Timeline for Hilbert scheme of relative subschemes of lenght 2
Current License: CC BY-SA 3.0
3 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 30, 2015 at 10:00 | comment | added | user3001 | Thank you for your help. I did not know length 2 subscheme were local complete intersection. Although in Kollar's book there is only the criterion for flatness (theorem I.2.15), it should indeed be easy to conclude from it. | |
| Jul 29, 2015 at 15:12 | comment | added | Jason Starr | Yes, you do have that. The standard obstruction theory for Hilbert schemes (as described in Artin's "Algebraization ... I" or in Chapter 1 of Koll'ar's "Rational curves ...") is actually an obstruction theory relative to a base $S$, at least as long as $\mathfrak{X}$ is flat over $S$ (otherwise there are some corrections). So the usual argument -- length $2$ schemes are LCI and the global obstruction is an $H^1$ that vanishes on zero dimensional schemes -- proves smoothness over $S$ of the relative Hilbert scheme. | |
| Jul 29, 2015 at 15:04 | history | asked | user3001 | CC BY-SA 3.0 |