Let, $S$ be a smooth hypersurface in $\mathbb{P}^{3}$ and let $C$ be a reducible curve inside $S$ and $C =C_1 \cup C_2 \cup ...\cup C_n$, be its decomposition in terms of irreducible components.
At this point my question is :Does there exist any relation in terms of short exact sequence relating $Pic (C_i),Pic(C)$ and $Pic(S)$?
Since $C_i$'s are irreducible curve,they can be thought of as effective divisor and therefore gives rise to line bundle on $S$,but that does not give me any precise relation.
Can somebody give me a reference discussing about it?
Any help from anyone is welcome.