Good Morning,

These questions are a result of trying to understand the proof of Proposition III.18 in Beauville's book 'Complex Algebraic Surfaces'.

Here is the setup - everything is smooth, projective, algebraic, complex.

For a curve $C$, let $E \to C$ be a rank 2 vector bundle, and let $S=\mathbb{P}_C(E)$ be the associated geometrically ruled surface; label the surjection $S \to C$ as $p$. Let $\mathcal{O}_S(1)$ be the tautological bundle of $S$.

Let $F (\cong \mathbb{P}^1)$ be a fibre of $p$. The first equality I don't understand is $F.\mathcal{O}_S(1)=1$. The 'rough' justification I have for this is that the tautological bundle corresponds to a section of $S\to C$, and hence meets $F$ exactly once with multiplicity 1, but this doesn't totally make sense to me. Why exactly does the tautological bundle correspond to a section? (Or if this is nonsense, how else may I see the equality?).

Secondly, there is the claim that $\mathcal{O}_S(1).p^* det(E)=deg(E)$. Could someone please explain why this is true?

I suspect that both of my problems are related to not understanding what $\mathcal{O}_S(1)$ is, so here is the construction of it that I'm using. Consider the pullback bundle $p^* E \to S$. Define a sub-line bundle $N$ of $p^* E$ as follows. A point $s \in S$ is actually equal to a one dimensional subspace of $E_{p(s)}$. Choose $N_s$ to be this one dimensional subspace of $(p^*E)_s = E_{p(s)}$. Then define $\mathcal{O}_S(1)$ by

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

This question is a bit vague, but could someone motivate this definition for me, or try to explain what is going on here? Is this the same as the pullback of a hyperplane in some projective embedding?

Thank you,

Robert

edit: not totally sure why the latex code isn't working at the end, apologies.