Let $X$ be a smooth projective complex surface and $V$ a globally generated vector bundle on $X$.
Suppose we have a vector bundle $E$ sitting in an exact sequence $$0\to V\to E\to O_X(C)\otimes A \to 0$$ where $C$ is a smooth curve, and $A$ a torsion sheaf on $X$ supported on $C$, whose restriction to $C$ is a line bundle. In particular $V$ and $E$ have the same rank and the map $V\to E$ is an isomorphism away from $C$. In particular, $E$ too is globally generated, away from $C$. The question is: what about the points of $C$?
Can we deduce from the sequence above that $E$ can fail to be globally generated only at the base points of $A$ ? (we may also assume that $O_C( C )$ is globally generated)
Edit As Mohan suggests this is not true if the sections of $C$ can't be lifted. So my question is: what precisely means that "the sections of $C$ can be lifted" and how to conclude for the base points of $E$ under this assumption?