also globally generated. Hence the minimal degree of a direct summand of $\mathcal{E}|_L(d_{\min})$ is non-negative. Note here that the minimal degree of a direct summand of $\mathcal{E}|_L(d_{\min})$ is $d_{\min}$ since $r>d$. Therefore $d_{\min}\geqq 0$.

An upper bound for $d_{\min}$ {#UpperBound}
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Let $X$, $G_0,\dots,G_m$, $G$, $\mathcal{E}$, $\mathcal{O}_X(1)$, and $d_{\min}$ be as in § \[Preliminaries\]. Assume that $\mathcal{E}$ be a nef vector bundle of rank $r$ as in § \[easyconstraints\]. In this section, we assume that the base field $K$ is of characteristic zero, and we give an upper bound for $d_{\min}$ in the cases (1), (2), and (3) as described in § \[easyconstraints\]. Let $\mathcal{O}_X(1)$, $d$, and $(a,b)$ be as in § \[easyconstraints\].

Let $\pi:\mathbb{P}(\mathcal{E})\to X$ be the projection, and let $H(\mathcal{E})$ be the tautological line bundle on $\mathbb{P}(\mathcal{E})$.

In Lemma \[KodairaOnProjectiveSpace\] below, in order to unify and shorten descriptions in $\mathbb{P}^n$ and $\mathbb{Q}^n$ ($n\geqq 3$), we denote by $c_1(X)$ the integer corresponding to $c_1(X)$ via the isomorphism ${\mathop{\rm Pic}\nolimits}X\cong \mathbb{Z}$ which sends the ample generator to $1$. However, needless to say, we must not use this abuse of notation in intersection formulas.

\[KodairaOnProjectiveSpace\] Let $X$ be a smooth Fano variety of dimension $n$ with ${\mathop{\rm Pic}\nolimits}X\cong \mathbb{Z}$, and let 