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!