Timeline for Examples or references for this claim about elliptic Calabi-Yau threefolds
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11 events
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| Dec 9, 2021 at 17:27 | comment | added | Basics | Ah. Thanks for the reference! | |
| Dec 8, 2021 at 22:57 | comment | added | Ennio Mori cone | Oh, but showing the existence of integral solutions of cubics is part of the arguments: See Heath-Brown & Wilson "Calabi-Yau threefolds with $\rho > 13$ for instance. | |
| Dec 7, 2021 at 10:17 | comment | added | Basics | @Ennio Mori cone, good points. But I am doubtful about "....defines a cubic equation in many variables, so there are integral solutions..." because generic cubic forms do not tend to have nontrivial integral solutions even if its rank is very large. This is a crucial difference between two-dimension and higher dimensions. | |
| Dec 6, 2021 at 23:00 | comment | added | Ennio Mori cone | Anyway, I guess they just mean that in many cases, the intersection form on $Pic(X)$ gives the existence of divisors with $D^3=0, D^2\neq 0$, which defined elliptic fibrations on some model. | |
| Dec 6, 2021 at 22:58 | comment | added | Ennio Mori cone | (If $D$ is not nef, Riemann-Roch gives that either $D$ or $-D$ is pseudoeffective, and most likely $D$ is movable in most cases. That means you may need to flop a few curves to get to a birational model where it is in fact nef). | |
| Dec 6, 2021 at 22:58 | comment | added | Ennio Mori cone | If $X$ is a CY3, with large Picard number, then there tend to be lots of divisors $D$ with $D^3=0$ - this is simply because the equation $D^3=0$ defines a cubic equation in many variables, so there are integral solutions. If $D$ is nef, then by abundance, some multiple $mD$ defines a map $X\to Y$ to a lower-dimensional variety (since $D^3=0$), and by adjunction, the general fiber is either a K3 surface or an elliptic curve (so in the latter case $X$ is elliptically fibered). These two conditions can be distinguished by looking at whether or not $D^2$ is zero or not). | |
| Dec 5, 2021 at 3:43 | history | edited | Basics | CC BY-SA 4.0 |
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| S Dec 4, 2021 at 13:52 | history | edited | Basics | CC BY-SA 4.0 |
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| S Dec 4, 2021 at 13:52 | history | suggested | CommunityBot |
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| Dec 4, 2021 at 7:37 | history | asked | Basics | CC BY-SA 4.0 |