$\operatorname{Ext}^q(\mathcal{S},\mathcal{E}(a+j,b+j))=0$ for all $q>0$ and $j\geqq 0$.

4.  $\operatorname{Ext}^q(\mathcal{S}(1,1),\mathcal{E}(a,b))=0$ for all $q>0$, if $2ab>c_2(\mathcal{E})$.

\(1) We have isomorphisms $$\operatorname{Ext}^q(\mathcal{O}(j,j),\mathcal{E}(a,b))
\cong H^q(\mathcal{E}(a-j,b-j))
\cong H^q(\mathbb{P}(\mathcal{E}), H(\mathcal{E})+\pi^*\mathcal{O}_{\mathbb{Q}^2}(a-j,b-j)).$$ We claim that the last cohomology group vanishes by the Kodaira vanishing theorem; indeed we have $$H(\mathcal{E})+\pi^*\mathcal{O}_{\mathbb{Q}^2}(a-j,b-j)-K_{\mathbb{P}(\mathcal{E})}
\cong (r+1)H(\mathcal{E})+\pi^*\mathcal{O}_{\mathbb{Q}^2}(2-j,2-j)$$ and $(r+1)H(\mathcal{E})+\pi^*\mathcal{O}_{\mathbb{Q}^2}(2-j,2-j)$ is ample by the Nakai-Moishezon criterion since $j<2$. Therefore the claim follows.

\(2) Note that $H(\mathcal{E})^{r+1}=c_1(\mathcal{E})^2-c_2(\mathcal{E})=2ab-c_2(\mathcal{E})$. Therefore if $2ab>c_2(\mathcal{E})$ then $H(\mathcal{E})$ is nef and big. Hence $H(\mathcal{E})+\pi^*\mathcal{O}_{\mathbb{Q}^2}(a-j,b-j)-K_{\mathbb{P}(\mathcal{E})}$ is nef and big if $j=2$. The result then follows from the Kawamata-Viehweg vanishing theorem.

\(3) Suppose that $\mathcal{S}\cong \mathcal{O}(1,0)$. We have isomorphisms $$\begin{split}
\operatorname{Ext}^q(\mathcal{S},\mathcal{E}(a+j,b+j))&\cong 
H^q(\mathbb{Q}^2, \mathcal{E}(a+j-1,b+j))\\
&\cong
H^q(\mathbb{P}(\mathcal{E}), H(\mathcal{E})
+
\pi^*\mathcal{O}_{\mathbb{Q}^2}(a+j-1,b+j)).
\end{split}$$ We show that the last cohomology group vanishes by the Kodaira vanishing theorem; we see that $H(\mathcal{E})+
\pi^*\mathcal{O}_{\mathbb{Q}^2}(a+j-1,b+j)-K_{\mathbb{P}(\mathcal{E})}
$ is isomorphic to $(r+1)H(\mathcal{E})+
\pi^*\mathcal{O}_{\mathbb{Q}^2}(j+1,j+2)$, and this line bundle is ample if $j\geqq 0$ by the Nakai-Moishezon criterion.

\(4) The proof is almost the same as (3); if $j=-1$, then $(r+1)H(\mathcal{E})+
\pi^*\mathcal{O}_{\mathbb{Q}^2}(j+1,j+2)$ is nef and big if so is $H(\mathcal{E})$. Now the result follows from the Kawamata-Viehweg vanishing theorem.

\[quadricResol\] Let $\mathcal{E}$ be a nef vector bundle of rank $r$ on $\mathbb{Q}^2$. Then we can take $(d_{1,\min},d_{2,\min})$ such that $d_{1,\min}\leqq a$ and $d_{2,\min}\leqq b$. Moreover we 