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
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
and the nth
symmetric product
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 =
or X = T
where Σ is a Riemann surface. These examples are
particularly nice because of the existence of a
which naturally induces
an action on
(See Chapter 7.) Moreover, for X =
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 ottsche’s formula for the Poincar´ e polynomials in Chapter 6.
Their generating function is given by


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