number two, that $H(\mathcal{S})+ p^*\mathcal{O}_{\mathbb{Q}^n}(j)-K_{\mathbb{P}(\mathcal{S})}$ is $p$-ample, and that it is $q$-ample if $j+s\geqq 0$. Therefore $H(\mathcal{S})+ p^*\mathcal{O}_{\mathbb{Q}^n}(j)-K_{\mathbb{P}(\mathcal{S})}$ is ample if $j\geqq -s$, and the claim follows.

\[quadricResol3\] Let $\mathcal{E}$ be a nef vector bundle of rank $r$ on a smooth hyperquadric $\mathbb{Q}^n$ of dimension $n\geqq 3$. Then $d_{\min}\leqq d$. Moreover if $n\geqq 4$ and $H(\mathcal{E})^{n+r-1}>0$ then $d_{\min}< d$.

This follows from Lemmas \[KodairaOnProjectiveSpace\] and \[KodairaOnQuadric\].

Finally we deal with the case $X=\mathbb{Q}^2$. We have $\operatorname{Ext}^q(G,\mathcal{E}(d',d'))=0$ for $d'\gg 0$ and $q>0$ by Serre’s vanishing theorem. Note that if $(d_1,d_2)$ is a pair of integers such that $\operatorname{Ext}^q(G,\mathcal{E}(d_1,d_2))=0$ for all $q>0$, then $\mathcal{E}(d_1,d_2)$ has the standard resolution with respect to $\mathcal{O}$, $\mathcal{O}(1,0)$, $\mathcal{O}(0,1)$, and $\mathcal{O}(1,1)$, which implies that $\operatorname{Ext}^q(G,\mathcal{E}(d_1',d_2'))=0$ for all $q>0$, all $d_1'\geqq d_1$, and all $d_2'\geqq d_2$. Then we define a pair $(d_{1,\min},d_{2,\min})$ of integers by the following property: $$\begin{split}
\operatorname{Ext}^q(G,\mathcal{E}(d_1',d_2'))&=0\textrm{ for all }q>0,\textrm{ all }d_1'\geqq d_{1,\min},\textrm{ and all }d_2'\geqq d_{2,\min},\\
\operatorname{Ext}^q(G,\mathcal{E}(d_{1,\min}-1,d_{2,\min}))&\neq 0\textrm{ for some }q>0,\\
\operatorname{Ext}^q(G,\mathcal{E}(d_{1,\min},d_{2,\min}-1))&\neq 0\textrm{ for some }q>0.
\end{split}$$

\[KodairaOnQuadricSurf\] Suppose that $X=\mathbb{Q}^2$. Let $\mathcal{S}$ be a spinor bundle $\mathcal{O}(1,0)$ or $\mathcal{O}(0,1)$. Then we have the following vanishing.

1.  $\operatorname{Ext}^q(\mathcal{O}(j,j), \mathcal{E}(a,b))=0$ for all $q>0$ and $j< 2$.

2.  $\operatorname{Ext}^q(\mathcal{O}(2,2), \mathcal{E}(a,b))=0$ for all $q>0$, if $2ab>c_2(\mathcal{E})$.

3.  