# Introduction

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

Example 10.1

Consider the moduli space of points of ${\displaystyle \mathbb {C} ^{2}}$ satisfying ${\displaystyle y=x^{2}}$ and ${\displaystyle y=0}$, that is, ${\displaystyle Z(x^{2},y)\subset \mathbb {C} ^{2}}$. As a set, this is clearly just the origin. However, the ring of functions is ${\displaystyle \mathbb {C} [x,y]/(y-x^{2},y)\cong \mathbb {C} [x]/(x^{2})}$, and as a scheme this is ${\displaystyle {\text{Spec }}\mathbb {C} [x]/(x^{2})}$. 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 ${\displaystyle Z(y-x^{2},y-t)}$, for some ${\displaystyle t}$, then we will have two closed points ${\displaystyle (x-{\sqrt {t}},t),(x+{\sqrt {t}},t)}$, which will be distinct if ${\displaystyle t\neq 0}$. We want to be able to see that setting ${\displaystyle t=0}$ still gives us more than just a point - it should give us a thickened point, or double point. So we need to consider schemes.