文档介绍：LECTURE 21
Parameter Spaces and Moduli Spaces
Parameter Spaces
We can now give a slightly expanded introduction to the notion of parameter space,
introduced in Lecture 4 and discussed occasionally since. This is a fairly delicate
subject, and one that is clearly best understood from the point of view of scheme
theory, so that in some sense this discussion violates our basic principle of dealing
only with topics that can be reasonably well understood on an elementary level.
Nevertheless, since it is one of the fundamental constructions of algebraic geometry,
and since the constructions can at least be described in an elementary fashion, we
will proceed. One unfortunate consequence of this sort of overreaching, however,
is that the density of unproved assertions, high enough in the rest of the text, will
reach truly appalling levels in this lecture.
With this understood, let us first say what we should mean by a parameter
space. The basic situation is that we are given a collection of subvarieties X,
of a projective space P”­for example, the set of all varieties of a given dimension
and degree or the subset of those with a given Hilbert polynomial. The problem is
then to give a bijection between this set {X,} and the points of an algebraic variety
2. Of course, not just any bijection will do; we want to choose a correspondence
that is reasonably natural, in the sense that as the point X varies continuously in
2, the coefficients of the defining equations of the varieties X c P”  should likewise
vary continuously, in whatever topology. Clearly, the first thing to do is to make
precise this requirement.
One way to do es from the construction of the universal hypersurface:
for example, we saw that if we associated to each hypersurface X c P”  of degree d
a point X E PN, then the subset of the product
” 
{(X, p): p E X} c PN x P
Parameter Spaces 267
is in fact a subvariety. In general, we can take the analogous s