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Concept
Hilbert scheme
Summary
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 . Hironaka's example shows that non-projective varieties need not have Hilbert schemes.
The Hilbert scheme of 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 that are flat over S. The closed subschemes of that are flat over S can informally be thought of as the families of subschemes of projective space parameterized by S. The Hilbert scheme breaks up as a disjoint union of pieces corresponding to the Hilbert polynomial of the subschemes of projective space with Hilbert polynomial P. Each of these pieces is projective over .
Grothendieck constructed the Hilbert scheme of -dimensional projective space as a subscheme of a Grassmannian defined by the vanishing of various determinants. Its fundamental property is that for a scheme , it represents the functor whose -valued points are the closed subschemes of that are flat over .
If is a subscheme of -dimensional projective space, then corresponds to a graded ideal of the polynomial ring in variables, with graded pieces . For sufficiently large all higher cohomology groups of with coefficients in vanish. Using the exact sequencewe have has dimension , where is the Hilbert polynomial of projective space. This can be shown by tensoring the exact sequence above by the locally flat sheaves , giving an exact sequence where the latter two terms have trivial cohomology, implying the triviality of the higher cohomology of . Note that we are using the equality of the Hilbert polynomial of a coherent sheaf with the Euler-characteristic of its sheaf cohomology groups.
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