Let $ X $ be a smooth threefold and $ C \subset X $ be a smooth (but not necessarily irreducible) curve with ideal sheaf $ \mathcal{I_C} $. I am looking for an answer to the question of when an element $ \xi \in \text{Ext}^1(\mathcal{I}_C, \mathcal{O}_X) $ is represented by a short exact sequence with a vector bundle in the middle. More generally, for a rank $ 1 $ coherent sheaf $ \mathcal{F} $ on $ X $, when is an extension $ \xi \in \text{Ext}^1(\mathcal{F}, \mathcal{O}_X) $ given by a vector bundle?
Here I'm not assuming anything about $ X $ but if you wish, take $ X $ to be proper/projective as I'm thinking of $ X $ as projective space in the examples I have.
Best Answer
Consider the natural homomorphism $$ \mathrm{Ext}^1(\mathcal{I}_C,\mathcal{O}_X) \to H^0(X,\mathcal{Ext}^1(\mathcal{I}_C,\mathcal{O}_X)). $$ Since $C \subset X$ is a local complete intersection of codimension 2, the sheaf $\mathcal{Ext}^1(\mathcal{I}_C,\mathcal{O}_X)$ is a line bundle on $C$; moreover, $$ \mathcal{Ext}^1(\mathcal{I}_C,\mathcal{O}_X) \cong \wedge^2 \mathcal{N}_{C/X}, $$ where $\mathcal{N}_{C/X}$ is the normal bundle of the curve. The extension is locally free if and only if the global section of $\wedge^2 \mathcal{N}_{C/X}$ corresponding to $\xi$ is nonzero at every point of $C$. In particular, a necessary condition is that the line bundle $\wedge^2 \mathcal{N}_{C/X}$ is trivial.
For the second question, the criterion is that $\mathcal{Ext}^i(\mathcal{F},\mathcal{O}_X) = 0$ for $i > 1$ and that $\mathcal{Ext}^1(\mathcal{F},\mathcal{O}_X)$ is generated by the global section corresponding to $\xi$.