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I am confused by Hilbert Scheme and Chow Scheme. Whenever you have a point in hilbert scheme, take its fiber in the universal family and take its fumdamental class, we get a point in Chow Scheme; and if we have a point in Chow scheme, take its fiber and then take its support, we get a point in Hilbert scheme. I know they should not be the same, but what is the difference between Hilbert scheme and Chow Scheme?

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    $\begingroup$ A given closed subset can have lots of different scheme structures. If you haven't already, try googling "Hilbert-Chow morphism". $\endgroup$
    – Lazzaro Campeotti
    Commented Nov 25, 2015 at 22:28

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Consider the Chow variety of $2$ points in $\mathbb P^1$. When the points collide, the support of that cycle is $1$ point. I.e. the map you attempted to define from the Chow variety to "the" Hilbert scheme goes to the wrong Hilbert scheme.

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  • $\begingroup$ can we just interpret the support of those two coincide points with multiplicity two? $\endgroup$
    – user42804
    Commented Nov 26, 2015 at 19:58
  • $\begingroup$ Read about the Hilbert scheme of 2 points in the plane. $\endgroup$
    – Allen Knutson
    Commented Nov 27, 2015 at 3:23

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