$\mathcal{S}^+$ and $\mathcal{S}^-$ are the spinor bundles on $\mathbb{Q}^{
    n}$. If $n\geqq 4$, then $\mathcal{O}_X(1)$ is the ample generator of ${\mathop{\rm Pic}\nolimits}X$, and if $n=2$, then $\mathcal{O}_X(1)$ is the ample line bundle $\mathcal{O}(1,1)$ of minimal degree.

Thus we always assume, in the rest of the paper, that if $X$ is as in (1), (2), or (3), then $(G_0,\dots,G_m)$ is as in (1), (2), or (3) respectively.

If ${\mathop{\rm Pic}\nolimits}X\cong \mathbb{Z}$, we denote by $d$ the integer such that $\mathcal{O}_X(d)\cong \det\mathcal{E}$, and if $X\cong \mathbb{Q}^2$, we denote by $(a,b)$ the pair of integers such that $\mathcal{O}_{\mathbb{Q}^2}(a,b)\cong \det\mathcal{E}$.

In this section, we give some easy constraints among $d_{\min}$ and $e_{l,j}$’s in the cases (1), (2), and (3) above.

\[firstConstraint\] We have the following constraints

1.  Suppose that $X=\mathbb{P}^n$.

    1.  If $j>d+d_{\min}$, then $e_{0,j}=0$.

    2.  If $d<r$, then $d_{\min}\geqq 0$.

2.  Suppose that $X=\mathbb{Q}^{n}$.

    1.  Suppose that $n$ is odd. If $j>d+d_{\min}+1$, then $e_{0,j}=0$.

    2.  Suppose that $n$ is even. If $j>d+d_{\min}+2$, then $e_{0,j}=0$.

    3.  Suppose that $n$ is even. If $e_{l,k}=0$ for all $k>2$, then $e_{l+1,k}=0$ for all $k>0$. In particular $e_{n,1}=0$ and $e_{n+1,0}=0$.

    4.  If $d<r$, then $d_{\min}\geqq 0$.

    5.  Suppose that $n=2$. If $\min \{a,b\}<r$, then $d_{\min}\geqq 0$.

The proofs 