References: Lectures on the Hilbert Schemes of Points on Surfaces by Hiraku Nakajima
Recall that the simplest Hilbert scheme \((\mathbb{A}^2)^{[n]}\) can be expressed as the space of \(n\)-dimensional cyclic \(\mathbb{C}[x,y]\)-module together with a cyclic vector. Equivalently,
\[ (\mathbb{A}^2)^{[n]} = \{(B_1, B_2, i) : [B_1,B_2]=0, \text{no submodule containing }i\} /\mathrm{GL}_n(\mathbb{C}) \]
It is also easy to pass the module to ideals, by just taking the annihilator of the cyclic vector, \(I = \{f\in\mathbb{C}[x,y] : f(B_1,B_2)i = 0\}\).
First Nakajima shows that the differential of the map \((B_1,B_2,i)\mapsto [B_1,B_2]\) has constant rank. Start by looking at the cokernel of the differential \(\mathrm{d}\varphi : V\to W\), which is \(W/\mathrm{Im}f\). Think about the dual space of the cokernel is the kernel of \(W^*\to (\mathrm{Im}f)^*\), and since there is a non-degenerate inner product, the trace form on \(W\), such subspace can be identified back to a subspace of \(W\). Thus we identify the cokernel in \(W\) as the subspace
\[\begin{align*} \{w: \mathrm{tr}(w \,\mathrm{d}f) = 0\} &= \{w: \mathrm{tr}(w ([\mathrm{d}B_1, B_2]+[B_1, \mathrm{d}B_2])) = 0, \forall \mathrm{d}B_1, \mathrm{d}B_2\} \\ &= \{w: \mathrm{tr}(\mathrm{d}B_1 [B_2, w]) + \mathrm{tr}(\mathrm{d}B_2 [w, B_1]) = 0, \forall \mathrm{d}B_1, \mathrm{d}B_2\} \\ &= \{w: [w,B_1] = [w,B_2] = 0\}. \end{align*}\]
Here we used an elementary fact that \(\mathrm{tr}(A[B,C])\) is a sign representation of \(S_3\) and is a cyclic invariant in \(A,B,C\).
The matrix description can be understood as a holomorphic symplectic quotient, and we have a holomorphic symplectic form \(\omega \in \Omega^2_{(\mathbb{A}^2)^{[n]}}\) that is everywhere non-degenerate. In fact
\[ \text{Symplectic Structure }X \Rightarrow \text{Symplectic Structure }X^{[n]}. \]
We can use the affine translation structure on \(\mathbb{C}^2\) to move every point such that the average of \(n\) points is the origin. This gives us a decomposition
\[(\mathbb{C}^2)^{[n]} = \mathbb{C}^2\times ((\mathbb{C}^2)^{[n]}/\mathbb{C}^2).\]
In terms of matrices, this sends
\[(B_1, B_2, i)\mapsto ((\mathrm{tr}B_1, \mathrm{tr}B_2), (B_1-\mathrm{tr}B_1, B_2-\mathrm{tr}B_2, i)).\]
And the set \((\mathbb{C}^2)^{[n]}/\mathbb{C}^2\) is identifiable with the subset of \((\mathbb{C}^2)^{[n]}\) consisting of traceless matrices.
We will see the existence of symplectic structure helps us estimate the dimension of fibres of the Hilbert-Chow morphism \(\pi: (\mathbb{C}^2)^{[n]}\to \mathrm{Sym}^n(\mathbb{C}^2)\).
The subvariety \(\pi^{-1}(n[0])\) lies in the subset \((\mathbb{C}^2)^{[n]}/\mathbb{C}^2\) and is isotropic with respect to the symplectic form on \((\mathbb{C}^2)^{[n]}/\mathbb{C}^2\). This tells us we must have
\[ \dim \pi^{-1}(n[0]) \le \frac{\dim (\mathbb{C}^2)^{[n]}/\mathbb{C}^2}{2} = n-1.\]
Moreover there exists at least one component of \(\pi^{-1}(n[0])\) of dimension \(n-1\).
Proof.
\[ \Phi_{t_1,t_2}(z_1,z_2) = (t_1z_1, t_2z_2). \]
Lifting to \((\mathbb{C}^2)^{[n]}\) this is the action of multiplying matrices by scalars, or changing the module structure by multiplying scalars.