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 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.

Contents

Hilbert scheme of points on a manifold

Generalized Kummer variety

References

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.
=== 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 $Sym^2\; M$ are locally isomorphic to
$\{Bbb\; C\}^2\; imes\; \{Bbb\; C\}^2/\{pm\; 1\}$. 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.

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.

References

★ Beauville, A., ''Varietes Kahleriennes dont la premiere classe de Chern est nulle.'' J. Diff. Geom. '18', pp. 755-782 (1983).

★

★ 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''