We set $L:=\mbox{det}(E)\in \mathrm{Pic}^d(C)$ and consider the determinant map $$\lambda :  \bigwedge^2 H^0(C, E)  {\rightarrow}H^0(C, L)$$ The evaluation map $H^0(C, E) \otimes \OO_C {\rightarrow}E$ induces a morphism $\phi_E:  x \mapsto E(x)\in G\bigl(2, H^0(C, E)^{\vee}\bigr) $. Following [@BV], [@M2] we have a commutative diagram

$$\label{diag}
\xymatrix@R=22pt@C=26pt{
C \ar[d]_{\phi_L} \ar[r]^{\phi_E \qquad} & G\bigl(2, H^0(C, E)^{\vee}\bigr) \ar@{^(->}[d] \\
\PP(H^0(C, L)^{\vee}) \ar[r]^{\PP(\lambda^{\vee})} &  \PP(\bigwedge^2 H^0(C, E)^{\vee})
}$$

where the vertical arrow on the right is the Plücker embedding and $\PP(\lambda^{\vee})$ is the map induced at the level of projective spaces by the map dual to $\lambda$. In order to estimate de number of sections of $L$ we will use the following lemma, which is a direct consequence of [@PR] Lemma 3.9. We formulate it in a way that is compatible with (\[diag\]).

\[PRlemma\] Let $E$ be a globally generated rank 2 vector bundle on $C$ without sub-pencils. Then $$\mathrm{dim}\bigl( \mathrm{Im}\ \lambda\bigr) \geq 2h^0(C, E)-3.$$ In particular, $h^0(C, L)\geq 2h^0(C, E)-3$  and  $\mathrm{dim }\ ( \mathrm{Im}\ \PP(\lambda^{\vee}))\geq 2h^0(C, E)-4$.

We identify $G(2, H^0(C, E))\subset \PP\bigl(\bigwedge^2 H^0(C, E)\bigr)$ with the set of decomposable tensors $s\wedge t$, where $s, t\in H^0(C, E)$. The assumption that $E$ carries no sub-pencils implies that  $\PP(\mbox{Ker}\ \lambda) \cap G(2, H^0(C, E))=\emptyset$, 