Let $E$ be a globally generated vector bundle on a surface $S$ of rank $r\geq 2$. By standard facts about degeneracy loci, for a general $V\in G(r,H^0(E))$ one has: 


(\*)the evaluation map $ev: V\otimes \mathcal{O}\_S\to E$ is injective and the cokernel is a line bundle supported on a smooth curve.


Now, let $E\_1$ be a subvector bundle of $E$ and assume $E\_1$ is globally generated as well. Is it possible to find $V\in G(r,H^0(E))$ which satisfies (\*) and such that $V\cap H^0(E\_1)$ has dimension equal to the rank of $E\_1$?



Globally generated subvector bundles and evaluation maps.
No. 


Suppose that $E$ has rank $2$, and write $V\_1=V\cap H^0(E\_1)$. Then you have an exact sequence
$$
0\to coker(V\_1\otimes \mathcal O\_S\to E\_1)\to coker(V\otimes \mathcal O\_S\to E)\to coker((V/V\_1)\otimes \mathcal O\_S\to E/E\_1)\to 0.
$$
If the line bundles $E\_1$ and $E/E\_1$ are such that every divisor corresponding to $E\_1$ has nonempty intersection with every divisor corresponding to $E/E\_1$ (for example when $S=\mathbb P^2$ and both $E\_1$ and $E/E\_1$ are the hyperplane bundle), then there must be a point in $S$ where $coker(V\otimes \mathcal O\_S\to E)$ is too big.


