# Introduction

A moduli space, loosely speaking, is a 'parameter' space: a space parametrizing certain objects on $X$ , so that the points of the moduli space $M$ correspond to objects on $X$ . However, we need to know about than just the points of $M$ , as the following example illustrates.

Example 10.1

Consider the moduli space of points of $\mathbb {C} ^{2}$ satisfying $y=x^{2}$ and $y=0$ , that is, $Z(x^{2},y)\subset \mathbb {C} ^{2}$ . As a set, this is clearly just the origin. However, the ring of functions is $\mathbb {C} [x,y]/(y-x^{2},y)\cong \mathbb {C} [x]/(x^{2})$ , and as a scheme this is ${\text{Spec }}\mathbb {C} [x]/(x^{2})$ . The scheme is the object which is going to give us information on the non-reduced structure, not the set of closed points.

We will want a moduli space to take into account deformations. I define this below, but in this example, if we take $Z(y-x^{2},y-t)$ , for some $t$ , then we will have two closed points $(x-{\sqrt {t}},t),(x+{\sqrt {t}},t)$ , which will be distinct if $t\neq 0$ . We want to be able to see that setting $t=0$ still gives us more than just a point - it should give us a thickened point, or double point. So we need to consider schemes.