Hilbert scheme
Encyclopedia
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 scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomial
s. The basic theory of Hilbert schemes was developed by .
of the Hilbert scheme is naturally isomorphic to the set of closed subschemes of Pn × S that are flat
over S. The closed subschemes of Pn × S that are flat over S can informally be thought of as the families of subschemes of projective space parameterized by S. The Hilbert scheme Hilb(n) breaks up as a disjoint union of pieces Hilb(n, P) corresponding to the Hilbert polynomial of the subschemes of projective space with Hilbert polynomial P. Each of these pieces is projective over Spec(Z).
defined by the vanishing of various determinant
s. Its fundamental property is that for a scheme T over S, it represents the functor whose T valued points are the closed subschemes of Pn ×S T that are flat over T.
If X is a subscheme of n-dimensional projective space, then X corresponds to a graded ideal IX of the polynomial ring S in n+1 variables, with graded pieces IX(m). For sufficiently large m, depending only on the Hilbert polynomial P of X, all higher cohomology groups of X with coefficients in O(m) vanish, so in particular IX(m) has dimension Q(m) − P(m), where Q is the Hilbert polynomial of projective space.
Pick a sufficiently large value of m. The Q(m) − P(m)-dimensional space IX(m) is a subspace of the Q(m)-dimensional space S(m), so represents a point of the Grassmannian G(Q(m) − P(m), Q(m)). This will give an embedding of the piece of the Hilbert scheme corresponding to the Hilbert polynomial P 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 from IX(m) ⊗ S(k) to 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.)
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, degree d subschemes) of a scheme of dimension n to have dimension dn, but if n≥3 its irreducible components can have much larger dimension.
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 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
points on a surface is
also smooth (Grothendieck). If
, it is a blow-up of
a singular subvariety on a symmetric square of
. It was used by Mark Haiman
in his proof
of the positivity of the coefficients of some Macdonald polynomial
s.
The Hilbert scheme of a smooth manifold of dimension
3 or more is usually not smooth.
be a complex Kähler
surface with
(K3 surface
or a torus). The canonical bundle of
is trivial, as follows from Kodaira classification of surfaces
. Hence
admits a holomorphic symplectic form. It was observed by Fujiki (for 
) and 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 Calabi-Yau theorem
. Hilbert schemes of points on K3
and a 4-dimensional torus give two series of examples of hyperkähler manifold
s: a Hilbert scheme of points on K3 and a generalized Kummer manifold.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomial
Hilbert polynomial
In commutative algebra, the Hilbert polynomial of a graded commutative algebra or graded module is a polynomial in one variable that measures the rate of growth of the dimensions of its homogeneous components...
s. The basic theory of Hilbert schemes was developed by .
Hilbert scheme of projective space
The Hilbert scheme Hilb(n) of Pn classifies closed subschemes of projective space in the following sense: For any locally Noetherian scheme S, the set of S-valued points-
-
-
- Hom(S, Hilb(n))
-
-
of the Hilbert scheme is naturally isomorphic to the set of closed subschemes of Pn × S that are flat
Flat morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,is a flat map for all P in X...
over S. The closed subschemes of Pn × S that are flat over S can informally be thought of as the families of subschemes of projective space parameterized by S. The Hilbert scheme Hilb(n) breaks up as a disjoint union of pieces Hilb(n, P) corresponding to the Hilbert polynomial of the subschemes of projective space with Hilbert polynomial P. Each of these pieces is projective over Spec(Z).
Construction
Grothendieck constructed the Hilbert scheme Hilb(n)S of n-dimensional projective space over a Noetherian scheme S as a subscheme of a GrassmannianGrassmannian
In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...
defined by the vanishing of various determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
s. Its fundamental property is that for a scheme T over S, it represents the functor whose T valued points are the closed subschemes of Pn ×S T that are flat over T.
If X is a subscheme of n-dimensional projective space, then X corresponds to a graded ideal IX of the polynomial ring S in n+1 variables, with graded pieces IX(m). For sufficiently large m, depending only on the Hilbert polynomial P of X, all higher cohomology groups of X with coefficients in O(m) vanish, so in particular IX(m) has dimension Q(m) − P(m), where Q is the Hilbert polynomial of projective space.
Pick a sufficiently large value of m. The Q(m) − P(m)-dimensional space IX(m) is a subspace of the Q(m)-dimensional space S(m), so represents a point of the Grassmannian G(Q(m) − P(m), Q(m)). This will give an embedding of the piece of the Hilbert scheme corresponding to the Hilbert polynomial P 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 from IX(m) ⊗ S(k) to 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.)
Variations
The Hilbert scheme Hilb(X)S is defined and constructed for any projective scheme X in a similar way. Informally, its points correspond to closed subschemes of X.Properties
determined for which polynomials the Hilbert scheme Hilb(n, P) is non-empty, and showed that if Hilb(n, P) 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, degree d subschemes) of a scheme of dimension n to have dimension dn, but if n≥3 its irreducible components can have much larger dimension.
Hilbert scheme of points on a manifold
"Hilbert scheme" sometimesrefers 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
-thsymmetric product of
. This morphism isbirational 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
Symmetric product of an algebraic curve
In mathematics, the n-fold symmetric product of an algebraic curve C is the quotient space of the n-fold cartesian productor Cn by the group action of the symmetric group on n letters permuting the factors. It exists as a smooth algebraic variety ΣnC; if C is a compact Riemann surface it is...
of C. It is smooth.
The Hilbert scheme of
points on a surface isalso smooth (Grothendieck). If
, it is a blow-up ofa singular subvariety on a symmetric square of
. It was used by Mark HaimanMark Haiman
Mark David Haiman is a mathematician at the University of California at Berkeley who proved theMacdonald positivity conjecture for Macdonald polynomials.-References:* Haiman's *Mark Haiman J. Amer. Math. Soc. 14 , 941–1006...
in his proof
of the positivity of the coefficients of some Macdonald polynomial
Macdonald polynomial
In mathematics, Macdonald polynomials Pλ are a family of orthogonal polynomials in several variables, introduced by...
s.
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 KählerKähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
surface with
(K3 surfaceK3 surface
In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....
or a torus). The canonical bundle of
is trivial, as follows from Kodaira classification of surfacesEnriques-Kodaira classification
In mathematics, the Enriques–Kodaira classification is a classification of compact complex surfaces into ten classes. For each of these classes, the surfaces in the class can be parametrized by a moduli space...
. Hence
admits a holomorphic symplectic form. It was observed by Fujiki (for ) and BeauvilleArnaud Beauville
Arnaud Beauville is a French mathematician, whose research interest is algebraic geometry.A native of Bologne, Haute-Marne, Beauville earned his doctorate from Paris Diderot University in 1977, with a thesis regarding Prym varieties and the Schottky problem, under supervision of Jean-Louis...
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
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
is hyperkähler
Hyperkähler manifold
In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp(k) In differential geometry, a hyperkähler...
, as follows from Calabi-Yau theorem
Calabi conjecture
In mathematics, the Calabi conjecture was a conjecture about the existence of good Riemannian metrics on complex manifolds, made by and proved by ....
. Hilbert schemes of points on K3
K3 surface
In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....
and a 4-dimensional torus give two series of examples of hyperkähler manifold
Hyperkähler manifold
In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp(k) In differential geometry, a hyperkähler...
s: a Hilbert scheme of points on K3 and a generalized Kummer manifold.