From Closed Sets to Functors

We have not yet settled the question which representable functors actually do arise from closed sets. The only answer to this in general is the following. 

First, we recall the definition of the Hilbert scheme in general (not necessarily of points, nor on a surface). Let X be a projective scheme over an algebraically closed field k and Ox 1) an ample line bundle on X. We consider the contravariant functor Hilbx from the category of schemes to the category of sets... 

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