CHAPTER 1

Hilbert scheme of points

In this chapter, we collect basic facts on the Hilbert scheme of points on a

surface. We do not assume the field k is C unless it is explicitly stated.

1.1. General Results on the Hilbert scheme

First, we recall the definition of the Hilbert scheme in general (not necessarily

of points, nor on a surface). Let X be a projective scheme over an algebraically

closed field k and OX (1) an ample line bundle on X. We consider the contravariant

functor HilbX from the category of schemes to the category of sets

HilbX : [Schemes] → [Sets],

which is given by

HilbX (U) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

Z ⊂ X × U

Z is a closed subscheme,

Z

i

→ X × U

π ↓ ↓ p2 : π is

flat⎪⎪

U = U

⎫

⎪

⎪

⎪

⎬

⎪

⎭

.

Namely, HilbX is a functor which associates a scheme U with a set of families of

closed subschemes in X parametrized by U. Let π : Z → U be the projection. For

u ∈ U, the Hilbert polynomial in u is defined by

Pu(m) = χ(OZu ⊗ OX (m)),

where Zu =

π−1(u).

Since Z is flat over U, Pu is independent of u ∈ U if U is

connected. Conversely, for each polynomial P , let HilbX

P

be the subfunctor of HilbX

which associates U with a set of families of closed subschemes in X parametrized by

U which has P as its Hilbert polynomial. Now the basic fact proved by Grothendieck

is the following theorem.

Theorem 1.1 (Grothendieck [50]). The functor HilbX

P

is representable by a

projective scheme HilbX

P

.

This means that there exists a universal family Z on HilbX

P

, and that every

family on U is induced by a unique morphism φ: U → HilbX

P

.

Moreover, if we have an open subscheme Y of X, then we have the correspond-

ing open subscheme HilbY

P

of HilbX

P

parametrizing subschemes in Y . In particular,

HilbY

P

is defined for a quasi-projective scheme Y .

The proof of this theorem is not given in this book. But we shall give a

concrete description when P is a constant polynomial and X is the aﬃne plane

A2

(see Theorem 1.9). Once this is established, one can prove the representatibity of

a similar functor for a nonsingular complex surface X by a patching argument (see

§1.5). In particular, we get HilbX

P

as a complex manifold, and this is practically

5

Hilbert scheme of points

In this chapter, we collect basic facts on the Hilbert scheme of points on a

surface. We do not assume the field k is C unless it is explicitly stated.

1.1. General Results on the Hilbert scheme

First, we recall the definition of the Hilbert scheme in general (not necessarily

of points, nor on a surface). Let X be a projective scheme over an algebraically

closed field k and OX (1) an ample line bundle on X. We consider the contravariant

functor HilbX from the category of schemes to the category of sets

HilbX : [Schemes] → [Sets],

which is given by

HilbX (U) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

Z ⊂ X × U

Z is a closed subscheme,

Z

i

→ X × U

π ↓ ↓ p2 : π is

flat⎪⎪

U = U

⎫

⎪

⎪

⎪

⎬

⎪

⎭

.

Namely, HilbX is a functor which associates a scheme U with a set of families of

closed subschemes in X parametrized by U. Let π : Z → U be the projection. For

u ∈ U, the Hilbert polynomial in u is defined by

Pu(m) = χ(OZu ⊗ OX (m)),

where Zu =

π−1(u).

Since Z is flat over U, Pu is independent of u ∈ U if U is

connected. Conversely, for each polynomial P , let HilbX

P

be the subfunctor of HilbX

which associates U with a set of families of closed subschemes in X parametrized by

U which has P as its Hilbert polynomial. Now the basic fact proved by Grothendieck

is the following theorem.

Theorem 1.1 (Grothendieck [50]). The functor HilbX

P

is representable by a

projective scheme HilbX

P

.

This means that there exists a universal family Z on HilbX

P

, and that every

family on U is induced by a unique morphism φ: U → HilbX

P

.

Moreover, if we have an open subscheme Y of X, then we have the correspond-

ing open subscheme HilbY

P

of HilbX

P

parametrizing subschemes in Y . In particular,

HilbY

P

is defined for a quasi-projective scheme Y .

The proof of this theorem is not given in this book. But we shall give a

concrete description when P is a constant polynomial and X is the aﬃne plane

A2

(see Theorem 1.9). Once this is established, one can prove the representatibity of

a similar functor for a nonsingular complex surface X by a patching argument (see

§1.5). In particular, we get HilbX

P

as a complex manifold, and this is practically

5