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 Alexander Grothendieck (1961). Hironaka's example shows that non-projective varieties need not have Hilbert schemes.

Hilbert scheme of projective space

The Hilbert scheme $Hilb (n)$ of $P n$ 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 $P n \times S$ that are flat over $S$ . The closed subschemes of $P n \times S$ 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 Grassmannian defined by the vanishing of various determinants. Its fundamental property is that for a scheme $T$ over $S$ , it represents the functor whose $T$ -valued points are the closed subschemes of $P n \times S T$ that are flat over $T$ .

If $X$ is a subscheme of $n$ -dimensional projective space, then $X$ corresponds to a graded ideal $I X$ of the polynomial ring $S$ in $n + 1$ variables, with graded pieces $I X (m)$ . 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 $I X (m)$ 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 $I X (m)$ is a subspace of the $Q (m)$ -dimensional space $S (m)$ , so represents a point of the Grassmannian $Gr (Q (m) - P (m), Q (m))$ . 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 $I X (m) \otimes S (k) \to S (k + m)$ has rank at most $dim(I X (k + m))$ 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

Macaulay (1927) determined for which polynomials the Hilbert scheme $Hilb (n, P)$ is non-empty, and Hartshorne (1966) 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, length $d$ subschemes) of a scheme of dimension $n$ to have dimension $dn$ , but if $n \geq 3$ its irreducible components can have much larger dimension.

Examples

The hilbert scheme of points of a closed point is just a point.
The hilbert scheme of degree k hypersurfaces in $\mathbb {P} ^{n}$ is given by the projectivization $\mathbb {P} (\Gamma ({\mathcal {O}}(k)))$ . For example, the hilbert scheme of degree 2 hypersurfaces in $\mathbb {P} ^{1}$ is $\mathbb {P} ^{2}$ with the universal hypersurface given by

${\text{Proj}}(k[x_{0},x_{1}][\alpha ,\beta ,\gamma ]/(\alpha x_{0}^{2}+\beta x_{0}x_{1}+\gamma x_{1}^{2}))\subseteq \mathbb {P} _{x_{0},x_{1}}^{1}\times \mathbb {P} _{\alpha ,\beta ,\gamma }^{2}$

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. (Fogarty 1968, 1969, 1973).

The Hilbert scheme $M [n]$ 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 $n = 2$ , it is obtained from $M \times M$ by blowing up the diagonal and then dividing by the $Z /2 Z$ action induced by $(x, y) \mapsto (y, x)$ . 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 $M$ be a complex Kähler surface with $c 1 = 0$ (K3 surface or a torus). The canonical bundle of $M$ is trivial, as follows from Kodaira classification of surfaces. Hence $M$ admits a holomorphic symplectic form. It was observed by Fujiki (for $n = 2$ ) and Beauville that $M [n]$ is also holomorphically symplectic. This is not very difficult to see, e.g., for $n = 2$ . Indeed, $M [2]$ is a blow-up of a symmetric square of $M$ . Singularities of $Sym 2 M$ are locally isomorphic to $C 2 \times C 2 /{\pm1}.$ The blow-up of $C 2 /{\pm1}$ is $T * P 1 (C)$ , 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 $M [n]$ . It is extended to the rest of $M [n]$ 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 manifolds: a Hilbert scheme of points on K3 and a generalized Kummer manifold.

References

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External links

Bertram, Aaron (1999), Construction of the Hilbert scheme, retrieved 2008-09-06
Bolognese, Barbara; Losev, Ivan, A general introduction to the Hilbert scheme of points on the plane (PDF), Archived from the original on 2017-08-30
Maclagan, Diane, Notes on Hilbert Schemes (PDF), Archived from the original on 2016-03-07

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