Timeline for Non-algebraizable Formal Scheme?
Current License: CC BY-SA 4.0
15 events
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| S Jul 4, 2022 at 7:18 | history | suggested | Z. M | CC BY-SA 4.0 |
Remove the LaTeX for an emphatic text; change Specf to Spf
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| Jul 3, 2022 at 21:44 | review | Suggested edits | |||
| S Jul 4, 2022 at 7:18 | |||||
| Nov 9, 2010 at 5:39 | vote | accept | jlk | ||
| Nov 9, 2010 at 5:33 | comment | added | jlk | @FP: Thanks! Those examples are very nice! | |
| Nov 8, 2010 at 9:04 | comment | added | Francesco Polizzi | Dear Brian and jlk, I do not see how to make an approximation argument work either. About Artin's examples, I looked for them but I could not find the exact reference. However, during my search I met further examples of non-algebraizable formal schemes, maybe (if you do not know them already) you could find them interesting. The references are [Hironaka-Matsumura, "Formal functions and formal embeddings" J. math. soc. Japan 20, Theorem 5.3.3 ] and [Hartshorne, Ample subvarieties of algebraic varieties, p. 205]. Regards, Francesco | |
| Nov 8, 2010 at 2:00 | comment | added | jlk | @BCnrd: Please let me know if you see an approximation argument or if you remember Artin's examples. I'd be interested in both. | |
| Nov 7, 2010 at 19:41 | comment | added | BCnrd | Dear Francesco: Do you see what such an approximation argument might be? I thought for a little bit and didn't see how to make it work. I have vague recollection that Artin constructed examples of non-algebraizable formal singularities (perhaps over any alg. closed field of char. 0?), but I don't remember anything more about that. | |
| Nov 7, 2010 at 14:15 | comment | added | Francesco Polizzi | @BCnrd Good point. Thank you for the remark! | |
| Nov 7, 2010 at 13:54 | comment | added | BCnrd | The question asked for non-algebraization as abstract (locally noetherian) scheme, not equipped with auxiliary structure (such as map to a specific affine scheme). By replacing $\mathbf{C}$ with $\mathbf{Q}$, perhaps an approximation argument can prove that if the above example admits algebraization as an abstract scheme then it also does as a proper flat scheme over the indicated base (hence a contradiction, as explained above). | |
| Nov 7, 2010 at 1:08 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
added 38 characters in body
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| Nov 6, 2010 at 19:59 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
added 235 characters in body; added 2 characters in body
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| Nov 6, 2010 at 19:56 | comment | added | Emerton | I guess so. The universal deformation of an abelian variety of dimension $> 1$ would be another, I guess. | |
| Nov 6, 2010 at 19:55 | comment | added | Francesco Polizzi | And into my mind too :-) I think it's a kind of standard example... | |
| Nov 6, 2010 at 19:52 | comment | added | Emerton | Yes, this is the first example that came to my mind! | |
| Nov 6, 2010 at 19:51 | history | answered | Francesco Polizzi | CC BY-SA 2.5 |