Timeline for Hilbert scheme of points on a complex surface
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Feb 24, 2010 at 3:30 | comment | added | Allen Knutson | @Peter: It's called the "Hilbert-Chow morphism", and more generally goes from Hilbert schemes to Chow varieties. This particular instance is described in detail in e.g. Brion & Kumar's book on Frobenius splitting, where they use the fact that it's a crepant resolution of singularities to show that the Hilbert scheme is Frobenius split. | |
| Feb 23, 2010 at 19:17 | comment | added | Peter Samuelson | Is there a natural map from Hilb_n(C^2) to $(\mathbb{C}^2)^n/S_n$ (or maybe to $((\mathbb{C}^\times)^2)^n/S_n$)? Something like (pair of matrices) goes to (their eigenvalues)? I've heard there's a map like this that's supposed to be a resolution of singularities, but I don't know the details, or what this means exactly... | |
| Jan 29, 2010 at 12:23 | comment | added | Allen Knutson | I'm guessing that the n points are sitting at k spots, so one can think of it as a disjoint union of k fat points, and Q_i is the coordinate ring of the ith fat point. So yeah, that's as a set; it would be some work from that definition even to understand the topology of what happens when points collide. | |
| Jan 29, 2010 at 5:39 | vote | accept | Vamsi | ||
| Jan 29, 2010 at 5:36 | comment | added | Vamsi | Thanks. The description in terms of matrices was concrete indeed But, how does one describe the hilbert scheme of X where X is a general complex 2-manifold? I mean, in Gottsche's slides (for a talk) it is written that atleast as a set it is the collection {(x1,Q1),(x2,Q2),....(xk,Qk)} where where xk is a point on X and Qk is "the quotient ring of holomorphic functions at xk" (any idea as to what this means?). | |
| Jan 29, 2010 at 3:35 | history | answered | Allen Knutson | CC BY-SA 2.5 |