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I was recently reading one of the papers of Kapranov on $\mathcal{M}_{0,n}$ and he says that one can see it as a subvariety of the Hilbert scheme parametrizing all subschemes of a certain projective space. What is the precise definition of this Hilbert scheme? Why it is sure it exists? Is it in EGA somewhere or does anybody have a reference?

flag	asked Mar 9 at 9:35 IMeasy 1,355●4●12

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don't take me bad... but, to obtain a Hilbert scheme, doesn't one need to fix the Hilbert polynomial? The Hilb-scheme parametrizing all subchemes is just the disjoint union of the different Hilb-schemes? – IMeasy Mar 9 at 17:47

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Yes, the Hilbert scheme classifying flat closed (finitely presented) subschemes of a projective space is a disjoint union indexed by the various Hilbert polynomials. The functor makes sense without reference to ample line bundles or Hilbert polynomials (and as such is an algebraic space when one goes beyond the setting of projective geometry). The Hilbert polynomial is a means of cutting it into (not obviously) finite-type pieces in the projective case; there is nothing like that available for Hilbert functors of more general proper finitely presented schemes in place of projective spaces. – pranavk Mar 9 at 18:25

Hey put your comment in an answer so that I validate it! – IMeasy Mar 9 at 20:44

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Dear IMeasy: My above comment can stand on its own merits without needing "validation". :) – pranavk Mar 10 at 5:17

I think he means so that he can accept it as the answer. – arsmath Mar 14 at 21:43

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Damien L · Accepted Answer · 2013-03-09 10:30:10Z UTC

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The construction is done in Geometry of Algebraic Curves by Arbarello et al.

answered Mar 9 at 10:30

Damien L
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Marc Palm · Answer 2 · 2013-03-09 09:48:50Z UTC

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There is a book by Nakajima "Lectures on Hilbert Schemes of points of surfaces". First chapter will do.

answered Mar 9 at 9:48

Marc Palm
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one “big” Hilbert scheme?

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