Let's $C \subset X$ be a smooth curve inside a three dimensional variety with split normal bundle $N\_C^X= \nu\_1 \oplus \nu\_2$. What is a locally free resolution of $\iota\_{\*}\mathcal{O}\_{C}$ ? 



How do we write a locally free resolution for...
In general, there is no straight way to write such a resolution (note by the way, that a resolution is in no way unique!). However, in some cases there is a distinguished resolution. For example, if $C$ is the zero locus of a global section of a rank 2 vector bundle $E$ then there is a Koszul resolution
$$
0 \to \det E^\* \to E^\* \to O\_X \to \iota\_\*O\_C \to 0.
$$
Another case, is when $C$ is the degeneration locus of a morphism $\phi:E \to F$ of vector bundles such that $rank(E) = rank(F) - 1$. Then there is a resolution 
$$
0 \to E\otimes L \to F\otimes L \to O\_X \to \iota\_\*O\_C \to 0,
$$
where $L =\det E\otimes (\det F)^{-1}$.


