of (1) (a), (2) (a), and (2) (b) are essentially the same; the only difference comes from the fact that there exists $\mathcal{S}$ or a pair of $\mathcal{S}^+$ and $\mathcal{S}^-$ between $\mathcal{O}$ and $\mathcal{O}(1)$, so that the numbering of $G_j$ differs. For simplicity, we only write explicitly the proof of (1) (a), but the reader will easily modify the proof for (2) (a), and (2) (b).

\(1) (a) Suppose that $e_{0,j}\neq 0$ for some $j> d+d_{\min}$. Then we have a non-zero map $\mathcal{O}(j)\to \mathcal{E}(d_{\min})$, which gives a non-zero map $\mathcal{O}_L(j)\to \mathcal{E}|_L(d_{\min})$ for a general line $L$ in $\mathbb{P}^n$. This implies that the maximal degree of a direct summand of $\mathcal{E}|_L(d_{\min})$ is at least $j$. On the other hand, since $\mathcal{E}$ is nef, the maximal degree of a direct summand of $\mathcal{E}|_L(d_{\min})$ is at most $d+d_{\min}$. This is a contradiction, since $j> d+d_{\min}$.

\(2) (c) This is because $e_{l+1,1}=\sum_{1<k}e_{l,k}\dim \operatorname{Hom}(G_1,G_k)=e_{l,2}\dim \operatorname{Hom}(\mathcal{S}^+,\mathcal{S}^-)$ and $\operatorname{Hom}(\mathcal{S}^+,\mathcal{S}^-)=0$.

The proofs of (1) (b), (2) (d), and (2) (e) are essentially the same; For simplicity, we only write explicitly the proof of (2) (d).

\(2) (d) We have a surjection $\mathcal{P}_0\to \mathcal{E}(d_{\min})$. Since $\mathcal{P}_0$ is globally generated, the restriction $\mathcal{E}|_L(d_{\min})$ to a line $L$ in $\mathbb{Q}^n$ is 