Let $X$ be a smooth projective variety over $\mathbb{C}$. The Hilbert scheme on $X$ parametrizes quotients $\mathcal{O}_X \to E$ with fixed Hilbert polynomial. Let us fix the Hilbert polynomial to lead to subschemes of codimension at least $2$. Then we can also look at the moduli space parametrizing semistable rank $1$ sheaves with corresponding Hilbert polynomial and trivial determinant, i.e. ideal sheaves.

Whenever we have a family of quotients, we can take the kernel to get a family of ideal sheaves. That should induce a bijective morphism. However, it seems unclear to me how to get an inverse morphisms.

Under what kind of hypotheses are these two moduli spaces the same? I am in particular interested in the case of Hilbert schemes of curves in $\mathbb{P}^3$.

**Edit:** Since I never wrote it down and it came up in the comments, let me write down what the functors are I am exactly talking about.

The Hilbert scheme represents the functor that maps $S$ to quotients $\mathcal{O}_{X \times S} \to E$ that are flat over the base $S$.

The moduli of semistable sheaves represents the functor that maps $S$ to flat families $E \in \operatorname{Coh}(X \times S)$ of semistable sheaves modulo tensoring with line bundles pulled back from $S$.

We then look at the corresponding connected components coming from fixing the Hilbert polynomial and ask whether they are the same.