Consider over $\mathbb{C}$. Let $(X,\mathcal{O}(1))$ be a smooth projective scheme with an ample polarisation. Let $P(t):=\chi(X,\mathcal{O}(t))$ denote the Hilbert polynomial of $\mathcal{O}\_X$. Choose a decomposition $P(t)=I(t)+Q(t)$ such that $I(t),Q(t)$ are also Hilbert polinomials. If a sheaf of ideals $\mathcal{I}\subseteq\mathcal{O}\_X$ has Hilbert polynomial $I(t)$, then the sheaf $\mathcal{O}\_X/\mathcal{I}$ has Hilbert polynomial $Q(t)$.


By smoothness, we have that $\mathcal{O}\_X$ is Gieseker stable. Then any ideal $\mathcal{I}\subseteq\mathcal{O}\_X$ is stable, because any subsheaf of a rank one pure sheaf is Gieseker stable.


Consider the following moduli scheme of stable sheaves and Hilbert scheme
$$\mathrm{M}^{I(t),\mathrm{s}}:=\{\textrm{stable sheaves on }X\textrm{ of Hilbert polynomial }I(t)\}\\
\mathrm{Hilb}\_X^{Q(t)}:=\{\textrm{subschemes }Z\hookrightarrow X\textrm{ s.t. }\mathcal{O}\_Z\textrm{ has Hilbert polynomial }Q(t)\}.$$


For a closed subschem $Z\hookrightarrow X$, the associated ideal $\mathcal{I}\_Z:=\ker(\mathcal{O}\_X\to \mathcal{O}\_Z)$ is stable of Hilbert polynomail $I(t)$, defining a point of $\mathrm{M}^{I(t),\mathrm{s}}$. This defines a morphism
$$\mathrm{Hilb}\_X^{Q(t)}\to \mathrm{M}^{I(t),\mathrm{s}},\quad (Z\hookrightarrow X)\mapsto \mathcal{I}\_Z. $$


The question is when is the morphism an isomorphism? It seems that it is an isomorphism when $X=\mathbb{P}^3$ and the Hilbert scheme parametrises curves according to discussions in this [MO](https://mathoverflow.net/questions/223618/hilbert-schemes-and-moduli-of-ideal-sheaves). Does it suffices to assume that $X$ is smooth, and $Z\hookrightarrow X$ has higher codimension?



When is the morphism from the Hilbert scheme to the moduli scheme of stable sheaves an isomorphism?
This works for any smooth projective variety $X$ under the assumption
$$
\mathrm{Pic}^0(X) = 0
$$
and any $Z$ of codimension at least 2. For the proof see Lemma B.5.6 in Kuznetsov, Alexander G.; Prokhorov, Yuri G.; Shramov, Constantin A. Hilbert schemes of lines and conics and automorphism groups of Fano threefolds. Jpn. J. Math. 13 (2018), no. 1, 109--185.


