In Butler' paper "Normal generation of vector bundles over a curve" (J.d.g.,1994). Proposition 1.4 said that $$prop^+(M_E)\leq \max\left\{-2,\frac{-prop^+(E)}{prop^+(E)-g}\right\}$$ where $E$ is a globally generated vector bundle containing no trivial summands over a curve $C$ of genus $g$ and $M_E$ is the kernel bundle of $E$ve. $prop^+(E)$ is defined as follows. $$prop^+(E):=\sup_S\{\mu(S)|S\subseteq E, S\ne E\},$$ where $\mu(S)$ is the slope of $S$.

If $E$ is not stable, then the above inequality follows from Butler's another inequality in Propostion 1.4 $$\mu^+(M_E)\leq \max\left\{-2,\frac{-\mu^+(E)}{\mu^+(E)-g}\right\},$$ where $\mu^+(E)$ is the maximal slope of $E$. So everything is OK.

My Question: How to prove the first inequality when $E$ is stable?

In this case, $prop^+(E)<\mu^+(E)$. Similar to the proof of the second inequality, one can find a stable subbundle $N\subseteq M_E$ with maximal slope and a vector bundle $F$ such that $M_{V,F}=N$ where $V\otimes \mathcal{O}_C\to F$ is a surjective map with a kernel $M_{V,F}$. If the induced map $F\to E$ is not surjective, then the proof is similar. Now we assume $F\to E$ is surjective. I don't know how to deal with this case.


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