Hilbert scheme
In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by (Alexander Grothendieck 1961). Hironaka's example shows that non-projective varieties need not have Hilbert schemes.
Hilbert scheme of projective space
The Hilbert scheme Hilb(n) of Pn classifies closed subschemes of projective space in the following sense: For any locally Noetherian scheme S, the set of S-valued points
of the Hilbert scheme is naturally isomorphic to the set of closed subschemes of Pn × S that are flat over S. The closed subschemes of Pn × S that are flat over S can informally be thought of as the families of subschemes of projective space parameterized by S. The Hilbert scheme Hilb(n) breaks up as a disjoint union of pieces Hilb(n, P) corresponding to the Hilbert polynomial of the subschemes of projective space with Hilbert polynomial P. Each of these pieces is projective over Spec(Z).
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