We want to investigate the following space
\[\mathrm{Hilb}^n(\mathbb{C}^2/\mathbb{Z}_2) = \{ I \subset \mathbb{C}[x,y]^{\mathbb{Z}_2} \mid \mathrm{codim\,}I = n \}\]
where \(\mathrm{Hilb}^n(\mathbb{C}^2/\mathbb{Z}_2)\) is the Hilbert scheme of \(n\) points on \(\mathbb{C}^2/\mathbb{Z}_2\).
In principle this is a Poisson manifold with singularities.
The Poisson structure \(\{,\}\) gives us a bivector field \(\pi\in \Gamma(\wedge^2 TM)\)
\[\pi(\mathrm{d}f, \mathrm{d}g) = \{f,g\}.\]
Therefore partial application of \(\pi\) gives a map \(\pi^\sharp: T^*M \to TM\), and we have \(\pi^\sharp(\mathrm{d}f) = \{f,\cdot\} = X_f\).
For example \(\mathbb{R}^{2n}=\{(q_1,\dots,q_n, p_1,\dots,p_n)\}\) has Poisson structure given by standard symplectic form \(\omega = \sum_{i=1}^n \mathrm{d}q_i \wedge \mathrm{d}p_i\).
One has
\[\{f,g\} = \sum_{i=1}^n \left( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right)\]
and therefore
\[\pi = \sum_{i=1}^n \frac{\partial}{\partial q_i} \wedge \frac{\partial}{\partial p_i}\]
Let’s check what happens in \(n=2\), let’s compute what happens near the ideal \(I=(x^4,xy,y^2)\), also denoted as point \(p\). The functions are generated by \(x^2, xy, y^2\), all functions are linear combinations of monomials coming from the products of these generators.
Let’s compute \(T^*_pM\), because we want to learn the rank of the Poisson structure.
I tried, but I did not succeed in doing so. There are some difficulties:
I need to understand the cotangent space T_p^*M for a particular ideal, I don’t know how to do that.
I don’t know what exactly is computing {f, g} for f, g doing? That seems weird to me because what are the functions on the space Hilb^n(..)? There might be more functions than just these generator looking functions.
Consider two possible directions:
Using quiver picture, I think we can compute that tangent space. It seems unclear to me, how do we obtain ideal description using this obscure bridge?
Using some combinatorial descriptions, but I have to find and establish such picture before we can proceed!