Let $X$ be a singular curve over an algebraic closed field $k$ with characteristic zero. Let $Z$ be the closed subset of singular points on $X$ and $U=X-Z$ be the smooth part, which is an open subset of $X$.


Let $\mathcal{L}$ be a line bundle on $U$. Could we always extend $\mathcal{L}$ to a line bundle on $X$, i.e. could we find a line bundle $\widetilde{\mathcal{L}}$ on $X$ such that $\widetilde{\mathcal{L}}|\_U\cong \mathcal{L}$? If not, do we have counter examples?



Could we extend any line bundle on the smooth part of a singular curve to a line bundle on the whole curve?
The answer is 'yes'. One way to argue this is to first find a Cartier divisor $D$ on $U$ whose associated line bundle is $\mathcal{L}$ (the existence of such a divisor is ensured, for instance, by [EGA IV$\_4$, 21.3.4 a)]), extend $D$ to a Cartier divisor $\widetilde{D}$ on the whole $X$ (e.g., by applying [EGA IV$\_4$, 21.9.4]), and then let $\widetilde{\mathcal{L}}$ be the line bundle associated to $\widetilde{D}$.


