Timeline for When is a formal deformation convergent?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Dec 8, 2018 at 15:03 | history | bounty ended | Saal Hardali | ||
| Dec 8, 2018 at 13:20 | history | edited | Piotr Achinger | CC BY-SA 4.0 |
added 2242 characters in body
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| Dec 8, 2018 at 12:51 | vote | accept | Saal Hardali | ||
| Dec 8, 2018 at 12:51 | comment | added | Saal Hardali | Thanks! The answer is more than complete now. Could you point to a specific reference for the positive answer? (the result of Elkik). | |
| Dec 8, 2018 at 12:07 | comment | added | Piotr Achinger | Of course! Take $y^2z =x(x-z)(x-\lambda z)$ but in $\mathbb{C}[[x,y,z]]$. There are also plane curve singularity examples, e.g. $xy(x+y)(x-\lambda y)$. | |
| Dec 8, 2018 at 11:18 | comment | added | Saal Hardali | Yes, I understand now, of course you are correct, sorry. This covers the proper-smooth case. Do you know a simple example of a surface singularity with a divergent formal deformation? Just to make this answer complete | |
| Dec 8, 2018 at 10:57 | comment | added | Piotr Achinger | (I didn't check that in this case $j(\lambda) = 256 (1-\lambda(1-\lambda)^3) \lambda^{-2} (1-\lambda)^{-2}$ is divergent, but most likely it is.) | |
| Dec 8, 2018 at 10:55 | comment | added | Piotr Achinger | The answer still stands: take something like $E\colon y^2 z = x(x-z)(x-\lambda z)$ where $\lambda = \sum n! t^n$. | |
| Dec 8, 2018 at 9:25 | comment | added | Saal Hardali | I changed the question in hope that it would be more precise so that I could understand the answer. I'm not sure how an elliptic curve can have a divergent $j$-invariant if it comes from a formal deformation. In particular I think both singular affine curves and complete non-singular curves can't be counterexamples. Could you be more precise please? Sorry about the abrupt edit of the question. | |
| Dec 8, 2018 at 9:13 | vote | accept | Saal Hardali | ||
| Dec 8, 2018 at 9:13 | |||||
| Dec 7, 2018 at 11:08 | vote | accept | Saal Hardali | ||
| Dec 8, 2018 at 9:11 | |||||
| Dec 5, 2018 at 18:18 | vote | accept | Saal Hardali | ||
| Dec 5, 2018 at 18:18 | |||||
| Dec 5, 2018 at 18:18 | vote | accept | Saal Hardali | ||
| Dec 5, 2018 at 18:18 | |||||
| Dec 5, 2018 at 14:43 | history | answered | Piotr Achinger | CC BY-SA 4.0 |