Let $C$ be a quasi-projective curve, $C\_i$ for $i=1,...,r$ are the irreducible components of $C$. Assume that $C\_i$ is non-singular and $F\_i$ locally free sheaf on $C\_i$ of the same rank for all $i$. Can we glue the sheaves $F\_i$ i.e., does there exists a locally free sheaf $F$ on $C$ such that $F|\_{C\_i} \cong F\_i$ for all $i$? 



Gluing locally free sheaves on curves
By induction one can assume that $k = 2$. Let $Z = C\_1 \cap C\_2$ be the scheme-theoretic intersection. Choose an isomorphism of $\varphi:F\_{1|Z} \to F\_{2|Z}$ (it exists since both are free of the same rank) and define $F$ from the exact sequence
$$
0 \to F \to F\_1 \oplus F\_2 \to F\_{1|Z} \to 0,
$$
where the second arrow is $(\varphi,1)$. The sheaf $F$ is locally free since the sequence is locally isomorphic to
$$
0 \to O\_C \to O\_{C\_1} \oplus O\_{C\_2} \to O\_Z \to 0.
$$


