Let $S$ be a smooth complex algebraic surface, $D$ be a divisor on $S$, and $\mathcal{E}$ be a locally free coherent sheaf on $S$ of rank $2$ (in practice, $\mathcal{E} \simeq \Omega_S$).
Let us consider the associated projective space bundle $P := \mathbb{P}(\mathcal{E})$. We have a projection morphism $\pi : P \rightarrow S$, and an invertible sheaf $\mathcal{O}(1)$ on $P$. Let $H$ be a hyperplane of $P$ corresponding to $\mathcal{O}(1)$.
My first question is that :
Is this $H$ isomorphic to $S$ ?
I want to calclate the cohomology $H^i(P, \pi^*(\mathcal{O}_S(D)) \otimes N_H^{\otimes k})$, where $N_H$ is the normal bundle of $H$ in $P$.
My second question is that:
How can I treat $\pi^*(\mathcal{O}_S(D))|_H$ and $N_H^{\otimes k}$? Is it correct that $\pi^*(\mathcal{O}_S(D))|_H \simeq \mathcal{O}_S(D)$?
More precisely, when I calculate the intersection number of $N_H^{\otimes k}$ with $\pi^*(\mathcal{O}_S(D))$ or $N_H^{\otimes k}$ itself, can I treat $N_H$ as $\bigwedge^2\mathcal{E}$ (if the first question is correct)?