Hilbert Scheme - Hilbert Scheme of Projective Space

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

Hom(S, Hilb(n))

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).