Abstract
Moduli spaces of objects associated to a space of \(X\) are interesting, the Hilbert scheme is a scheme parametrizing closed subschemes of \(X\) with a fixed Hilbert polynomial. In particular the simplest case is the Hilbert scheme of points, which parametrizes the closed subschemes of \(X\) of fixed finite length. We will look at examples of Hilbert schemes of points on surfaces and discuss some of their relations to quiver varieties.
Hilbert Schemes and Hilbert Scheme Of Points
Hilbert Chow Morphism
\((\mathbb{C}^2)^{[n]}\)
Their Algebraic Description, And
Their Quiver Variety Description
Introduction To Hilbert Schemes Of Points On Surfaces
Let \(X\) be a projective scheme over an algebraically closed field \(k=\overline{k}\) for example \(k=\mathbb{C}\).
The Hilbert functor \(\mathrm{Hilb}_X: {\bf Sch}_k^{op}\to {\bf Set}\) maps \(U\) to the set of closed subschemes \(Z\subset X\times U\) flat over \(U\),
\[\mathrm{Hilb}_X(U) := \{ Z\subset X\times U : Z \text{ is closed and flat over } U \}.\]
But this is a too big functor, to get a fine moduli space, we can restrict to a subfunctor parametrizing all subschemes of \(X\) with a fixed Hilbert Polynomial.
\[P_u(m) := \chi(\mathcal{O}_{Z_u} \otimes \mathcal{O}_X(m))\]
\[\mathrm{Hilb}_X^{P}(U) := \{ Z\subset X\times U : Z \text{ is closed and flat over } U, \chi(Z_u) = P \}.\]
(Grothendieck) This functor is representable by a projective scheme, which is also denoted as \(\mathrm{Hilb}_X^P\).
\[ \mathrm{Hilb}_X^P(\cdot) \cong \mathrm{Hom}_{{\bf Sch}}(\cdot, \mathrm{Hilb}_X^P)\]
Which tells us that every family of closed subschemes of \(X\) with Hilbert polynomial \(P\) over a scheme \(U\) is a morphism \(U\to \mathrm{Hilb}_X^P\), and such family is a pullback of the universal family over \(\mathrm{Hilb}_X^P\).
The case when we choose \(P=n\) be a constant polynomial, we obtain the Hilbert scheme of points on \(X\), which we denote as \(\mathrm{Hilb}^n(X)\) or \(X^{[n]}\).
This is the scheme parametrizing the \(0\)-dimensional closed subschemes of \(X\) of length \(n\).
This can be \(n\) distinct points scattered on \(X\).
Or points with multiplicities and extra data.
Example
Consider \(X= \mathbb{C}^2\), then \(\mathrm{Hilb}^n(\mathbb{C}^2)\) is the scheme parametrizing the \(0\)-dimensional closed subschemes of \(\mathbb{C}^2\) of length \(n\).
\[\begin{align*} (\mathbb{C}^2)^{[n]} &= \{ Z\subset \mathbb{C}^2 : Z \text{ is a 0-dimensional closed subscheme of length } n \} \\ &= \{ Z\subset \mathbb{C}^2 : \mathrm{length}\mathcal{O}_Z = n \} \\ &= \{ I\subset \mathbb{C}[x,y] : I \text{ is an ideal of codim } n, \dim \mathbb{C}[x,y]/I = n \} \\ \end{align*}\]
Examples Of Points In \((\mathbb{C}^2)^{[n]}\)
If \(p_1 = (x_1, y_1)\) and \(p_2 = (x_2, y_2)\) are two distinct points on \(\mathbb{C}^2\), we can obviously form the ideal defining the union of these two points
\[\begin{align*} I(p_1\cup p_2) &= I(p_1) \cap I(p_2) = (x-x_1, y-y_1) \cap (x-x_2, y-y_2) \\ &= \{f\in \mathbb{C}[x,y] : f(x_1, y_1) = f(x_2, y_2) = 0\} \end{align*}\]
This ideal \(I(p_1\cup p_2)\) defines a 0-dimensional closed subscheme of length \(2\).
What happens when \(p=p_1=p_2\), the two equations \(f(p_1)=f(p_2)=0\) collapsed into one equation and does not define a codimension \(2\) ideal. If we are looking for higher order constraints, we could constraint the differential of \(f\) to vanish on a specific direction. This gives a \(\mathbb{P}^1\) family in \((\mathbb{C}^2)^{[2]}\),
\[\mathbb{P}^1 = \mathbb{P}(T_p\mathbb{C}^2) \to (\mathbb{C}^2)^{[2]}\] \[ [\partial]=[a\partial_x+b\partial_y]\mapsto I_{p,[\partial]} \{f\in \mathbb{C}[x,y] : f(x,y)=0, (df)_p(a\partial_x + b\partial_y) = 0\}\]
Actually, \((\mathbb{C}^2)^{[2]}\) is a blow-up of \(\mathbb{C}^2\times \mathbb{C}^2\) along the diagonal, then quotient by \(S_2\).
