Timeline for Is the set of surfaces over Spec Z with ample canonical sheaf empty
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Jan 18, 2013 at 20:07 | vote | accept | Ariyan Javanpeykar | ||
| Jan 15, 2013 at 18:58 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
added 180 characters in body
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| Jan 15, 2013 at 18:54 | comment | added | Sándor Kovács | Other than what you mention, you could look at Vojta's papers. His conjectures on various height estimates are considered higher dimensional analogous of Mordell, which, I am sure you know, is closely related to Shafarevich. There is a chance that one can get a sort of boundedness statement from some height estimates (this is what happens for curves) and then you need rigidity to get finiteness. As the example in my answer shows, in the geometric case you need more than just non-isotriviality for rigidity, but this could be a place where the number field case is different. | |
| Jan 15, 2013 at 17:05 | comment | added | Ariyan Javanpeykar | Thank you very much. This affirms my expectation that we don't know the answer to Q1. I do expect the answer to be negative (because I "believe" in the analogy.) Do you know of any references where the arithmetic Shafarevich conjecture is studied except for the papers by Faltings and André concerning abelian varieties, curves and K3 surfaces and the survey papers by Zarhin, Parshin, Szpiro, etc. ? For instance, do you know whether the arithmetic Shafarevich conjecture is known for surfaces of Kodaira dimension one (=elliptic fibrations)? | |
| Jan 14, 2013 at 21:24 | history | answered | Sándor Kovács | CC BY-SA 3.0 |