Author: Eiko

Time: 2025-02-16 12:30:56 - 2025-02-16 12:30:56 (UTC)

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 $k=\bar{k}$ for example $k=\mathbb{C}$.

The Hilbert functor ${\mathrm{Hilb}}_X:{\mathbf{S}\mathbf{c}\mathbf{h}}_k^{op}\to \mathbf{S}\mathbf{e}\mathbf{t}$ 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 ${\mathrm{Hilb}}_X^P$.

  $${\mathrm{Hilb}}_X^P(\cdot )\cong {\mathrm{Hom}}_{\mathbf{S}\mathbf{c}\mathbf{h}}(\cdot ,{\mathrm{Hilb}}_X^P)$$

  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 $n$.

$$\begin{array}{rl}({\mathbb{C}}^2{)}^{[n]}& =\{Z\subset {\mathbb{C}}^2:Z\text{ is a 0-dimensional closed subscheme of length }n\}\\ & =\{Z\subset {\mathbb{C}}^2:\mathrm{length}{\mathcal{O}}_Z=n\}\\ & =\{I\subset \mathbb{C}[x,y]:I\text{ is an ideal of codim }n,\dim \mathbb{C}[x,y]/I=n\}\end{array}$$

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

  $$\begin{array}{rl}I(p_1\cup p_2)& =I(p_1)\cap I(p_2)=(x-x_1,y-y_1)\cap (x-x_2,y-y_2)\\ & =\{f\in \mathbb{C}[x,y]:f(x_1,y_1)=f(x_2,y_2)=0\}\end{array}$$

  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 $S_2$.

  $$({\mathbb{C}}^2{)}^{[2]}={\mathrm{Bl}}_{\mathrm{\Delta }}({\mathbb{C}}^2\times {\mathbb{C}}^2)/S_2$$

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,

$${\mathrm{Sym}}^n(X)=\{\sum _{i=1}^n[p_i]:p_i\in X\}$$

The symmetric power is a scheme

$${\mathrm{Sym}}^n(X)=X^n/S_n=\mathrm{Spec}(((\mathbb{C}[X]{)}^{\otimes n}{)}^{S_n})$$

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

$$S_v^{n}X=\{\sum v_i[p_i]:p_i\in X\text{ distinct}\}$$

$${\mathrm{Sym}}^n(X)=\underset{v⊢n}{⨆}{S}_v^{n}X$$

• The open stratum $S_{(1,\dots ,1)}^{n}X$ 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 ${\mathrm{Sym}}^{n}X$, the Hilbert-Chow morphism is an isomorphism.

  $$\pi :{\mathrm{Hilb}}_0^n(X)\stackrel{\sim }{\to }{S}_0^n(X)$$

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 $\dim X=1$, $\dim \mathbb{P}(T_{p}X)=0$, 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 $I$,

$$\begin{array}{rl}& \{I\subset \mathbb{C}[x,y]\}\\ \leftrightarrow & \{(M,m)\mid M=\mathbb{C}[x,y]m,{\dim }_{\mathbb{C}}M=n\}/\cong \\ \leftrightarrow & \{(k^n,i)\mid i:k\to k^n\text{ cyclic},B_i\in \mathrm{End}(k^n),[B_1,B_2]=0\}/\mathrm{GL}(k^n),\end{array}$$

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 $(B_1,B_2,i)$.

  $$g\cdot (B_1,B_2,i)=(g{B}_1{g}^{-1},g{B}_2{g}^{-1},gi)$$

• 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:

    $$B_1=\left(\begin{array}{cc}{x}_1& \\ & x_2\end{array}\right),B_2=\left(\begin{array}{cc}{y}_1& \\ & y_2\end{array}\right),i=\left(\begin{array}{c}1\\ 1\end{array}\right).$$

  • $p=p_1=p_2$ with multiplicity $2$:

    $$B_1=\left(\begin{array}{cc}x& a\\ & x\end{array}\right),B_2=\left(\begin{array}{cc}y& b\\ & y\end{array}\right),i=\left(\begin{array}{c}1\\ 1\end{array}\right).$$

• The Hilbert-Chow morphism under this data is given by

  $$(B_1,B_2,\mathrm{\_})\mapsto \sum [({\lambda }_i,{\mu }_i)]$$

  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)

$$\mathrm{Rep}(Q,(1,n))/{/}_{\chi }\mathrm{GL}(n)=({\mathbb{C}}^2{)}^{[n]}\mathrm{Rep}(Q,(1,n))//\mathrm{GL}(n)={\mathrm{Sym}}^n({\mathbb{C}}^2)$$

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 $\bar{Q}$ is

$$\omega ((B_1,B_2,i,j),(B_1^{\prime },B_2^{\prime },i^{\prime },j^{\prime }))=\mathrm{tr}(B_1{B}_2^{\prime }-B_2{B}_1^{\prime })+\mathrm{tr}(i{j}^{\prime }-i^{\prime }j)$$

The moment map is

$$\mu :T^{\ast }\mathrm{Rep}(Q,(n,1))\to {\mathfrak{g}}^{\ast }=\mathrm{End}({\mathbb{C}}^n)$$ $$(B_1,B_2,i,j)\mapsto [B_1,B_2]+ij$$

Then we have, upon choosing the stability $\theta =(n,-1)$, on the semi-stable locus $j=0$,

$${\mathfrak{M}}_{\theta }(Q,(1,n))={\mu }_{Q,(1,n)}^{-1}(0)/{/}_{\chi }\mathrm{GL}(n)=({\mathbb{C}}^2{)}^{[n]}$$ $${\mathfrak{M}}_0(Q,(1,n))={\mu }_{Q,(1,n)}^{-1}(0)//\mathrm{GL}(n)={\mathrm{Sym}}^n({\mathbb{C}}^2)$$

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.

$${\mathfrak{M}}_{\theta }(Q_{\mathrm{\Gamma }},(1,n\delta ))={\mathrm{Hilb}}^n({\mathbb{C}}^2/\mathrm{\Gamma })$$

where $\mathrm{\Gamma }$ is a finite subgroup of $\mathrm{SL}(2,\mathbb{C})$, $Q_{\mathrm{\Gamma }}$ is the McKay quiver of $\mathrm{\Gamma }$, and $\delta$ is the isotropic imaginary root of $\mathrm{\Gamma }$.

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 $n$.