Moduli spaces of objects associated to a space \(X\) can be interesting because
They can inherit some of the structures of \(X\)
They some times reveal hidden structures of \(X\).
The main example is the Hilbert scheme parametrizing certain \(0\)-dimensional subschemes of length \(n\) of smooth projective surface \(X\), which we denote by \(\mathrm{Hilb}^n(X)\) or \(X^{[n]}\).
Most points in \(X^{[n]}\) are unordered set of \(n\) distinct points. But the Hilbert scheme consists of more complicated subschemes where these points collide. For example we could have a pair of collided two points \(x\) and a \(1\)-dimensional subspace of \(T_xX\), marking the tangent direction of the collision. This is also considered as a point in the Hilbert scheme.
When \(\dim X=1\), there is only one tangent direction so actually \(X^{[n]}=\mathrm{Sym}^n X\) is just the ordinary \(n\)-th symmetric product of \(X\) (which for affine schemes, it is defined as \(\mathrm{Spec}((\mathbb{C}[X]^{\otimes n})^{S_n})\)).
When \(\dim X=2\), the Hilbert scheme is smooth and presents a resolution of singularities (Fogarty) \[ X^{[n]} \to \mathrm{Sym}^n X.\]
When \(\dim X\ge 3\), the Hilbert scheme is not smooth in general.
The Hilbert scheme of points always come with a natural morphism
\[\pi : \mathrm{Hilb}^n(X) \to \mathrm{Sym}^n X\]
which basically maps a subscheme to its support, counting multiplicities,
\[Z \mapsto \sum_{x\in Z} \mathrm{length}(Z_x) [x].\]
There is a natural stratification of the symmetric power \(S^n X\) given by all partitions of \(n\), let \(n = v_1 + \cdots + v_k\) be a partition of \(n\), then we have a stratum
\[ S^n_vX = \left\{ \sum v_i [x_i] : x_i \text{ distinct} \right\},\]
\[ S^n X = \bigcup S^n_vX.\]
The starting example is the case of affine plane. By definition,
\[ (\mathbb{C}^2)^{[n]} = \{ I \subset \mathbb{C}[x,y] \mid \dim_\mathbb{C}\mathbb{C}[x,y]/I = n \}.\]
By considering the correspondence between cyclic \(\mathbb{C}[x,y]\)-module of dimension \(n\) with a given cyclic vector and the annihilator ideal \(I\),
\[\begin{align*} & \{ I\subset \mathbb{C}[x,y] \} \\ \leftrightarrow & \{ (M, m) \mid M = \mathbb{C}[x,y]m, \dim_\mathbb{C}M = n \}/\cong \\ \leftrightarrow &\{ (k^n, i) \mid i: k\to k^n, B_i\in \mathrm{End}(k^n), [B_1, B_2]=0 \}/\mathrm{GL}(k^n), \end{align*}\]
where in the last description we need to keep the cyclic property, i.e. \(k^n\) is generated by the vector mapped by \(i\). We have the following handy description of \((\mathbb{C}^2)^{[n]}\),
\[ (\mathbb{C}^2)^{[n]} = \left\{ (B_1, B_2, i) \mid B_i\in \mathrm{End}(k^n), [B_1, B_2]=0, k^n \text{ is stable} \right\}/\mathrm{GL}(k^n)\]
where \(i: k\to k^n\), the group \(\mathrm{GL}(k^n)\) acts as
\[ g\cdot (B_1, B_2, i) = (gB_1g^{-1}, gB_2g^{-1}, gi),\]
and the stability condition means that any \(k[B_1,B_2]\)-submodule of \(k^n\) generated by \(\mathrm{Im}i\) must equal to \(k^n\). This quotient gives a smooth variety since the action is free (can be proved by utilizing stability condition and utilize the submodule \(\ker(g-1)\)).
The quotient space \((\mathbb{C}^2)^{[n]}= \{(B_1,B_2,i)\}/\mathrm{GL}(k^n)\) carries a flat family of zero-dimensional subschemes of \(\mathbb{C}^2\) determined by (the kernel of) the natural surjection
\[\mathbb{C}[x,y]\to k^n\]
where \(x,y\) acts as \(B_1,B_2\) with cyclic vector \(i(1)\) on \(k^n\).
