I encounter a problem when I read compact complex surfaces.Let $X$ be a projective surface,$E$ is a vector bundle of rank 2 over $X$, $S \subset \mathbb{P}(E)$ is an irreducible subvariety such that $\pi |\_S :S\rightarrow X$ is birational. It say that $\mathcal{O}\_{\mathbb{P}(E)}(1) \otimes \mathcal{O}\_{\mathbb{P}(E)}(-S)$ is trivial on all fibres of $\mathbb{P}(E)$, why? 



Sections over projective bundles
Since $\pi|\_S$ is birational, it follows that $S$ is a section of $\pi$. Hence we can write $$\mathcal{O}\_{\mathbb{P}(E)}(S) \cong \mathcal{O}\_{\mathbb{P}(E)}(1) \otimes \pi^\* \mathcal{L},$$ 
where $\mathcal{L}$ is a line bundle on $X$. This implies $$\mathcal{O}\_{\mathbb{P}(E)}(1) \otimes \mathcal{O}\_{\mathbb{P}(E)}(-S) \cong \pi^\* \mathcal{L}^{-1}.$$ 
Since $\pi^\*{\mathcal{L}}^{-1}$ comes from the base $X$ of the projective bundle $\mathbb{P}(E) \to X$, it is trivial when restricted to any fibre.


