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).
Read more about this topic: Hilbert Scheme
Famous quotes containing the words scheme and/or space:
“Your scheme must be the framework of the universe; all other schemes will soon be ruins.”
—Henry David Thoreau (1817–1862)
“In bourgeois society, the French and the industrial revolution transformed the authorization of political space. The political revolution put an end to the formalized hierarchy of the ancien regimé.... Concurrently, the industrial revolution subverted the social hierarchy upon which the old political space was based. It transformed the experience of society from one of vertical hierarchy to one of horizontal class stratification.”
—Donald M. Lowe, U.S. historian, educator. History of Bourgeois Perception, ch. 4, University of Chicago Press (1982)