can take $(d_{1,\min},d_{2,\min})$ such that $d_{1,\min}\leqq a-1$ and $d_{2,\min}\leqq b-1$, if $2ab>c_2(\mathcal{E})$.

Maximal degree subbundles of a nef vector bundle 
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Let $X$ be as in § \[Preliminaries\], and let $\mathcal{E}$ be as in § \[Preliminaries\]. Assume that $\mathcal{E}$ is a nef vector bundle of rank $r$ as in § \[easyconstraints\].

\[detsub\] Suppose that there exists a non-zero morphism $\varphi:\det\mathcal{E}\to \mathcal{E}$. Then $\varphi$ makes $\det\mathcal{E}$ a subbundle of $\mathcal{E}$.

Let $s$ be a non-zero element of $H^0(\mathcal{E}\otimes (\det\mathcal{E})^{\vee})$ corresponding to $\varphi$, and suppose that the zero locus $(s)_0$ of $s$ is not empty. Take a curve $C$ such that $C\cap (s)_0\neq \emptyset$ and that $C$ is not contained in $(s)_0$, and let $\pi:\tilde{C}\to C$ be the normalization. Then $\mathcal{O}_{\tilde{C}}(\pi^*((s)_0\cap C))$ is a subbundle of $
\pi^*(\mathcal{E}\otimes (\det\mathcal{E})^{\vee})$. This implies that $\pi^*\mathcal{E}$ has a quotient bundle of negative degree, which contradicts that $\mathcal{E}$ is nef. Therefore $(s)_0$ is empty and $\varphi$ makes $\det\mathcal{E}$ a subbundle of $\mathcal{E}$.

\[dsubExist\] Suppose that $H^1(\det\mathcal{E})=0$ and that every nef vector bundle with trivial determinant is isomorphic to a direct sum of copies of $\mathcal{O}$. Then $\operatorname{Hom}(\det\mathcal{E},\mathcal{E})\neq 0$ implies that $\mathcal{E}\cong \mathcal{O}^{\oplus r-1}\oplus\det\mathcal{E}$.

If $\operatorname{Hom}(\det\mathcal{E},\mathcal{E})\neq 0$, then, by Lemma \[detsub\], there exists an exact sequence $$0\to 