Moduli scheme of subschemes of a scheme, represents the flat-family-of-subschemes functor
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 Alexander Grothendieck (1961). 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 S, the set of S-valued points
formula_3
of the Hilbert scheme is naturally isomorphic to the set of closed subschemes of formula_4 that are flat over S. The closed subschemes of formula_4 that are flat over S can informally be thought of as the families of subschemes of projective space parameterized by S. The Hilbert scheme formula_1 breaks up as a disjoint union of pieces formula_7 corresponding to the Hilbert scheme of the subschemes of projective space with Hilbert polynomial P. Each of these pieces is projective over formula_8.
Construction as a determinantal variety.
Grothendieck constructed the Hilbert scheme formula_1 of formula_10-dimensional projective formula_2 space as a subscheme of a Grassmannian defined by the vanishing of various determinants. Its fundamental property is that for a scheme formula_12, it represents the functor whose formula_12-valued points are the closed subschemes of formula_14 that are flat over formula_12.
If formula_16 is a subscheme of formula_10-dimensional projective space, then formula_16 corresponds to a graded ideal formula_19 of the polynomial ring formula_20 in formula_21 variables, with graded pieces formula_22. For sufficiently large formula_23 all higher cohomology groups of formula_16 with coefficients in formula_25 vanish. Using the exact sequenceformula_26we have formula_27 has dimension formula_28, where formula_29 is the Hilbert polynomial of projective space. This can be shown by tensoring the exact sequence above by the locally flat sheaves formula_30, giving an exact sequence where the latter two terms have trivial cohomology, implying the triviality of the higher cohomology of formula_31. Note that we are using the equality of the Hilbert polynomial of a coherent sheaf with the Euler-characteristic of its sheaf cohomology groups.
Pick a sufficiently large value of formula_23. The formula_33-dimensional space formula_22 is a subspace of the formula_35-dimensional space formula_36, so represents a point of the Grassmannian formula_37. This will give an embedding of the piece of the Hilbert scheme corresponding to the Hilbert polynomial formula_38 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 "IX"("m") ⊗ "S"("k") → "S"("k" + "m") has rank at most dim("IX"("k" + "m")) for all positive k, which is equivalent to the vanishing of various determinants. (A more careful analysis shows that it is enough just to take "k" 
 1.)
Universality.
Given a closed subscheme formula_39 over a field with Hilbert polynomial formula_40, the Hilbert scheme H=Hilb("n", "P") has a universal subscheme formula_41 flat over formula_42 such that
Tangent space.
The tangent space of the point formula_48 is given by the global sections of the normal bundle formula_57; that is,
formula_58
Unobstructedness of complete intersections.
For local complete intersections formula_59 such that formula_60, the point formula_61 is smooth. This implies every deformation of formula_59 in formula_16 is unobstructed.
Dimension of tangent space.
In the case formula_64, the dimension of formula_42 at formula_66 is greater than or equal to formula_67.
In addition to these properties, Francis Sowerby Macaulay (1927) determined for which polynomials the Hilbert scheme formula_7 is non-empty, and Robin Hartshorne (1966) 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 d points (more precisely dimension 0, length d subschemes) of a scheme of dimension n to have dimension dn, but if "n" ≥ 3 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_20, let formula_71 and letformula_72be the functor sending a relative scheme formula_73 to the set of isomorphism classes of the setformula_74where the equivalence relation is given by the isomorphism classes of formula_75. This construction is functorial by taking pullbacks of families. Given formula_76, there is a family formula_77 over formula_78.
Representability for projective maps.
If the structure map formula_79 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_80 defined over a scheme formula_20. Then, the Hilbert functor is defined as
formula_82
sending "T" to 
formula_83.
This functor is not representable by a scheme, but by an algebraic space. Also, if formula_84, and formula_85 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_86 of degree formula_87, there is a scheme formula_88 in formula_89 parameterizing formula_90 where formula_42 is a formula_92-plane in formula_2, meaning it is a degree one embedding of formula_94. For smooth surfaces in formula_95 of degree formula_96, the non-empty Fano schemes formula_88 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_10-points of a scheme formula_16, typically denoted formula_100. For a Riemann surface X, formula_101. For formula_102 there is a nice geometric interpretation where the boundary loci formula_103 describing the intersection of points can be thought of parametrizing points along with their tangent vectors. For example, formula_104 is the blowup formula_105 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_107. For example, the Hilbert scheme of degree 2 hypersurfaces in formula_108 is formula_102 with the universal hypersurface given by
formula_110
where the underlying ring is bigraded.
Hilbert scheme of curves and moduli of curves.
For a fixed genus formula_111 algebraic curve formula_112, the degree of the tri-tensored dualizing sheaf formula_113 is globally generated, meaning its Euler characteristic is determined by the dimension of the global sections, so
formula_114.
The dimension of this vector space is formula_115, hence the global sections of formula_113 determine an embedding into formula_117 for every genus formula_111 curve. Using the Riemann-Roch formula, the associated Hilbert polynomial can be computed as
formula_119.
Then, the Hilbert scheme
formula_120
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
formula_121,
where formula_122 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. (Fogarty 1968, 1969, 1973).
The Hilbert scheme formula_123 of n points on M is equipped with a natural morphism to an n-th symmetric product of M. This morphism is birational for M of dimension at most 2. 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). If formula_124, it is obtained from formula_125 by blowing up the diagonal and then dividing by the formula_126 action induced by formula_127. 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 M be a complex Kähler surface with formula_128 (K3 surface or a torus). The canonical bundle of M is trivial, as follows from the Kodaira classification of surfaces. Hence M admits a holomorphic symplectic form. It was observed by Akira Fujiki (for formula_124) and Arnaud Beauville that formula_123 is also holomorphically symplectic. This is not very difficult to see, e.g., for formula_124. Indeed, formula_132 is a blow-up of a symmetric square of M. Singularities of formula_133 are locally isomorphic to formula_134. The blow-up of formula_135 is formula_136, 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_123. It is extended to the rest of formula_123 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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