I'm reading Beauville's book, Complex Algebraic Surfaces, and I'm trying to understand an affirmation in a proposition that characterizes the Picard group of a geometrically ruled surface.


First, let $S= \mathbb{P}\_C(E)$ be a geometrically ruled surface over a curve $C$, $p: S \rightarrow C$ the structure map. The tautological bundle is constructed as the cokernel of the inclusion $i: N \rightarrow p^\*E$, where $N$ is the line bundle such that $N\_s$ is the line corresponding to $s$ at $E\_{p(s)}$. That is , we have the exact sequence


$$0 \rightarrow N \rightarrow p^\*E \rightarrow \mathcal{O}\_S(1) \rightarrow 0 $$


Then, there is a proposition: if $H$ is the divisor corresponding to $\mathcal{O}\_S(1)$, then
$$\mathrm{Pic}(S) = p^\* \mathrm{Pic}(C) \oplus \mathbb{Z} \cdot H $$


My problem with this proof is with its first affirmation: if $F$ is a fiber of $p$, then $F \cdot H = 1$. I know that $F \cdot H = \deg (H|\_F)$. As $F\simeq \mathbb{P}^1$, I bet that I need to prove that $\mathcal{O}\_S(1)|\_F = \mathcal{O}\_F(1)$. I can see it if, for some embedding $S\subset \mathbb{P}^n$, in fact $\mathcal{O}\_S(1)$ is the sheaf of hyperplanes. But I'm not sure about it. Is it true? Or do I need a different approach?



Intersection of the tautological bundle with a fiber of a geometrically ruled surface
By [**1**, Chapter II, Proposition 7.12], the surjective map of sheaves $p^\*E \to \mathcal{O}\_S(1)$ gives a morphism of schemes
$C \to \mathbb{P}(E)$ over $C$, namely a section of $p \colon \mathbb{P}(E) \to C$.


Moreover, the image of this section corresponds to a divisor in $|\mathcal{O}\_S(1)|$, see [$1$, Chapter V Propositions 2.6 and 2.8].


Then, every element $H \in |\mathcal{O}\_S(1)|$ satisfies $HF=1$.


**Reference.**


**[1]** R. Hartshorne, *Algebraic Geometry*, Springer 1977.