\[(\mathbb{C}^2)^{[2]} = \operatorname{Bl}_{\Delta}(\mathbb{C}^2\times \mathbb{C}^2)/S_2\]
The Symmetric Power
There is a simpler model of points with multiplicities, the \(n\)-th symmetric of \(X\) is defined as the set of formal sums of points in \(X\) with \(n\) multiplicities,
\[ \mathrm{Sym}^n(X) = \left\{ \sum_{i=1}^n [p_i] : p_i\in X \right\}\]
The symmetric power is a scheme
\[ \mathrm{Sym}^n(X) = X^n/S_n = \mathrm{Spec}\left(((\mathbb{C}[X])^{\otimes n})^{S_n}\right)\]
where \(S_n\) acts on \(X^n\) by permuting the factors.
There is a natural stratification of \(\mathrm{Sym}^n(X)\) given by all partitions of \(n\). Let \(n=v_1+\cdots+v_k\) be a partition of \(n\), we have a stratum
\[ S_v^nX = \left\{ \sum v_i[p_i] : p_i\in X \text{ distinct} \right\}\]
\[ \mathrm{Sym}^n(X) = \bigsqcup_{v\vdash n} S_v^nX\]
- The open stratum \(S^n_{(1,\dots,1)}X\) is non-singular if \(X\) is non-singular.
Hilbert-Chow Morphism
The Hilbert Scheme of points always come with a natural morphism
\[\pi:\mathrm{Hilb}^n(X)\to \mathrm{Sym}^n(X)\]
mapping a subscheme to its support with multiplicities.
\[ J\mapsto \mathrm{Spec}(A/J)\] \[ Z \mapsto \sum_{p\in Z} \mathrm{length}(Z_p)[p]\]
Basically this forgets the extra data associated to points with multiplicities, only remembers the support and multiplicities.
Outside the singular locus of \(\mathrm{Sym}^nX\), the Hilbert-Chow morphism is an isomorphism.
\[ \pi : \mathrm{Hilb}^n_0(X)\xrightarrow{\sim} S^n_0(X) \]
For example
\(\pi(I_{p_1\cup p_2}) = [p_1]+[p_2]\)
\(\pi(I_{p,[\partial]}) = 2[p]\)
Hilbert Scheme Of Points On Surfaces
When \(\dim X=1\), \(\dim \mathbb{P}(T_pX)=0\), there is no dimension left for extra data, so actually \(X^{[n]} = \mathrm{Sym}^n(X)\) is just the ordinary symmetric power.
When \(\dim X=2\) smooth, the Hilbert scheme is smooth and presents a resolution of singularities
\[X^{[n]} \to \mathrm{Sym}^n(X)\]
When \(\dim X\ge 3\), the Hilbert scheme is generally singular even if \(X\) is smooth.
Algebraic Description Of \((\mathbb{C}^2)^{[n]}\)
By definition, any ideal of codimension \(n\) naturally gives a module of dimension \(n\) over \(\mathbb{C}[x,y]\),
\[ (\mathbb{C}^2)^{[n]} = \{ I \subset \mathbb{C}[x,y] \mid \dim_\mathbb{C}\mathbb{C}[x,y]/I = n \}.\]
\[ I \mapsto \mathbb{C}[x,y]/I\]
The right hand side is a cyclic module of dimension \(n\) over \(\mathbb{C}[x,y]\).
By considering the correspondence between cyclic \(\mathbb{C}[x,y]\)-module of dimension \(n\) with a given cyclic vector and the annihilator ideal \(I\),
\[\begin{align*} & \{ I\subset \mathbb{C}[x,y] \} \\ \leftrightarrow & \{ (M, m) \mid M = \mathbb{C}[x,y]m, \dim_\mathbb{C}M = n \}/\cong \\ \leftrightarrow & \{ (k^n, i) \mid i: k\to k^n \text{ cyclic}, B_i\in \mathrm{End}(k^n), [B_1, B_2]=0 \}/\mathrm{GL}(k^n), \end{align*}\]
We have the following algebraic description of \((\mathbb{C}^2)^{[n]}\),
\[(\mathbb{C}^2)^{[n]} = \{(B_1,B_2,i) \mid i\in \mathrm{Hom}(\mathbb{C},\mathbb{C}^n), B_i\in \mathrm{End}(\mathbb{C}^n), [B_1, B_2]=0, \mathbb{C}^n \text{ is stable} \}/\mathrm{GL}(\mathbb{C}^n)\]
Where the group \(\mathrm{GL}(\mathbb{C}^n)\) acts by conjugation on the triple \((B_1, B_2, i)\).
\[g\cdot (B_1, B_2, i) = (gB_1g^{-1}, gB_2g^{-1}, gi)\]
The stability condition means that any \(k[B_1,B_2]\)-submodule of \(\mathbb{C}^n\) generated by \(i\) must equal to \(\mathbb{C}^n\).
