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?
Unless the sections from $C$ can be lifted to $E$, why do you expect this to be true? For example, for any curve $Y$ of positive genus and a point $p\in Y$, we have an exact sequence, $0\to\mathcal{O}_Y\to \mathcal{O}_Y(p)\to k(p)\to 0$ and the section from $k(p)$ does not lift. If you want a surface example, take the same and pull it back to $f:X=Y\times Z\to Y$, $Z$ another curve. If you want a rank 2 bundle example, just add an $\mathcal{O}$ to the last two terms. Then you have $0\to \mathcal{O}_{Y\times Z}^2\to\mathcal{O}_{Y\times Z}\oplus f^*\mathcal{O}_Y(p)\to \mathcal{O}_{\{p\}\times Z}\to 0$. The line bundle on $\{p\}\times Z$ is globally generated, but the middle rank 2 bundle is not.