Recall of what we did, and what the question is.
We were studying the Hilbert scheme of points on ADE singularities, i.e. the scheme
\[X=\mathrm{Hilb}^n(\mathbb{C}^2/\Gamma)\]
where \(\Gamma\) is a finite subgroup of \(\mathrm{SL}_2(\mathbb{C})\), and \(n\) is a positive integer.
Let \(X\) be Poisson and \(Y\subset X\) be a locally closed Hamiltonian invariant subset.
We have that \(\overline{Y}\subset X\) is closed under Hamiltonian action.
Let \(\Gamma = \mathbb{Z}_2 = \{\pm 1\}\) be the simplest non-trivial case.
Then we know that, denote \(R = \mathbb{C}[x,y]^\Gamma = \mathbb{C}[x^2,xy,y^2]\), the Hilbert scheme of points on \(\mathbb{C}^2/\mathbb{Z}_2\) admits a description as the set of ideals of codimension \(n\) in the ring \(R\):
\[\mathrm{Hilb}^n(\mathbb{C}^2/\mathbb{Z}_2) = \{ I \subset \mathbb{C}[x^2,xy,y^2] \mid \dim_\mathbb{C}(R/I) = n \}.\]
Prove that the closures of symplectic leaves \(\overline{\{\text{Symplectic Leaves}\}}\subset \mathrm{Hilb}^n(\mathbb{C}^2/\Gamma)\) are exactly given by
\[S_{2(n-k^2)}=\left\{I: \mathrm{codim\,}I = n, I\subset \mathfrak{m}^k\right\} ,\quad\mathfrak{m}= (x^2,xy,y^2) . \]
which will guarantee that
\[n = \mathrm{codim\,}I \ge \mathrm{codim\,}\mathfrak{m}^k = k^2.\]
Find an explicit description of these closures, either in terms of ideals or geometric description / picture.
Give conceptual reason for the decomposition.
Think about \(\mathrm{Hilb}^n(\mathbb{C}^2/\mathbb{Z}_3)\) instead.
What we know:
We know from the quiver picture about all the possible stratifications, their dimensions, and Hasse diagram of the Hilbert scheme of points on \(\mathbb{C}^2/\Gamma\).
We know by the quiver picture that the Hilbert scheme of points on \(\mathbb{C}^2/\mathbb{Z}_2\) has two strata
\[\mathrm{Hilb}^2(\mathbb{C}^2/\mathbb{Z}_2) = S_4 \sqcup S_2.\]
where
\(S_4\) is the four dimensional stratum consisting of two distinct points away from origin (which has degree of freedom \(4\)), or collided points at the origin that does not lie inside the cone (whose dimension is expected to be \(\mathbb{P}^2\), which is \(2\) complex dimension and \(4\) real dimensions).
\(S_2\) is the closure of the set of points of the form \(\{0, p\}\) where \(p\in \mathbb{C}^2\) is a point away from the origin. Taking closure gets you collided points at the origin whose ‘direction’ lies inside the cone, and is parametrized by \(\mathbb{P}^1\).
Can we find explicit description of these ideals? Can we determine which ideals lie in which strata?
For ideals that has a component at origin, and for ideals that are concentrated at the origin (collided), what is the condition for them to lie inside the cone?
Again \(\Gamma = \mathbb{Z}_2\) and we let \(n\ge 2\). From the quiver picture again we predict a stratification of the form
\[\mathrm{Hilb}^n(\mathbb{C}^2/\mathbb{Z}_2) = S_{2n} \sqcup S_{2(n-1^2)}\sqcup S_{2(n-2^2)} \sqcup \cdots \sqcup S_{2(n-\lfloor\sqrt{n}\rfloor^2)}.\]
It is almost clear that \(S_{2n}\) should be the closure of \(n\) distinct points away from the origin, taking closure might give you some extra collisions that are actually smooth
\[S_{2n} = \overline{\pi^{-1}(\Delta^c)}\]
where \(\Delta^c\) is the complement of all the diagonals inside \(\mathrm{Sym}^n(\mathbb{C}^2/\mathbb{Z}_2)\).
(Warn) It is then reasonable to guess that \(S_{2(n-1)}\) is the closure of points of the form \(\{0, p_2, \ldots, p_{n}\}\) where \(p_i\in \mathbb{C}^2\) are distinct points away from the origin.
