Hilbert scheme on arvau

Wikipedia / 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 (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.

Hilbert scheme of projective space

classifies closed subschemes of projective space in the following sense: For any locally Noetherian scheme , the set of -valued points

that are flat over that are flat over can informally be thought of as the families of subschemes of projective space parameterized by corresponding to the Hilbert scheme of the subschemes of projective space with Hilbert polynomial .

Construction as a determinantal variety

space as a subscheme of a Grassmannian defined by the vanishing of various determinants.

of the polynomial ring vanish. Using the exact sequence is the Hilbert polynomial of projective space. This can be shown by tensoring the exact sequence. Note that we are using the equality of the Hilbert polynomial of a coherent sheaf with the Euler-characteristic of its sheaf cohomology groups.

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. (A more careful analysis shows that it is enough just to take .)

Universality

, the Hilbert scheme such that

Tangent space

; that is,

Unobstructedness of complete intersections

is smooth. This implies every deformation is unobstructed.

Dimension of tangent space

In addition to these properties, is non-empty, and 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 (more precisely dimension 0, length subschemes) of a scheme of dimension to have dimension , but if its irreducible components can have much larger dimension.

Functorial interpretation

and let to the set of isomorphism classes of the setwhere the equivalence relation.

Representability for projective maps

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

. Then, the Hilbert functor is defined as

sending T to

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. are smooth and zero-dimensional. This is because lines on smooth surfaces have negative self-intersection.

Hilbert scheme of points

. For a Riemann surface of the diagonal modulo the symmetric action.

Degree d hypersurfaces

with the universal hypersurface given by

where the underlying ring is bigraded.

Hilbert scheme of curves and moduli of curves

is globally generated, meaning its Euler characteristic is determined by the dimension of the global sections, so

The dimension of this vector space curve. Using the Riemann-Roch formula, the associated Hilbert polynomial can be computed as

Then, the Hilbert scheme

parameterizes all genus g curves. Constructing this scheme is the first step in the construction of the moduli stack of algebraic curves. The other main technical tool are GIT quotients, since this moduli space is constructed as the quotient

is the sublocus of smooth curves in the Hilbert scheme.

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. .

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 (a dimension-1 complex manifold) is isomorphic to a symmetric power of . It is smooth.

The Hilbert scheme of points on a surface. This 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 (K3 surface or a torus). 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 is a blow-up of a symmetric square 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.

References

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