Timeline for every abelian scheme quotient of a Picard scheme?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Feb 2, 2013 at 21:47 | comment | added | Ben Wieland | I doubt that this is true for some ring of integers in a number field, where Bertini's theorem fails. But you're happy to exclude this case. | |
| Feb 2, 2013 at 9:51 | answer | added | user19475 | timeline score: 1 | |
| Nov 22, 2012 at 14:22 | history | bounty ended | CommunityBot | ||
| Nov 19, 2012 at 20:20 | comment | added | Matthieu Romagny | Liu, good point! Of course Poonen's Bertini theorems involve sections with hypersurfaces of large degree. I thought that in the classical proof that abelian varieties are quotients of jacobians, it was important to cut by hyperplanes. Now that I think about it again, I see that it is not. | |
| Nov 19, 2012 at 15:24 | history | edited | user19475 | CC BY-SA 3.0 |
added 50 characters in body
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| Nov 15, 2012 at 21:17 | comment | added | Qing Liu | @Timo: the statement should hold over local (maybe noetherian) rings because Bertini's theorem is true in this case. | |
| Nov 15, 2012 at 21:15 | comment | added | Qing Liu | @Matthieu: it is true over finite fields. The proof, as in the classical case, relies on Bertini's theorem over finite fields and it is proved by Gabber and by Poonen's around 2000. | |
| Nov 15, 2012 at 17:10 | comment | added | Matthieu Romagny |
As far as I see from the literature, the question is not settled already if $X$ is the spectrum of a finite field.
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| Nov 15, 2012 at 16:12 | comment | added | Laurent Moret-Bailly | Does your definition of abelian schemes include "projective over $X$"? This at least is necessary. (I assume that by "curve" you mean "smooth proper curve"). | |
| Nov 15, 2012 at 15:02 | comment | added | user19475 | and jmilne.org/math/CourseNotes/AV.pdf p. 116. | |
| Nov 15, 2012 at 15:01 | comment | added | user19475 | See also modular.math.washington.edu/edu/Fall2003/252/lectures/10-17-03/… | |
| Nov 15, 2012 at 15:00 | comment | added | user19475 | Yes, I know this, but what I really need is Abelian schemes. | |
| Nov 15, 2012 at 14:35 | comment | added | user25309 | something related : in "Algebraic Groups and Class Fields" (J.P Serre), it is proved (VII.13) that on a algebraically closed field, every abelian variety is isomorphic to the quotient of a product of jacobians (of curves) by a connected subgroup. | |
| Nov 15, 2012 at 14:17 | history | bounty started | CommunityBot | ||
| Jan 16, 2012 at 20:30 | history | edited | user19475 | CC BY-SA 3.0 |
edited title
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| Jan 16, 2012 at 20:05 | history | asked | user19475 | CC BY-SA 3.0 |