# When is the morphism from the Hilbert scheme to the moduli scheme of stable sheaves an isomorphism?

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. Does it suffices to assume that $$X$$ is smooth, and $$Z\hookrightarrow X$$ has higher codimension?

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.
• I find this MO answer saying that if we replace $\mathrm{M}^{I(t),\mathrm{s}}$ by the moduli scheme of stable sheaves with trivial determinant, then we have an isomorphism with the Hilbert scheme. Aug 15, 2023 at 3:06