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, 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.

Hilbert scheme of projective space

The Hilbert scheme of classifies closed subschemes of projective space in the following sense: For any locally Noetherian scheme, the set of -valued points
of the Hilbert scheme is naturally isomorphic to the set of closed subschemes of that are flat over. The closed subschemes of that are flat over can informally be thought of as the families of subschemes of projective space parameterized by. 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. Each of these pieces is projective over.

Construction

Grothendieck constructed the Hilbert scheme of -dimensional projective space over a Noetherian scheme as a subscheme of a Grassmannian defined by the vanishing of various determinants. Its fundamental property is that for a scheme over, 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, depending only on the Hilbert polynomial of, all higher cohomology groups of with coefficients in vanish, so in particular has dimension, where is the Hilbert polynomial of projective space.
Pick a sufficiently large value of. The -dimensional space is a subspace of the -dimensional space, so represents a point of the Grassmannian. This will give an embedding of the piece of the Hilbert scheme corresponding to the Hilbert polynomial into this Grassmannian.
It remains to describe the scheme structure on this image, in other words to describe enough elements for the ideal corresponding to it. Enough such elements are given by the conditions that the map has rank at most for all positive, which is equivalent to the vanishing of various determinants.

Properties

Universality

Given a closed subscheme over a field with Hilbert polynomial, the Hilbert scheme has a universal subscheme flat over such that
The tangent space of the point is given by the global sections of the normal bundle ; that is,

Unobstructedness of complete intersections

For local complete intersections such that, the point is smooth. This implies every deformation of in is unobstructed.

Dimension of tangent space

In the case, the dimension of at is greater than or equal to.
In addition to these properties, determined for which polynomials the Hilbert scheme is non-empty, and showed that if is non-empty then it is linearly connected. So two subschemes of projective space are in the same connected component of the Hilbert scheme if and only if they have the same Hilbert polynomial.
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. For example, one might expect the Hilbert scheme of points of a scheme of dimension to have dimension, but if its irreducible components can have much larger dimension.

Functorial interpretation

There is an alternative interpretation of the Hilbert scheme which leads to a generalization of relative Hilbert schemes parameterizing subschemes of a relative scheme. For a fixed base scheme, let and let
be the functor sending a relative scheme to the set of isomorphism classes of the set
where the equivalence relation is given by the isomorphism classes of. This construction is functorial by taking pullbacks of families. Given, there is a family over.

Representability for projective maps

If the structure map 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.

Relative Hilbert scheme for maps of algebraic spaces

In its greatest generality, the Hilbert functor is defined for a finite type map of algebraic spaces defined over a scheme. Then, the Hilbert functor is defined as
sending
This functor is not representable by a scheme, but by an algebraic space. Also, if, and is a finite type map of schemes, their Hilbert functor is represented by an algebraic space.

Examples of Hilbert schemes

Fano schemes of hypersurfaces

One of the motivating examples for the investigation of the Hilbert scheme in general was the Fano scheme of a projective scheme. Given a subscheme of degree there is a scheme in parameterizing where is a -plane in, meaning it is a degree one embedding of. For smooth surfaces in of degree, the non-empty Fano schemes are smooth an zero dimensional. This is because lines on smooth surfaces have negative self-intersection.

Hilbert scheme of points

Another common set of examples are the Hilbert schemes of -points of a scheme, typically denoted. For there is a nice geometric interpretation where the boundary loci describing the intersection of points can be thought of parametrizing points along with their tangent vectors. For example, is the blowup of the diagonal modulo the symmetric action.

Degree d hypersurfaces

The Hilbert scheme of degree k hypersurfaces in is given by the projectivization. For example, the Hilbert scheme of degree 2 hypersurfaces in is with the universal hypersurface given by
where the underlying ring is bigraded.

Hilbert scheme of points on a manifold

"Hilbert scheme" sometimes refers to the punctual Hilbert scheme of 0-dimensional subschemes on a scheme. Informally this can be thought of as something like finite collections of points on a scheme, though this picture can be very misleading when several points coincide.
There is a Hilbert–Chow morphism from the reduced Hilbert scheme of points to the Chow variety of cycles taking any 0-dimensional scheme to its associated 0-cycle..
The Hilbert scheme of points on is equipped with a natural morphism to an -th symmetric product of. This morphism is birational for of dimension at most 2. For of dimension at least 3 the morphism is not birational for large : 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 is isomorphic to a symmetric power of. It is smooth.
The Hilbert scheme of points on a surface is also smooth. If, it is obtained from by blowing up the diagonal and then dividing by the action induced by. It was used by Mark Haiman in his proof of the positivity of the coefficients of some Macdonald polynomials.
The Hilbert scheme of a smooth manifold of dimension 3 or more is usually not smooth.

Hilbert schemes and hyperkähler geometry

Let be a complex Kähler surface with . The canonical bundle of is trivial, as follows from the Kodaira classification of surfaces. Hence admits a holomorphic symplectic form. It was observed by Akira Fujiki and Arnaud Beauville that is also holomorphically symplectic. This is not very difficult to see, e.g., for. Indeed, is a blow-up of a symmetric square of. Singularities of are locally isomorphic to The blow-up of is, and this space is symplectic. This is used to show that the symplectic form is naturally extended to the smooth part of the exceptional divisors of. It is extended to the rest of by Hartogs' principle.
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.