Timeline for "un-nil-ifying" ideals via deformation
Current License: CC BY-SA 2.5
5 events
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| Dec 5, 2010 at 6:17 | comment | added | Sándor Kovács | I guess this is sort of the point. If the general fiber is the disjoint union of two components, then the nilpotent part of "$2X$" would have a nowhere zero section, so it would have to be trivial. | |
| Dec 5, 2010 at 6:17 | comment | added | Sándor Kovács |
Right. If $X=\mathbb P^1$, then this example gives ${\rm Proj}\, k[x,y]\times {\rm Spec}\, k[\varepsilon]/(\varepsilon^2)$ which is obviously different from ${\rm Proj}\, k[x,y,z]/(z^2)$ because the latter is not a product since the normal bundle of a line in $\mathbb P^2$ is not trivial.
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| Dec 5, 2010 at 4:59 | history | edited | Allen Knutson | CC BY-SA 2.5 |
deleted 7 characters in body
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| Dec 5, 2010 at 3:16 | comment | added | Francesco Polizzi | In fact, when you take $X=\mathbb{P}^1$ you get a non-reduced scheme with arithmetic genus $-1$, so not all nilpotent structures are allowed (for instance, neither the one in my answer, whose arithmetic genus is $-2$, nor $\textrm{Proj} k[x,y,z]/(z^2)$, whose arithmetic genus is $0$). | |
| Dec 5, 2010 at 1:51 | history | answered | Sándor Kovács | CC BY-SA 2.5 |