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 $0\in\mathbb C^2$.



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> **Question 1**. What locus does $Coh^n(\mathbb C^2)\_0\subset Coh^n(\mathbb C^2)$ correspond to in $C\_n/GL\_n$?
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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 $(A',B')\in C\_i/GL\_i$.



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> **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?
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Thank you for any help!



Coherent sheaves on $\mathbb C^2$ and commuting matrices
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 $(A,B)$.


