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.
The Hilbert scheme formula_1 of formula_2 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 formula_4 that are flat over . The closed subschemes of formula_4 that are flat over can informally be thought of as the families of subschemes of projective space parameterized by . The Hilbert scheme formula_1 breaks up as a disjoint union of pieces formula_7 corresponding to the Hilbert polynomial of the subschemes of projective space with Hilbert polynomial . Each of these pieces is projective over formula_8.
Construction.
Grothendieck constructed the Hilbert scheme formula_9 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 formula_10 that are flat over .
If is a subscheme of -dimensional projective space, then corresponds to a graded ideal formula_11 of the polynomial ring in formula_12 variables, with graded pieces formula_13. For sufficiently large , depending only on the Hilbert polynomial of , all higher cohomology groups of with coefficients in vanish, so in particular formula_13 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. (A more careful analysis shows that it is enough just to take .)
Properties.
Universality.
Given a closed subscheme formula_15 over a field with Hilbert polynomial formula_16, the Hilbert scheme has a universal subscheme formula_17 flat over formula_18 such that
Tangent space.
The tangent space of the point formula_24 is given by the global sections of the normal bundle formula_33; that is,
Unobstructedness of complete intersections.
For local complete intersections formula_35 such that formula_36, the point formula_37 is smooth. This implies every deformation of formula_35 in formula_21 is unobstructed.
Dimension of tangent space.
In the case formula_40, the dimension of formula_18 at formula_42 is greater than or equal to formula_43.
In addition to these properties, determined for which polynomials the Hilbert scheme formula_7 is non-empty, and showed that if formula_7 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.
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 formula_46, let formula_47 and letformula_48be the functor sending a relative scheme formula_49 to the set of isomorphism classes of the setformula_50where the equivalence relation is given by the isomorphism classes of formula_51. This construction is functorial by taking pullbacks of families. Given formula_52, there is a family formula_53 over formula_54.
Representability for projective maps.
If the structure map formula_55 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 formula_56 defined over a scheme formula_46. Then, the Hilbert functor is defined as
sending "T" to 
This functor is not representable by a scheme, but by an algebraic space. Also, if formula_60, and formula_61 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 formula_62 of degree formula_63, there is a scheme formula_64 in formula_65 parameterizing formula_66 where formula_18 is a formula_68-plane in formula_2, meaning it is a degree one embedding of formula_70. For smooth surfaces in formula_71 of degree formula_72, the non-empty Fano schemes formula_64 are smooth and 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 formula_74-points of a scheme formula_21, typically denoted formula_76. For formula_77 there is a nice geometric interpretation where the boundary loci formula_78 describing the intersection of points can be thought of parametrizing points along with their tangent vectors. For example, formula_79 is the blowup formula_80 of the diagonal modulo the symmetric action.
Degree d hypersurfaces.
The Hilbert scheme of degree k hypersurfaces in formula_2 is given by the projectivization formula_82. For example, the Hilbert scheme of degree 2 hypersurfaces in formula_83 is formula_77 with the universal hypersurface given by
where the underlying ring is bigraded.
Hilbert scheme of curves and moduli of curves.
For a fixed genus formula_86 algebraic curve formula_87, the degree of the tri-tensored dualizing sheaf formula_88 is globally generated, meaning its Euler characteristic is determined by the dimension of the global sections, so
The dimension of this vector space is formula_90, hence the global sections of formula_88 determine an embedding into formula_92 for every genus formula_86 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
where formula_97 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. .
The Hilbert scheme formula_98 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 is also smooth (Grothendieck). If formula_99, it is obtained from formula_100 by blowing up the diagonal and then dividing by the formula_101 action induced by formula_102. 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 surface with formula_103 (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 (for formula_99 and Arnaud Beauville that formula_98 is also holomorphically symplectic. This is not very difficult to see, e.g., for formula_99. Indeed, formula_107 is a blow-up of a symmetric square of . Singularities of formula_108 are locally isomorphic to formula_109. The blow-up of formula_110 is formula_111, 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 formula_98. It is extended to the rest of formula_98 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.