Timeline for Proving that the Hilbert scheme of points on $\mathbb C^2$ is smooth
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 6, 2014 at 6:55 | vote | accept | aglearner | ||
| Dec 12, 2013 at 15:26 | comment | added | David E Speyer | The proof Allen sketches is written out in detail in chapter 18.2 of Miller and Sturmfels <i>Combinatorial Commutative Algebra</i>. It's elementary, but it's not easy. | |
| Dec 12, 2013 at 14:30 | comment | added | aglearner | Allen, thank you this is interesting. I wonder why this does not work in dimension 3. As far as I remember, even the dimension of the Hilbert scheme of points of $\mathbb C^3$ is unknown for large number of points | |
| Dec 12, 2013 at 7:38 | comment | added | Allen Knutson | I think there's a proof that goes like this. Let $T = (\mathbb C^\times)^2$ act on the plane hence on the Hilbert scheme. Then any point has a $T$-fixed point in the closure of its $T$-orbit, hence it's enough to check smoothness at the $T$-fixed points. For each of those, you can figure out what $Hom_R(I,R/I)$ looks like as a $T$-representation, and in particular that it is $2n$-dimensional. Which is to say, no $Ext^{i>0}$ is considered. | |
| Dec 12, 2013 at 3:21 | answer | added | Sasha | timeline score: 2 | |
| Dec 12, 2013 at 2:07 | comment | added | Daniel Litt | There are many proofs of this result--one using Ext groups is in FGA Explained, in the chapter on Hilbert schemes of points. It's not actually that bad, in my opinion. I believe one can also prove this result (more elementarily) using the description of the Hilbert scheme as a space of commuting matrices. | |
| Dec 12, 2013 at 0:01 | history | edited | aglearner | CC BY-SA 3.0 |
added 63 characters in body
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| Dec 11, 2013 at 23:46 | comment | added | aglearner | Depends on what is meant by "know". I red the definition 20 times over the span of my life. I can calculate these groups in some simple examples. But since I still have not read Hartchorne beyond section 1, maybe the honest answer should be: NO. | |
| Dec 11, 2013 at 23:37 | comment | added | Will Sawin | Do you know about $Ext$ groups? | |
| Dec 11, 2013 at 22:34 | history | asked | aglearner | CC BY-SA 3.0 |