Talk:Hilbert scheme

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)[reply]

"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)[reply]

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)[reply]

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)[reply]

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)[reply]

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

Organized overview of construction

Source: http://matwbn.icm.edu.pl/ksiazki/bcp/bcp36/bcp36111.pdf

Title: ELEMENTARY INTRODUCTION TO REPRESENTABLE FUNCTORS AND HILBERT SCHEMES

Author: STEIN ARILD STRØMME

The construction of the Hilbert scheme should be organized in a clearer manner. This can be done by introducing the ideas of the construction then splitting up those ideas into two sections. The main idea is to take the ideal sheaf of a subscheme with a fixed hilbert polynomial, identify that with a quotient of O_X, and identify that with a point in the grassmannian of points in H^0(O_X(m)) for some m large enough.

m-regularity

proposition 4.3 gives the result for all ideal sheaves. It is an induction argument on the dimension of the projective space, and an application of Serre's vanishing theorem for coherent sheaf cohomology.

flattening stratification

General results

Construction

For projective space

Fitting ideal of a map gives ideal of Hilbert scheme for X = P^n_S, assuming it exists
Reduce base case to Z
Embed Hilbert functor to grassmann functor, use m-regularity + grassmannian
Use ideas from this construction to universal sequence on grassmann variety, then build a correspondence of P^n x Gr
This gives a closed subscheme W, with a hilbert polynomial, so the flattening stratification of Gr has a stratum associated to this subscheme W. The associated subfunctor of Gr is the subfunctor for hilb, hence it is a scheme

remarks

Bottom of page 11 about dimension of hilbert schemes
Mumford's pathology
Vakil's Murphey's law

More examples needed

Add hilbert scheme of curves https://arxiv.org/pdf/math/9805101.pdf http://users.ictp.it/~pub_off/lectures/lns001/Looijenga/Looijenga.pdf
https://arxiv.org/abs/1603.00785
Hilbert scheme of points and moduli of stable sheaves https://arxiv.org/pdf/math/0304302.pdf
Rational Curves on Algebraic Varieties - https://link.springer.com/book/10.1007/978-3-662-03276-3
Add discussion of moduli of stable sheaves and relation to hilbert scheme Wundzer (talk) 01:54, 26 February 2020 (UTC)[reply]
http://www.mat.uniroma3.it/users/sernesi/hilbdott07.pdf — Preceding unsigned comment added by Wundzer (talk • contribs) 00:06, 30 March 2020 (UTC)[reply]

This page should discuss the results from https://arxiv.org/pdf/math/0411469.pdf — Preceding unsigned comment added by 128.138.65.151 (talk) 21:51, 30 August 2017 (UTC)[reply]

Bayer's Conjecture

Multigraded Hilbert scheme - https://arxiv.org/abs/math/0201271 proves Bayer's conjecture which defines the Hilbert scheme as a determinental variety in the grassmannian.
https://www.math.columbia.edu/~bayer/papers/Bayer-thesis.pdf