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.

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[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 hyperkähler geometry

Let M be a complex Kähler surface with c1 = 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 Sym2M 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, 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.

[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 Kn − 1 which is called a generalized Kummer variety. When n = 1, Kn is a Kummer surface associated with T. Chern numbers of Kn were computed here.

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