1.2. HILBERT SCHEME OF POINTS ON THE PLANE 9

with H → H, they glue together to define a morphism φ: U → H. By construction

φ∗H

is Z. The uniqueness of such a homomorphism is also clear.

Remark 1.10. (1) For k = C, we will give another argument to show the

smoothness of H in Corollary 3.42. The essential point is the observation that H

is a hyper-K¨ ahler quotient. Then we will observe that the smoothness of H is a

formal consequence of the free-ness of the GLn(k)-action, and that the smoothness

of H follows from the existence of a slice for a compact Lie group action, instead

of Luna’s slice theorem. Thus the proof will become natural and elementary.

(2) The above proof gives the representability of the functor HilbX

P

in the

special case when X = A2, P = n without invoking Theorem 1.1. In a preliminary

version of this book, we use Theorem 1.1 and then identify H with HilbX

P

as a set.

The above refinement is due to M.A. de Cataldo and L. Migliorini [22].

(3) As shown in the proof of the theorem, the ideal I corresponding to (B1,B2,i)

is given by

I = {f(z) ∈ k[z1,z2] | f(B1,B2)i(1) = 0}.

Note that this can be written as

I = {f(z) ∈ k[z1,z2] | f(B1,B2) = 0},

by the stability condition.

(4) The above description gives the tangent space of

(A2)[n]

at the point cor-

responding to (B1,B2,i). Let us consider the complex

Hom(kn,kn)

d1

−→

Hom(kn,kn)

⊕

Hom(kn,kn)

⊕

kn

d2

−→

Hom(kn,kn),

(1.11)

where d1(ξ) =

⎛

⎝[ξ,

[ξ, B1]

B2]⎠,

ξi

⎞

d2

⎛

⎝C2⎠

C1

I

⎞

= [B1,C2]+[C1,B2]. The homomorphism d1 is

the derivative of the GLn(k)-action, and d2 is that of the map (B1,B2,i) → [B1,B2].

We have shown that the cokernel of d2 has dimension n. It is also easy to see the

kernel of d1 is trivial by the stability condition. The above construction of H implies

that the tangent space is the middle cohomology group of the above complex. Note

also that it has dimension 2n.

Now let us examine some examples.

Example 1.12. (1) First, we consider the case of n = 1. We write B1 =

λ, B2 = μ ∈ k. From the stability condition, i must be non-zero. Hence we may

assume i = 1, after applying the action of GL1(k) if necessary. The corresponding

ideal is given by

I = {f(z) ∈ k[z1,z2] | f(λ, μ) = 0}.

Therefore it corresponds to the point (λ, μ) ∈

k2,

and this gives the description

(A2)[1]

∼

=

A2.

(2) Next, we consider the case when n = 2. Suppose either B1 or B2 has two

distinct eigenvalues. It is easy to see that we can assume B1 =

λ1 0

0 λ2

and

with H → H, they glue together to define a morphism φ: U → H. By construction

φ∗H

is Z. The uniqueness of such a homomorphism is also clear.

Remark 1.10. (1) For k = C, we will give another argument to show the

smoothness of H in Corollary 3.42. The essential point is the observation that H

is a hyper-K¨ ahler quotient. Then we will observe that the smoothness of H is a

formal consequence of the free-ness of the GLn(k)-action, and that the smoothness

of H follows from the existence of a slice for a compact Lie group action, instead

of Luna’s slice theorem. Thus the proof will become natural and elementary.

(2) The above proof gives the representability of the functor HilbX

P

in the

special case when X = A2, P = n without invoking Theorem 1.1. In a preliminary

version of this book, we use Theorem 1.1 and then identify H with HilbX

P

as a set.

The above refinement is due to M.A. de Cataldo and L. Migliorini [22].

(3) As shown in the proof of the theorem, the ideal I corresponding to (B1,B2,i)

is given by

I = {f(z) ∈ k[z1,z2] | f(B1,B2)i(1) = 0}.

Note that this can be written as

I = {f(z) ∈ k[z1,z2] | f(B1,B2) = 0},

by the stability condition.

(4) The above description gives the tangent space of

(A2)[n]

at the point cor-

responding to (B1,B2,i). Let us consider the complex

Hom(kn,kn)

d1

−→

Hom(kn,kn)

⊕

Hom(kn,kn)

⊕

kn

d2

−→

Hom(kn,kn),

(1.11)

where d1(ξ) =

⎛

⎝[ξ,

[ξ, B1]

B2]⎠,

ξi

⎞

d2

⎛

⎝C2⎠

C1

I

⎞

= [B1,C2]+[C1,B2]. The homomorphism d1 is

the derivative of the GLn(k)-action, and d2 is that of the map (B1,B2,i) → [B1,B2].

We have shown that the cokernel of d2 has dimension n. It is also easy to see the

kernel of d1 is trivial by the stability condition. The above construction of H implies

that the tangent space is the middle cohomology group of the above complex. Note

also that it has dimension 2n.

Now let us examine some examples.

Example 1.12. (1) First, we consider the case of n = 1. We write B1 =

λ, B2 = μ ∈ k. From the stability condition, i must be non-zero. Hence we may

assume i = 1, after applying the action of GL1(k) if necessary. The corresponding

ideal is given by

I = {f(z) ∈ k[z1,z2] | f(λ, μ) = 0}.

Therefore it corresponds to the point (λ, μ) ∈

k2,

and this gives the description

(A2)[1]

∼

=

A2.

(2) Next, we consider the case when n = 2. Suppose either B1 or B2 has two

distinct eigenvalues. It is easy to see that we can assume B1 =

λ1 0

0 λ2

and