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.

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).

Grothendieck constructed the Hilbert scheme Hilb(n)_S of n-dimensional projective space over a Noetherian scheme S as a subscheme of a Grassmannian defined by the vanishing of various determinants. Its fundamental property is that for a scheme T over S, it represents the functor whose T-valued points are the closed subschemes of Pⁿ ×_ST that are flat over T.

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