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
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