Let $C$ be a projective curve (over an algebraically closed field, not necessarily of characteristic zero) which is smooth except for exact one node. Let $\pi:\tilde{C} \to C$ be its normalization. Let $\mathcal{V}$ be a locally free sheaf over $\tilde{C}$. Is it true that the push forward, $\pi\_\*(\mathcal{V})$ is a locally free sheaf? If not, are there any general conditions (for example on the rank of $\mathcal{V}$) so that this holds true?



Normalization of a curve and push forward of vector bundles
Quoting answer\_bot: "No and yes for the rank zero vector bundle." In other words, the answer is "no" except for one case: the rank zero vector bundle. That means the answer is already "no" for *every* rank 1 vector bundle. To user46578, I recommend that you contemplate what happens for the trivial rank 1 vector bundle on $\widetilde{C}$. If you still have trouble, perhaps you should ask on Math StackExchange.


