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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!

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1 Answer 1

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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 $(A,B)$.

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  • $\begingroup$ thanks a lot for your answer! What do you mean by (A',B') being a sub quotient of (A,B)? I do not see how to get to $(A',B')$ starting from $(A,B)$. I am also unsure how the eigenvalue "give me" the support of $F$, since $A$ and $B$ do not necessarily have the same eigenvalues... $\endgroup$
    – Brenin
    Mar 28, 2015 at 17:49
  • $\begingroup$ Choose a basis in which both $A$ and $B$ are uppertriangular. Let $a_1,\dots,a_n$ be the diagonal entries of $A$ and $b_1,\dots,b_n$ those of $B$. Then the support is $(a_1,b_1) + \dots + (a_n,b_n)$. A subquotient means that on $\mathbb{C}^n$ there is a filtration preserved by $A$ and $B$ such that $(A',B')$ is its subquotient. $\endgroup$
    – Sasha
    Mar 28, 2015 at 18:01
  • $\begingroup$ So I have a filtration $V=V_n\supset V_{n-1}\supset \cdots\supset V_1\supset V_0=0$ such that $A\cdot V_i\subseteq V_i$ and $B\cdot V_i\subseteq V_i$. What is its subquotient? (Sorry but I do not want to misunderstand your answer). Moreover, it seems to me that nilpotency is a necessary but not sufficient condition for $\textrm{Supp}(F)=\{(0,0)\}$. Indeed, $A$ (or $B$) can be nilpotent without having only zeros on the diagonal. $\endgroup$
    – Brenin
    Mar 28, 2015 at 19:03
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    $\begingroup$ An upper triangular matrix is nilpotent if and only if its diagonal is zero. A subquotient is $V_k/V_l$. $\endgroup$
    – Sasha
    Mar 28, 2015 at 19:27

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