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?
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1$\begingroup$ 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? $\endgroup$– IMeasyCommented Mar 9, 2013 at 17:47
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1$\begingroup$ 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. $\endgroup$– user30379Commented Mar 9, 2013 at 18:25
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$\begingroup$ Hey put your comment in an answer so that I validate it! $\endgroup$– IMeasyCommented Mar 9, 2013 at 20:44
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1$\begingroup$ Dear IMeasy: My above comment can stand on its own merits without needing "validation". :) $\endgroup$– user30379Commented Mar 10, 2013 at 5:17
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$\begingroup$ welll yes.... it is actually what I meant! $\endgroup$– IMeasyCommented Mar 15, 2013 at 13:59
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2 Answers
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The construction is done in Geometry of Algebraic Curves by Arbarello et al.
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There is a book by Nakajima "Lectures on Hilbert Schemes of points of surfaces". First chapter will do.
