One idea which has been coming up a lot in the SMSTC course on Algebraic Geometry, lectured by Clark Barwick, has been the idea of the functor of points. Here, we’ll try and motivate that definition with some examples.

Throughout, we work over an algebraically field $k$, and so schemes and morphisms are relative to $k$.

Let $X$ be a scheme. Associated to this, we have a functor $h_X : \mathrm{Sch}/k^{\mathrm{op}} \to \mathrm{Sets}$, given by $$h_X(Y) = \mathrm{Mor}(Y, X)$$ and this is called the functor of points.

If $R$ is a $k$-algebra, then we write $X(R) = h_X(\mathrm{Spec}(R))$, and these are called the $R$-valued points of $X$.

Group schemes

A group scheme $G$ is a scheme $G$, along with morphisms $m : G \times G \to G$, $i : G \to G$ and $e : \bullet \to G$, satisfying diagrams which looks like the group axioms. More concretely, if $R$ is any $k$-algebra, then $G(R)$ is a group.

Why do we want to think about things in this way? One reason is that we can now define the multiplicative group, and the additive group, and so on, and work on these things uniformly. For example, we can then define $\mathrm{GL}_n$ as a group scheme, where $\mathrm{GL}_n(R)$ is the space of invertible $(n \times n)$-matrices with entries in $R$. This can be seen as formalising the intuitive notion of “everything we do here is an algebraic operation, so everything should work”.

Tangent space

We can think of $\mathrm{Spec}(k[\epsilon]/\epsilon^2)$ as a point with a preferred direction, and so points in $X(k[\epsilon]/\epsilon^2)$ can be considered as tangent vectors on $X$.

Vector bundles

Let $\mathbb{CP}^n$ denote complex projective $n$-space. That is, the space of lines in $\mathbb C^{n+1}$ through the origin. This has a tautological line bundle $$\mathcal O(-1) = \{(p, l) \in \mathbb C^{n+1} \times \mathbb{CP}^n \mid p \in l\} \to \mathbb{CP}^n$$

Let $X$ be a topological space, $L \to X$ a complex line bundle. Assume that $X$ is compact Hausdorff, and so we have sections $s_0, \dots s_n$, such that at each $x \in X$, at least one of the $s_i$ is non-zero. Choose a metric $h$ on $L$, then we have a map $f \colon L \to X \times \mathbb C^{n+1}$, given by $$f(x, \xi) = (x, h(s_0(x), \xi), \dots, h(s_n(x), \xi))$$ Thus, we have embedded $L$ as a subbundle of a trivial bundle. Thus, we can define a map $\phi : X \to \mathbb{CP}^n$ by $$\phi(x) = f(L_x)$$ By construction, $\phi^*\mathcal{O}(-1) = L$.

Any map into $\mathbb{CP}^n$ will induce such a line bundle, and so we have a correspondence $$\{\text{maps }X \to \mathbb{CP}^n\} \leftrightarrow \{\text{line bundles }L \to X\}$$ This is not a bijection, as homotopic maps will pull back the same line bundle, but we will treat it as a correspondence for now.

Returning to algebraic geometry, the left hand side becomes $h_{\mathbb P^n}(X)$, and the right hand side becomes line bundles which are generated by $(n+1)$ global sections. Thus, the functor $h_{\mathbb P^n}$ is giving us information about (globally generated) line bundles.

More generally, we can replace projective space with an appropriate Grassmannian, and we get vector bundles of higher rank.

Moduli

Finally, we come to the topic of moduli. Say we are interested in some objects of type $\star$ (substistute $\star$ for what you want, e.g. curves, or vector bundles, or …). We want to put it into some form of “moduli space” $M$ say. What do we want this to do? Often, in algebraic geometry, we consider families of objects, e.g. families of conics, cubics and so on. We can represent this as a morphism $X \to S$, where the fibres are all of type $\star$. Then as a map of sets, we have a map $S \to M$.

What we want instead is for this to be a map of varieties/schemes(/stacks…). Once we have this, we get a correspondence $$h_M(S) \leftrightarrow \{\text{families of }\star\text{ over }S\}$$

Thus, starting by considering families of objects, we have (quite naturally) ended up with the functor of points on the left hand side. So far, we have assumed that $M$ actually exists, but one can still make sense of the functor without assuming $M$ makes sense, and this is a starting point to (modern) moduli theory.