Parameter Spaces and Moduli 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.