Let $S$ be a projective bundle $\mathbb{P}(E)$ over the projective line $\mathbb{P}^1$. Assume we are working over $\mathbb{C}$ and that rank$E=2$. Let $p:S \to \mathbb{P}^1$ be the projection morphism and $f:E \to \mathbb{P}^1$ the vector bundle morphism.
I've seen two different statement about $\mathcal{O}_{S}(1)$: in some books I found written that there is a canonical injective morphism $$0 \to \mathcal{O}_S(-1) \to p^*E$$
which if I correctly understood, should be described in this way: the fiber of $(\mathcal{O}_S(-1)\mathcal{O}_S(-1))_x$ is identified with $Span(x) \subset E_{p(x)}$ with $y$.
Given that $p^*E_x=E_{p(x)}$ we have a natural inclusion.
Other times, I 've seen declaring that there is a canonical morphism $p^*E \to \mathcal{O}_S(1)$ which I really am not able to see how to define.
I'm asking this also because of this fact. I'm trying to study intersection theory on the Hirzebruch surface $ S_n=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}(0) \bigoplus \mathcal{O}_{\mathbb{P}^1}(n))$. Here, there is divisor $D$ defined as the image of the section $$\sigma : \mathbb{P}^1 \to S_n$$ $$x \to [1 \ \ 0]_x .$$
If I have correctly understood, the line bundle associated to $D$ is $\mathcal{O}_S(1)$ and we would like to prove that $D^2=n$. Now,in the proof I was looking at (which is the one taken from Vakil notes about algebraic surfaces), he uses the first sequence to get that $\mathcal{O}_S(-1) \cong p^*\Lambda^2E^* \otimes Q$ for some invertible sheaf $Q$. The problem is that calculating $D^2$ now gives $deg(\Lambda^2E^*)$ which is $-n$ and so I'm getting pretty confused and I can't see how to make the proof work