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 basic theory of Hilbert schemes was developed by Alexander Grothendieck (1961).
Hironaka's example shows that non-projective varieties need not have Hilbert schemes.
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 can informally be thought of as the families of subschemes of projective space parameterized by S. The Hilbert scheme
space as a subscheme of a Grassmannian defined by the vanishing of various determinants.
This can be shown by tensoring the exact sequence above by the locally flat sheaves
In addition to these properties, Francis Sowerby Macaulay (1927) determined for which polynomials the Hilbert scheme
Hilbert schemes can have bad singularities, such as irreducible components that are non-reduced at all points.
They can also have irreducible components of unexpectedly high dimension.
This construction is functorial by taking pullbacks of families.
is projective, then this functor is represented by the Hilbert scheme constructed above.
Generalizing this to the case of maps of finite type requires the technology of algebraic spaces developed by Artin.
[1] In its greatest generality, the Hilbert functor is defined for a finite type map of algebraic spaces
is a finite type map of schemes, their Hilbert functor is represented by an algebraic space.
This is because lines on smooth surfaces have negative self-intersection.
[3] Another common set of examples are the Hilbert schemes of
there is a nice geometric interpretation where the boundary loci
For example, the Hilbert scheme of degree 2 hypersurfaces in
with the universal hypersurface given by where the underlying ring is bigraded.
is globally generated, meaning its Euler characteristic is determined by the dimension of the global sections, so The dimension of this vector space is
is the sublocus of smooth curves in the Hilbert scheme.
For M of dimension at least 3 the morphism is not birational for large n: the Hilbert scheme is in general reducible and has components of dimension much larger than that of the symmetric product.
The Hilbert scheme of points on a curve C (a dimension-1 complex manifold) is isomorphic to a symmetric power of C. It is smooth.
The Hilbert scheme of n points on a surface is also smooth (Grothendieck).
This was used by Mark Haiman in his proof of the positivity of the coefficients of some Macdonald polynomials.
The canonical bundle of M is trivial, as follows from the Kodaira classification of surfaces.
This is used to show that the symplectic form is naturally extended to the smooth part of the exceptional divisors of
A holomorphically symplectic, Kähler manifold is hyperkähler, as follows from the Calabi–Yau theorem.
Hilbert schemes of points on the K3 surface and on a 4-dimensional torus give two series of examples of hyperkähler manifolds: a Hilbert scheme of points on K3 and a generalized Kummer surface.