Let $X$ be a projective surface over $\mathbb{C}$.
Let $E$ be a rank $2$ vector bundle whose general global section vanishes on finite points. (i.e. Codim $Z(s) = 2$ for general $s \in H^0(X, E)$)
In addition, we assume that $\text{dim}_\mathbb{C} H^0(X,E) \geq 2$.
Question
$E$ is generically global generated ?
(We say $E$ is generically global generated if the evaluation map $ ev: H^0(E) \otimes \mathcal{O}_X \rightarrow E$ is surjective on an open set $U \subseteq X $)
A general section of $E$, say $s_1$, gives an exact sequence $$ 0 \to \mathcal{O}_X \to E \to \det(E) \to \mathcal{O}_Z \to 0, $$ where $Z = Z(s_1)$. The cohomology exact sequence $$ 0 \to H^0(X,\mathcal{O}_X) \to H^0(X,E) \to H^0(X,\det(E)) $$ implies that the second section of $E$, say $s_2$, projects to a nontrivial global section of the line bundle $\det(E)$. This latter section is nonzero on an open subset $U \subset X$, and then a diagram chase shows that $E$ is generated by $s_1$ and $s_2$ on $U$.