Timeline for Albanese variety over non-perfect fields
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Apr 26, 2021 at 12:19 | history | edited | Matthieu Romagny | CC BY-SA 4.0 |
brakes --> break
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| Feb 3, 2017 at 2:27 | vote | accept | Thomas Geisser | ||
| Feb 3, 2017 at 2:25 | vote | accept | Thomas Geisser | ||
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| Feb 3, 2017 at 2:25 | vote | accept | Thomas Geisser | ||
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| Feb 1, 2017 at 4:05 | vote | accept | Thomas Geisser | ||
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| Feb 1, 2017 at 4:05 | vote | accept | Thomas Geisser | ||
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| Jan 31, 2017 at 18:48 | history | edited | YCor |
edited tags
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| Jan 31, 2017 at 18:26 | answer | added | nfdc23 | timeline score: 14 | |
| Jan 31, 2017 at 15:47 | answer | added | Daniel Loughran | timeline score: 14 | |
| Jan 31, 2017 at 15:22 | comment | added | Kevin Buzzard | So if $L/K$ is a separable degree two extension of (perfect) fields, what is the Albanese for $Spec(L)$ regarded as a $K$-scheme? I am still unclear about what the definition is or in what generality it is supposed to work; I can't see it in Serre or in Wikipedia. You say "every author who quotes him states those hypothesis" -- can you give an explicit example of such a quote? All the references explicitly mentioned so far seem to assume alg closed base field (and they're all written in the old language too! Do you know anything written post-60s?) | |
| Jan 31, 2017 at 10:05 | comment | added | Thomas Geisser | And regarding Kevin's question: the albanese of C is again C, which is a torsor under its Jacobian. | |
| Jan 31, 2017 at 10:03 | comment | added | Thomas Geisser | I should point out that there are two notions of Albanese variety: The one by Lang is universal for rational maps to abelian varieties, and the one of Serre (the one I am talking about) is universal for morphisms to torsors under abelian varieties. | |
| Jan 31, 2017 at 9:39 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
added 14 characters in body; edited title
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| Jan 31, 2017 at 9:26 | comment | added | Kevin Buzzard | Lang p52 "Chow's projective construction of the Jacobian...belongs to absolute geometry. Here of course, we have...completely lost track of the integral nature of parameters on which the curve may depend." But maybe here he's talking about the construction of the Jacobian. Note also p45 -- "if we do not work over an algebraically closed constant field, we must replace the mappings of $V$ into abelian varieties by mappings into principal homogeneous spaces" -- that's all I can find about non-alg-closed fields. It might all be there -- I'm no expert -- but I can't see it explicitly. | |
| Jan 31, 2017 at 9:19 | comment | added | Kevin Buzzard | I am not sure I'd be so optimistic about Lang's construction. I don't think he can be working over an arbitrary field -- for example consider a genus 1 curve $C$ over the rationals with no rational points. As far as I can see this would have no universal map to an abelian variety over the rationals -- but Lang's definition does not mention torsors. The Wikipedia article also does not stray from alg closed field. In fact I have a question for the OP -- what is the actual definition of the Albanese for varieties over a non-algebraically closed fields? Is the Albanese of $C$ above also $C$? | |
| Jan 31, 2017 at 8:52 | comment | added | abx | Lang (Abelian variety, chap II, §3) constructs the Albanese variety over any field, if I understand correctly his language. | |
| Jan 31, 2017 at 6:43 | history | asked | Thomas Geisser | CC BY-SA 3.0 |