High Quality Content by WIKIPEDIA articles! 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 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). The Hilbert scheme HilbPn of n-dimensional projective space classifies closed subschemes of projective space in the following sense: For any locally Noetherian scheme S, the set of S-valued points Hom(S,HilbPn) of the Hilbert scheme is naturally isomorphic to the set of closed subschemes of PnxS that are flat over S. The closed subschemes of PnxS that are flat over S can informally be thought of as the families of subschemes of projective space parameterized by S.

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