective schemes is birational and has trivial canonical bundle, but typically they are not isomorphic. | |
| Dec 29, 2016 at 16:32 | comment | added | user47305 | If you try to flop a line on a quintic like that, the result is not going to be projective. | |
| Dec 29, 2016 at 14:42 | comment | added | Jason Starr | Oops, I read your question too quickly. Namikawa's examples are more profound: they have equal Hodge structures, but they are not birational. Birationally equivalent but non-isomorphic Calabi-Yau's are much easier, e.g., blow up one of the 2875 lines in a "sufficiently general" quintic threefold (many of these do exist over finite fields), and then blow down the exceptional divisor $E\cong \mathbb{P}^1\times \mathbb{P}^1$ in the "other direction". | |
| Dec 29, 2016 at 14:40 | comment | added | Jason Starr | There are famous examples of Namikawa, see the following MO post: mathoverflow.net/questions/140695/open-torelli-problems/140741 | |
| Dec 29, 2016 at 14:36 | review | First posts | |||
| Dec 29, 2016 at 15:00 | |||||
| Dec 29, 2016 at 14:32 | history | asked | user102981 | CC BY-SA 3.0 |