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

Example 10.1

Consider the moduli space of points of satisfying and , that is, . As a set, this is clearly just the origin. However, the ring of functions is , and as a scheme this is . 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 , for some , then we will have two closed points , which will be distinct if . We want to be able to see that setting still gives us more than just a point - it should give us a thickened point, or double point. So we need to consider schemes.