Timeline for "un-nil-ifying" ideals via deformation
Current License: CC BY-SA 2.5
20 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 5, 2010 at 11:12 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
edited body; Post Made Community Wiki
|
| Dec 5, 2010 at 8:17 | comment | added | Francesco Polizzi | @Emerton: thanks for clarifying my comment. @Kevin: I apologize for having not been clear, sometimes I forget that what is "standard notation" for me might not be standard for people working on different subjects :-) | |
| Dec 5, 2010 at 4:22 | comment | added | Emerton | Dear Kevin, In Francesco's example as written, $L$ is a line on a smooth cubic surface $S$ in $\mathbb P^3$, so it is locally principal on $S$, cut out (locally) by some equation $f = 0$. Now consider the Cartier divisor on $S$ whose (local) equation is $f^2 = 0$. This is the scheme $2 L$. This notation makes sense for any Cartier divisor on any scheme. | |
| Dec 5, 2010 at 3:14 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
added 11 characters in body
|
| Dec 5, 2010 at 2:58 | comment | added | Francesco Polizzi | @kevin: I'm sorry, in order to give a counterexample it is necessary to consider $2L$, where the nilpotent structure is given by taking $L$ a smooth surface in $\mathbb{P}^3$. In fact, Hartshorne's results on rigidity apply only in this case. I suspect that the scheme $\textrm{Proj}k[x,y,z]/(z^2)$ provides a counterexample too, in the sense that it cannot be smoothable to the union of two disjoint curves, but I have no proof of this. | |
| Dec 5, 2010 at 2:53 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
added 185 characters in body
|
| Dec 5, 2010 at 1:47 | comment | added | Kevin H. Lin | I'm very sorry, I still don't know what you mean... | |
| Dec 5, 2010 at 0:48 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
edited body
|
| Dec 5, 2010 at 0:41 | comment | added | Francesco Polizzi | It is just a line "counted twice". | |
| Dec 5, 2010 at 0:38 | comment | added | Kevin H. Lin | I am not familiar with this notation $2L$... | |
| Dec 5, 2010 at 0:36 | comment | added | Francesco Polizzi | Mariano, you are completely right and my first answer was not correct. I edited it, now it should be ok (at least, I hope :) ) | |
| Dec 5, 2010 at 0:33 | history | undeleted | Francesco Polizzi | ||
| Dec 5, 2010 at 0:33 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
added 605 characters in body; added 4 characters in body
|
| Dec 5, 2010 at 0:21 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
deleted 446 characters in body
|
| Dec 4, 2010 at 23:39 | history | deleted | Francesco Polizzi | ||
| Dec 4, 2010 at 23:36 | comment | added | Mariano Suárez-Álvarez | In any case, your "For the sake of simplicity" simplified the problem waaaay too much! :) There are many more interesting ways in which a scheme can be non-reduced than the way the fiber $q^{-1}(0)$. | |
| Dec 4, 2010 at 23:35 | comment | added | Mariano Suárez-Álvarez | WOuldn't $p_n(z)$ need to be $z^n$? | |
| Dec 4, 2010 at 23:30 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
added 144 characters in body
|
| Dec 4, 2010 at 23:22 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
added 113 characters in body; added 5 characters in body
|
| Dec 4, 2010 at 23:16 | history | answered | Francesco Polizzi | CC BY-SA 2.5 |