Hilbert scheme

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In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some fixed scheme. In fact such a parameter space breaks up into pieces, each piece corresponding to a Hilbert polynomial. The basic theory of Hilbert schemes was developed by Alexander Grothendieck around 1960.

1 Hilbert scheme of points on a manifold
2 Hilbert schemes and hyperkaehler geometry
3 Generalized Kummer variety
4 References

[edit] Hilbert scheme of points on a manifold

In most recently published papers, "Hilbert scheme" refers to the Hilbert scheme of points on a manifold, that is, a classifying space of ideals of finite codimension.

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(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 a blow-up of a singular subvariety on a symmetric square of $M$ . It was used by Mark Haiman in his proof of the positivity of the coefficients of some Macdonald polynomials.

The Hilbert scheme of a manifold of dimension 3 or more is usually not smooth.

[edit] Hilbert schemes and hyperkaehler geometry

Let $M$ be a complex Kaehler 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 $S y m 2 M$ are locally isomorphic to ${\Bbb C}^2 \times {\Bbb C}^2/\{\pm 1\}$ . The blow-up of ${\Bbb C}^2/\{\pm 1\}$ is $T^*{\Bbb C} P^1$ , 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, Kaehler manifold is hyperkaehler, as follows from Calabi-Yau theorem. Hilbert schemes of points on K3 and a 4-dimensional torus give two series of examples of hyperkahler manifolds: a Hilbert scheme of points on K3 and a generalized Kummer manifold.

[edit] Generalized Kummer variety

Let $T$ be a compact complex torus of complex dimension 2, $T [n]$ its Hilbert scheme. A 2-sheeted covering of $T [n]$ is a product of $T$ and an irreducible hyperkaehler manifold $K n - 1$ which is called a generalized Kummer variety. When $n = 1$ , $K n$ is a Kummer surface associated with $T$ . Chern numbers of $K n$ were computed here.

[edit] References

Beauville, A., Varietes Kahleriennes sont la premiere classe de Chern est nulle. J. Diff. Geom. 18, pp. 755-782 (1983).
I. Dolgachev, "Hilbert scheme" SpringerLink Encyclopaedia of Mathematics (2001)
A. Grothendieck, Les Schémas de Hilbert, Séminaire Bourbaki, t. 13, 1960/61, no. 221.
Fundamental Algebraic Geometry: Grothendieck's FGA Explained (OUP 2006)
David Mumford, Lectures on Curves on an Algebraic Surface