TITLE: Coherent sheaves on $\mathbb C^2$ and commuting matrices
QUESTION [6 upvotes]: Let $V$ be an $n$-dimensional complex vector space. The stack $Coh^n(\mathbb C^2)$ of coherent sheaves on $\mathbb C^2$ supported on $n$ points (not necessarily distinct) is equivalent to the stack quotient $C_n/GL_n$, where $C_n\subset End(V)^2$ is the variety of couples of commuting matrices. 
In $Coh^n(\mathbb C^2)$ there lives the substack $Coh^n(\mathbb C^2)_0$ of sheaves supported at $0\in\mathbb C^2$.

Question 1. What locus does $Coh^n(\mathbb C^2)_0\subset Coh^n(\mathbb C^2)$ correspond to in $C_n/GL_n$?

When I pick a coherent sheaf $F\in Coh^n(\mathbb C^2)$, I can thus let it correspond to a couple of commuting matrices $(A,B)$, up to $GL_n$. Now, let $s\in\textrm{Supp}(F)$ be a point of multiplicity $i$, say. If I restrict $F$ to $s$ I get a new sheaf
$$
F|_s\in Coh^i(\mathbb C^2)_s\cong Coh^i(\mathbb C^2)_0,
$$
which will correspond to a point $(A',B')\in C_i/GL_i$.

Question 2. How is the point $(A,B)\in C_n/GL_n$ related to $F|_s$? In other words, how are $(A,B)$ and $(A',B')$ related?

Thank you for any help!

REPLY [8 votes]: For any commuting pair of matrices there is a basis in which both are upper triangular. The eigenvalues give you $n$ points of $\mathbb{C}^2$ and this recovers the support of the corresponding sheaf. This leads to the following answers:
Question 1. Both $A$ and $B$ should be nilpotent.
Question 2. $(A',B')$ is a subquotient of $(A,B)$.