This is the same as computing the annihilator ideal of the cyclic vector \(i(1)\), \(I = \mathrm{ann}(i(1))\). Note that this depends on the \(\mathbb{C}[x,y]\) module structure of \(k^n\) as well as \(i(1)\), but if the module together with the cyclic vector \((M,v)\) are isomorphic, then it should give out the same ideal, thus establishing the bijection between points in \((\mathbb{C}^2)^{[n]}\) and the cyclic \(\mathbb{C}[x,y]\)-module of dimension \(n\) with a given cyclic vector.
The category \((M,v)\) contains morphisms, but their isomorphism classes corresponds to the points in \((\mathbb{C}^2)^{[n]}\). To be precise we should be thinking about the category of finite dimensional \(\mathbb{C}[x,y]\)-modules with a cyclic vector, so the corresponding category in the geometry side should be something like all of \((\mathbb{C}^2)^{[n]}\) taking together. It seems that by contracting all these isomorphisms, we have lost a lot of information, maybe there is a possible viewpoint for higher categorical structure here.
The matrix \(B_1\) encodes the action of \(x\) and the matrix \(B_2\) encodes the action of \(y\), so \((B_1,B_2)\) together actually encodes something similar to \(n\) points \((x,y)\) in \(\mathbb{C}^2\). Under the action of \(\mathrm{GL}(k^n)\), we see that it does not distinguish the order of these points (as expected). One can think of \((x,y)\) as two linear morphisms \((\mathbb{C}\to \mathbb{C})^2\), and the \(B_1, B_2\) is somehow the amalgamation of \(n\) such points without order.
Since \([B_1,B_2]=0\), we know from linear algebra that we can simultaneously upper-triangulize them, so they can be simultaneously put into a form exposing their eigenvalues \(x_1,\dots,x_n\) and \(y_1,\dots,y_n\). In this case the Hilbert-Chow morphism maps the point represented by \((B_1,B_2,i)\) to
\[[(B_1,B_2,i)]\mapsto \sum_{i=1}^n [(x_i,y_i)].\]
The fact that the Hilbert scheme is a fine moduli space means that the functor of points is representable by a scheme, and any family of \(0\)-dim length \(n\) subschemes of \(\mathbb{C}^2\) gives a morphism \(U\to (\mathbb{C}^2)^{[n]}\). This family is obtained by a pullback of the universal family over \((\mathbb{C}^2)^{[n]}\).
Consider the cute example where \(n=2\). When the two points are distinct, say \((x_1,y_1), (x_2,y_2)\), then the subscheme is given by the annihiling ideal
\[ I_{(x_1,y_1),(x_2,y_2)} = \{ f\in \mathbb{C}[x,y] \mid f(x_1,y_1)=f(x_2,y_2)=0 \}.\]
when these two points collide, the equations \(f(p_1)=0, f(p_2)=0\) collapsed to one equation and does not give a codimension \(2\) ideal. Instead we are looking at higher order constraints, in this case we can constraint the differential of \(f\) to vanish on certain direction, giving a \(\mathbb{P}^1\) family in \((\mathbb{C}^2)^{[2]}\),
\[ \mathbb{P}(T_p\mathbb{C}^2)\to (\mathbb{C}^2)^{[2]}, \quad (a : b) \mapsto \{ f \mid f(x,y)=0, (d f)_p|_{a\partial_x+b\partial_y}=0 \}.\]
So in fact \((\mathbb{C}^2)^{[2]}\) is a blow-up of \(\mathbb{C}^2\times \mathbb{C}^2\) along the diagonal and quotient by \(S_2\).
Can we have some how a generating function for all of the \((\mathbb{C}^2)^{[n]}\), something like
\[ \frac{1}{1-\mathbb{C}^2 x} = \sum_{n=0}^\infty (\mathbb{C}^2)^{[n]} x^n.\]
I think one difficulty in finding such expression is that \((\mathbb{C}^2)^{[n]}\) is obtained as a quotient, a set of isomorphism classes in a certain category, not just a product or some tensored structure. Maybe we should look at the category directly instead of the isomorphism classes.