(Warn) Similarly we can understand that \(S_{2(n-k^2)}\) is the closure of points of the form \(\{0, \dots, 0, p_{k+1}, \ldots, p_{n}\}\) where \(p_i\in \mathbb{C}^2\) are distinct points away from the origin.
(Warn) Strangely this list ends at \(k=\lfloor\sqrt{n}\rfloor\), that means if you putting more points at the origin, they will be able to escape to the cone and belongs to one of the strata.
Monomial ideals are ideals generated by monomials. They give us more combinatorial picture, and are the fixed points of \(\mathbb{C}^*\) action on the Hilbert scheme of points.
The monomial ideals \(I\) are in correspondence with the skew partitions of \(n\), some pattern in the lattice \(L=(0,2)\mathbb{N}+ (1,1)\mathbb{N}+ (2,0)\mathbb{N}\). For any partition \(\lambda\subset L\) such that \(|\lambda|=n\), we can associate \(I_\lambda\) to be the ideal generated by the monomials outside \(\lambda\).
\[\mathrm{Part}_L(n) \to \mathrm{MonIdeal}(n) : \lambda \mapsto I_\lambda = \langle x^a y^b \mid (a,b) \notin \lambda \rangle.\]
Codimension of the monomial ideal \(I_\lambda\) is given by the size of the partition \(\lambda\), i.e.
\[\mathrm{codim\,}I_\lambda = |\lambda| = n.\]
Example. For \(n=2\) the partitions are \(\{(0,0),(0,2)\}, \{(0,0), (1,1)\}, \{(0,0),(2,0)\}\), and the corresponding ideals are
\[I_{\{(0,0),(0,2)\}} = \langle x^2,xy \rangle,\quad I_{\{(0,0),(1,1)\}} = \langle x^2,y^2 \rangle,\quad I_{\{(0,0),(2,0)\}} = \langle xy,y^2 \rangle.\]
Intersection. When we intersect two monomial ideals \(I,J\) corresponding to \(\lambda(I)\) and \(\lambda(J)\), the ideal \(I\cap J\) corresponds to the enlarged partition \(\lambda(I)\cup\lambda(J)\), which is the union of the two skew partitions.
\[I_{\lambda_1} \cap I_{\lambda_2} = I_{\lambda_1 \cup \lambda_2}.\]
Sum. When we sum two monomial ideals \(I_1, I_2\) corresponding to \(\lambda_1, \lambda_2\), the ideal \(I_1 + I_2\) corresponds to the partition \(\lambda_1 \cap \lambda_2\), which is the intersection of the two skew partitions.
\[I_{\lambda_1} + I_{\lambda_2} = I_{\lambda_1 \cap \lambda_2}.\]
\(\mathfrak{m}\) and all power \(\mathfrak{m}^k\) are monomial ideals, they play an important role here, since
they are Hamiltonian invariant (Note: Haven’t proved precisely),
Moreover, they have an interesting picture in the lattice, they are the lower left triangles, with codimension \(1+3+...+2k-1 = k^2\). Let us denote this triangle as \(T_k\).
Intersection of \(I\) with \(\mathfrak{m}^k\).
To compute the codimension of \(I\cap \mathfrak{m}^k\), we note that
\[I_\lambda \cap \mathfrak{m}^k = I_{\lambda \cup T_k}\]
therefore its codimension rises if \(\lambda\) does not contain \(T_k\). Otherwise, if \(\lambda\supset T_k\), we have \(I_\lambda\cap \mathfrak{m}^k = I_\lambda\) and therefore \(I_\lambda\subset \mathfrak{m}^k\).
Corollary. These monomial ideals \(I\) all has component at the origin, since all \(\lambda\) contain \((0,0) = T_1\) therefore \(I\subset \mathfrak{m}\).
This means that these monomial ideals will not lie in every stratum, in particular this proves they cannot lie in the open stratum \(S_{2n}\).
But they look promising as they seem to cover at least one example of points in every lower strata \(S_{2(n-k^2)}\).
This gives us a way to count the number of monomial ideals (i.e. \(\mathbb{C}^*\) fixed points) in each stratum \(S_{2(n-k^2)}\).
We want to know if all the three monomial ideals lie in the stratum \(S_2\).
How do we know if they lie in the smooth stratum or not?
I’m thinking, can we find a representation of \(Q_\Gamma\) whose annihilator is the ideal? That representation will tell us if the ideal is smooth or not.