To recover the ideal \(I\) from the data \((B_1,B_2,i)\), we can take the annihilator of the cyclic vector \(i\).
\[I_{(B_1,B_2,i)} = \{ f\in \mathbb{C}[x,y] : f(B_1,B_2)i = 0\in \mathbb{C}^n \}\]
For example, the points in \((\mathbb{C}^2)^{[2]}\) can be described as matrices
\(p_1 = (x_1, y_1)\) and \(p_2 = (x_2, y_2)\) distinct:
\[B_1 = \begin{pmatrix} x_1 & \\ & x_2 \end{pmatrix}, \quad B_2 = \begin{pmatrix} y_1 & \\ & y_2 \end{pmatrix},\quad i = \begin{pmatrix} 1 \\ 1 \end{pmatrix}.\]
\(p=p_1=p_2\) with multiplicity \(2\):
\[B_1 = \begin{pmatrix} x & a \\ & x \end{pmatrix}, \quad B_2 = \begin{pmatrix} y & b \\ & y \end{pmatrix},\quad i = \begin{pmatrix} 1 \\ 1 \end{pmatrix}.\]
The Hilbert-Chow morphism under this data is given by
\[(B_1,B_2, \_)\mapsto \sum [(\lambda_i,\mu_i)]\]
where \((\lambda_i,\mu_i)\) are the paired-eigenvalues of \(B_1\) and \(B_2\).
Quiver Variety Description Of \((\mathbb{C}^2)^{[n]}\)
It is clear that the above algebraic description gives us a way to describe \((\mathbb{C}^2)^{[n]}\) as a quiver variety
If we denote this quiver by \(Q\), we have that (ordinary GIT, not Nakajima’s quiver variety)
\[ \mathrm{Rep}(Q,(1,n))/\!\!/_\chi \mathrm{GL}(n) = (\mathbb{C}^2)^{[n]} \quad \mathrm{Rep}(Q,(1,n))/\!\!/\mathrm{GL}(n) = \mathrm{Sym}^n(\mathbb{C}^2) \]
and the canonical morphism
\[ \mathrm{Rep}(Q,(1,n))/\!\!/_\chi \mathrm{GL}(n) \to \mathrm{Rep}(Q,(1,n))/\!\!/\mathrm{GL}(n)\]
becomes the Hilbert-Chow morphism. This is not Nakajima’s quiver variety as it is not a double of any quiver.
Nakajima’s Quiver Variety Description
Actually this is almost Nakajima’s quiver variety, if we choose the (framed) quiver \(Q\) to be
whose double \(\overline{Q}\) is
\[\omega((B_1,B_2,i,j),(B_1',B_2',i',j')) = \mathrm{tr}(B_1B_2'-B_2B_1') + \mathrm{tr}(ij'-i'j)\]
The moment map is
\[\mu: T^*\mathrm{Rep}(Q,(n,1)) \to \mathfrak{g}^* = \mathrm{End}(\mathbb{C}^n)\] \[(B_1,B_2,i,j) \mapsto [B_1,B_2]+ij\]
Then we have, upon choosing the stability \(\theta=(n,-1)\), on the semi-stable locus \(j=0\),
\[\mathfrak{M}_\theta(Q,(1,n)) = \mu^{-1}_{Q,(1,n)}(0)/\!\!/_\chi \mathrm{GL}(n) = (\mathbb{C}^2)^{[n]}\] \[\mathfrak{M}_0(Q,(1,n)) = \mu^{-1}_{Q,(1,n)}(0)/\!\!/\mathrm{GL}(n) = \mathrm{Sym}^n(\mathbb{C}^2)\]
What If
Recall that McKay correspondence says there is a one-to-one correspondence between McKay graphs of finite subgroups of \(\mathrm{SL}(2,\mathbb{C})\) and extended ADE Dynkin diagrams.
If we replace the framed Jordan quiver \(Q\) with a framed extended ADE quiver, we obtain the Hilbert Scheme of Points on Kleinian singularities.
\[ \mathfrak{M}_\theta(Q_\Gamma, (1,n\delta)) = \mathrm{Hilb}^n(\mathbb{C}^2/\Gamma)\]
where \(\Gamma\) is a finite subgroup of \(\mathrm{SL}(2,\mathbb{C})\), \(Q_\Gamma\) is the McKay quiver of \(\Gamma\), and \(\delta\) is the isotropic imaginary root of \(\Gamma\).
Extra Remarks For Myself
Speaking of monomial ideals, it seems that we have very interesting skew partitions. (draw a picture). The monomial ideals of length \(n\) is in bijection with the set of skew partitions boxes of size \(n\).