Introduction

Moduli spaces parametrizing objects associated with a given space X are a

rich source of spaces with interesting structures. They usually inherit some of the

structures of X, but sometimes even more: they have more structures than X

has, or pull out hidden structures of X. The purpose of these notes is to give

an example of these phenomena. We study the moduli space parametrizing 0-

dimensional subschemes of length n in a nonsingular quasi-projective surface X

over C. It is called the Hilbert scheme of points, and denoted by

X[n].

An example of a 0-dimensional subscheme is a collection of distinct points. In

this case, the length is equal to the number of points. When some points collide,

more complicated subschemes appear. For example, when two points collide, we

get infinitely near points, that is a pair of a point x and a 1-dimensional subspace

of the tangent space TxX. This shows the difference between

X[n]

and the nth

symmetric product

SnX,

on which the information of the 1-dimensional subspace

is lost.

However, when X is 1-dimensional, we have a unique 1-dimensional subspace

in TxX. In fact, the Hilbert scheme X[n] is isomorphic to SnX when dim X = 1.

When X is 2-dimensional, X[n] is smooth and there is a morphism π : X[n] →

SnX which is a resolution of singularities by a result of Fogarty [31]. This presents

a contrast to Hilbert schemes for dim X 2, which, in general, have singularities.

As we mentioned at the beginning, X[n] inherits structures from X. First of

all, it is a scheme. It is projective if X is projective. These facts follow from

Grothendieck’s construction of Hilbert schemes. A nontrivial example is a result

by Beauville [11]: X[n] has a holomorphic symplectic form when X has one. When

X is projective, X has a holomorphic symplectic form only when X is a K3 surface

or an abelian surface by classification theory. We also have interesting noncompact

examples: X =

C2

or X = T

∗Σ

where Σ is a Riemann surface. These examples are

particularly nice because of the existence of a

C∗-action,

which naturally induces

an action on

X[n].

(See Chapter 7.) Moreover, for X =

C2,

we shall construct a

hyper-K¨ ahler structure on X[n] in Chapter 3.

These structures of X[n], discussed in the first half of this note, are inherited

from X. We shall begin to study newly arising structures in later chapters. They

appear when we consider the components X[n] all together. We will encounter this

phenomenon first in G¨ ottsche’s formula for the Poincar´ e polynomials in Chapter 6.

Their generating function is given by

∞

n=0

qnPt(X[n])

=

∞

m=1

(1 +

t2m−1qm)b1(X)(1

+

t2m+1qm)b3(X)

(1 − t2m−2qm)b0(X)(1 − t2mqm)b2(X)(1 − t2m+2qm)b4(X)

,

1

Moduli spaces parametrizing objects associated with a given space X are a

rich source of spaces with interesting structures. They usually inherit some of the

structures of X, but sometimes even more: they have more structures than X

has, or pull out hidden structures of X. The purpose of these notes is to give

an example of these phenomena. We study the moduli space parametrizing 0-

dimensional subschemes of length n in a nonsingular quasi-projective surface X

over C. It is called the Hilbert scheme of points, and denoted by

X[n].

An example of a 0-dimensional subscheme is a collection of distinct points. In

this case, the length is equal to the number of points. When some points collide,

more complicated subschemes appear. For example, when two points collide, we

get infinitely near points, that is a pair of a point x and a 1-dimensional subspace

of the tangent space TxX. This shows the difference between

X[n]

and the nth

symmetric product

SnX,

on which the information of the 1-dimensional subspace

is lost.

However, when X is 1-dimensional, we have a unique 1-dimensional subspace

in TxX. In fact, the Hilbert scheme X[n] is isomorphic to SnX when dim X = 1.

When X is 2-dimensional, X[n] is smooth and there is a morphism π : X[n] →

SnX which is a resolution of singularities by a result of Fogarty [31]. This presents

a contrast to Hilbert schemes for dim X 2, which, in general, have singularities.

As we mentioned at the beginning, X[n] inherits structures from X. First of

all, it is a scheme. It is projective if X is projective. These facts follow from

Grothendieck’s construction of Hilbert schemes. A nontrivial example is a result

by Beauville [11]: X[n] has a holomorphic symplectic form when X has one. When

X is projective, X has a holomorphic symplectic form only when X is a K3 surface

or an abelian surface by classification theory. We also have interesting noncompact

examples: X =

C2

or X = T

∗Σ

where Σ is a Riemann surface. These examples are

particularly nice because of the existence of a

C∗-action,

which naturally induces

an action on

X[n].

(See Chapter 7.) Moreover, for X =

C2,

we shall construct a

hyper-K¨ ahler structure on X[n] in Chapter 3.

These structures of X[n], discussed in the first half of this note, are inherited

from X. We shall begin to study newly arising structures in later chapters. They

appear when we consider the components X[n] all together. We will encounter this

phenomenon first in G¨ ottsche’s formula for the Poincar´ e polynomials in Chapter 6.

Their generating function is given by

∞

n=0

qnPt(X[n])

=

∞

m=1

(1 +

t2m−1qm)b1(X)(1

+

t2m+1qm)b3(X)

(1 − t2m−2qm)b0(X)(1 − t2mqm)b2(X)(1 − t2m+2qm)b4(X)

,

1