Talk:Hilbert scheme

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It would be nice to have a table for Chern numbers, Betti numbers and other invariants of Hilbert scheme of K3 and generalized Kummer. -- Tiphareth 05:25, 26 December 2006 (UTC)

"Birational morphism" is a morphism which is defined and bijective in a Zariski open set. The morphism from a Hilbert scheme of points to a symmetric product is birational, which is obvious. It is called a Hilbert-Chow morphism here.

The correction by R.e.b. "This morphism is birational if n is 1 or 2" is based on a wrong notion of birationality I guess. -- Tiphareth 16:40, 4 January 2007 (UTC)

You are overlooking the fact that the Hilbert scheme is usually reducible for manifolds of dimension at least 3. In general the Hilbert scheme and the symmetric product are not only not birational, but do not even have the same dimension. R.e.b. 04:14, 5 January 2007 (UTC)

Thank you for the correction!

It's birational within the scope of the definition

I cited above, but you are right that since it's

only one of the components, this statement is

misleading. --Tiphareth 04:42, 5 January 2007 (UTC)

If you want to define "birational" for reducible schemes you need to add the condition that the open sets are dense, otherwise it is not an equivalence relation. R.e.b. 15:44, 5 January 2007 (UTC)

Sure. Thanks. -- Tiphareth 05:58, 6 January 2007 (UTC)