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Let $X$, $G_0,\dots,G_m$, $\mathcal{E}$, $\mathcal{O}_X(1)$, $d_{\min}$, $e_{l,j}$, $e_{l,j}'$, and $\mathcal{P}_l
$ be as in § \[Preliminaries\] for $0\leqq j\leqq l\leqq m$. In the rest of this paper, we always assume that $\mathcal{E}$ is a nef vector bundle of rank $r$ and that $\mathcal{O}_X(1)$ is a “suitable" ample line bundle on $X$, e.g., an ample line bundle of “minimal degree". Then it is natural to consider the following

What constraints does the condition that $\mathcal{E}$ is nef impose on (or among) $d_{\min}$, $e_{l,j}$’s, and $e_{l,j}'$’s ? Find good constraints on (or among) them.

The rest of this paper addresses this problem in the following cases:

1.  $X$ is a projective space $\mathbb{P}^n$, $\mathcal{O}_X(1)$ is the ample generator of ${\mathop{\rm Pic}\nolimits}X$, and $(G_0,\dots, G_m)$ is equal to $(\mathcal{O},\mathcal{O}(1),\dots,\mathcal{O}(n))$;

2.  $X$ is an odd dimensional smooth hyperquadric $\mathbb{Q}^n$ with $n\geqq 3$, the field $K$ is of characteristic zero, $\mathcal{O}_X(1)$ is the ample generator of ${\mathop{\rm Pic}\nolimits}X$, and $(G_0,\dots, G_m)$ is equal to $(\mathcal{O},\mathcal{S},\mathcal{O}(1),\dots,
    \mathcal{O}(n-1)
    )$, where $\mathcal{S}$ is the spinor bundle on $\mathbb{Q}^{
    n}$;

3.  $X$ is an even dimensional smooth hyperquadric $\mathbb{Q}^{
    n}$, the field $K$ is of characteristic zero, and $(G_0,\dots, G_m)$ is equal to $(\mathcal{O},\mathcal{S}^+,\mathcal{S}^-, \mathcal{O}(1),\dots,
    \mathcal{O}(
    n-1))$, where 