Timeline for Does smooth and proper over $\mathbb Z$ imply rational?
Current License: CC BY-SA 3.0
11 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Dec 11, 2012 at 6:56 | comment | added | Will Sawin | Why do you believe this? Do you have a reference of some sort? | |
| Dec 10, 2012 at 19:30 | comment | added | Ben Wieland | I believe what happens with Enriques surfaces is the same as with abelian varieties: the covering map may cease to be etale, but the total space remains smooth over the same base. I don't know about the general case. | |
| Dec 9, 2012 at 21:35 | comment | added | Will Sawin | Why would the finite covers of fake projective planes be unramified over every prime? If they are ramified at some prime, why are they smooth? Enriques surfaces also have a degree 2 cover. It's defined canonically so it's unramified outside 2, but I don't see why it should be unramified at 2. | |
| May 17, 2012 at 15:52 | comment | added | Ben Wieland | Here is my summary of the discussion: (1) I hallucinated the conjecture. (2) The cohomological conjecture (and thus the birational conjecture) is almost certainly false in dimension 11+, but all known counterexamples are stacks. (3) In dimensions 3-10, the cohomological conjecture might be true, but no one offered an opinion on the birational conjecture. | |
| May 17, 2012 at 15:40 | vote | accept | Ben Wieland | ||
| May 16, 2012 at 8:10 | answer | added | Dan Petersen | timeline score: 10 | |
| May 16, 2012 at 6:59 | comment | added | Kevin Buzzard | ...small. However the Galois representation attached to the Ramanujan Delta function works fine, which is perhaps some hint that if $p+q=11$ then there might be some non-trivial cohomology... | |
| May 16, 2012 at 6:58 | comment | added | Kevin Buzzard | I'm not so sure that it's reasonable to suggest that too many more of the cohomology groups vanish. One way to convince yourself that Fontaine's results might be true are that if you consider the etale cohomology of the variety in degree $i$ then there'll be a contribution from $H^q(\Omega^p)$ if $p+q=i$. But the associated $\ell$-adic Galois representation would be unramified away from $\ell$ and crystalline at $\ell$. Such representations, other than powers of the cyclotomic character (which contributes to the $p=q$ case) are hard to come by if $i$ is small, because the weights are too... | |
| May 16, 2012 at 4:57 | comment | added | temp | Interesting question. I think by the word "propriety" you mean properness. But Enriques surfaces have the Hodge diamond like the one you mentioned in the 2nd paragraph (And they are not simply connected, regarding the other question you just answered). I don't know if there are smooth Enriques surfaces over $\mathbb{Z}$ though. | |
| May 16, 2012 at 4:34 | history | asked | Ben Wieland | CC BY-SA 3.0 |