Author: Eiko

Time: 2025-02-20 16:01:06 - 2025-02-20 16:01:06 (UTC)

Let $T=({\mathbb{C}}^{\times }{)}^2$ be the algebraic torus acting on $\mathbb{C}[x,y]$ by

$$(t_1,t_2)\cdot f(x,y)\mapsto f(t_{1}x,t_{2}y).$$

We have that the fixed functions are generated by monomials. So fixed points in $({\mathbb{C}}^2{)}^{[n]}$ are given by monomial ideals in $\mathbb{C}[x,y]$ of codimension $n$.

Monomial Ideals In $\mathbb{C}[x,y]$

Of length $n$ can be described by a partition of $n$, there is a correspondence between monomial ideals and partitions of $n$.

$$I\mapsto \lambda =\{(i,j)\in {\mathbb{N}}^2\mid x^i{y}^j\notin I\},\lambda \mapsto I_{\lambda }=⟨x^i{y}^j\mid (i,j)\notin \lambda ⟩.$$

i.e. the ideals are generated by monomials outside the partition. In fact, the set of monomials in $\lambda$ provides a basis $B_{\lambda }$ for the quotient $\mathbb{C}[x,y]/I_{\lambda }$.

Affine Open Set

For a partition $\lambda$ of $n$, the affine open set $U_{\lambda }$ is the set of monomial ideals of codimension $n$ that is indeed generated by the images of monomials in $\lambda$.

$$U_{\lambda }=\{I\in ({\mathbb{C}}^2{)}^{[n]}\mid B_{\lambda }\text{ is a basis for }\mathbb{C}[x,y]/I\}.$$

So for each point $I$ in this open set $I\in U_{\lambda }$ we can expand

$$x^r{y}^s=\sum _{(i,j)\in \lambda }{c}_{ij}^{rs}{x}^i{y}^j+I$$

Example. For $\lambda =\{(0,0),(1,0)\}$ the basis is $B_{\lambda }=\{1,x\}$, $I_{\lambda }=(x^2,y)$. The ideal $I=(x^2,y+\alpha x)$ is in $U_{\lambda }$ and we can expand like

• $c_{00}^{00}=c_{10}^{10}=1$

• $y=-\alpha x+I,c_{00}^{10}=-\alpha$

• $xy=-\alpha x^2+I=0+I,c_{\ast \ast }^{11}=0$

By multiplying we have interesting relations

$$\begin{array}{rl}{x}^{r+r^{\prime }}{y}^{s+s^{\prime }}& =\sum _{(i,j),(i^{\prime },j^{\prime })\in \lambda }{c}_{ij}^{rs}{c}_{i^{\prime }{j}^{\prime }}^{r^{\prime }{s}^{\prime }}{x}^{i+i^{\prime }}{y}^{j+j^{\prime }}+I\\ & =\sum _{(i,j),(i^{\prime },j^{\prime })\in \lambda }{c}_{ij}^{rs}{c}_{i^{\prime }{j}^{\prime }}^{r^{\prime }{s}^{\prime }}\sum _{(k,l)\in \lambda }{c}_{kl}^{i+i^{\prime },j+j^{\prime }}{x}^k{y}^l+I\\ & =\sum _{(k,l)}\left(\sum _{(i,j),(i^{\prime },j^{\prime })}{c}_{ij}^{rs}{c}_{i^{\prime }{j}^{\prime }}^{r^{\prime }{s}^{\prime }}{c}_{kl}^{i+i^{\prime },j+j^{\prime }}\right)x^k{y}^l+I\end{array}$$

This gives formula

$$c_{kl}^{r+r^{\prime },s+s^{\prime }}=\sum _{(i,j),(i^{\prime },j^{\prime })\in \lambda }{c}_{ij}^{rs}{c}_{i^{\prime }{j}^{\prime }}^{r^{\prime }{s}^{\prime }}{c}_{kl}^{i+i^{\prime },j+j^{\prime }}.$$

In particular if we take $(r^{\prime },s^{\prime })=(1,0)\in \lambda$

$$\begin{array}{rl}{c}_{kl}^{r+1,s}& =\sum _{(i,j),(i^{\prime },j^{\prime })\in \lambda }{c}_{ij}^{rs}{c}_{i^{\prime }{j}^{\prime }}^{10}{c}_{kl}^{i+i^{\prime },j+j^{\prime }}\\ & =\sum _{(i,j)\in \lambda }{c}_{ij}^{rs}{c}_{kl}^{i+1,j}\end{array}$$

This gives the ring of algebraic functions $\mathbb{C}[c_{ij}^{rs}{]}_{(i,j)\in \lambda }$ on $U_{\lambda }$, it should have some relations so it is of dimension $2n$.

Local Structure of Hilbert Schemes

Since for an ideal $I\in U_{\lambda }$, the coordinates $c_{ij}^{rs}$ for $(r,s)\in \lambda$ are constants, and the other coordinates are zero then $I=I_{\lambda }$, we know that the maximal ideal corresponding to $I$ is

$${\mathfrak{m}}_{\lambda }:=⟨c_{ij}^{rs}\mid (r,s)\notin \lambda ⟩.$$