Let $X$ be a smooth projective variety. I have seen people identify the Hilbert scheme of $n$ points on $X$ with the moduli space of stable sheaves with Chern character $(1,0,....,-n)$ (say on $\mathbb{P}^2$).
On the other hand, this identification clearly cannot work for all $X$, say $X=\mathbb{P}^1$, so we may need to assume dimension of $X$ is at least $2$.
My question: 1. what are the conditions needed for us to make the above identification?
- Let $Y_1,Y_2$ be two distinct length $n$ subschemes on $X$, how to we prove the two ideal sheaves $I_{Y_1},I_{Y_2}$ are not isomorphic?
Thanks for the help!
1) The first condition is $\dim(X) \ge 2$, the second is $Pic^0(X) = 0$. For a proof see Lemma B.5.6 in https://arxiv.org/abs/1605.02010
2) If $I = I_Y$ then $Y$ is the support of the cokernel of the natural map $I \to I^{\vee\vee}$. The proof of (1) is just the same argument applied to a family of sheaves.