I am reading the book Complex algebraic surfaces by Beauville, the chapter on Ruled surfaces. I have some doubts.
Let $C$ be a smooth curve. $E$ be a rank 2 vector bundle on $C$. Let $X=\mathbb{P}(E)$ the ruled surface associated to $E$. $p:X\rightarrow C$ is the projection morphism.
We have the canonical short exact sequence on $X$, $0\rightarrow N\rightarrow p^*E\rightarrow O_X(1)\rightarrow 0$.
Here $N$ is the line bundle whose fiber over a point associated to a one dimensional subspace of the corresponding fiber of $E$ is the subspace itself. $O_X(1)$ is the quotient.
1) why is $O_X(1)$ a line bundle.
2) Also it says if $F$ is the class of a fiber of $p$ in $Pic\,X$ then $F.O_X(1)=1$. Why is this so?
3) What are the sections of $O_X(1)$? That is there a divisor $D$ on $X$ such that $O_X(1)=O_X(D)$? Is this like bundle ample? Or nef, big?
1) Because the map $N \to p^*E$ is an embedding at every point of $X$.
2) Because, when you restrict the sequence to a fiber, you get the Euler's sequence $$ 0 \to O_F(-1) \to O_F \oplus O_F \to O_F(1) \to 0. $$
3) By pushing forward to $C$ one gets $H^0(X,O_X(1)) = H^0(C,E)$, so it has sections if and only if $E$ has. Furthermore, $O_X(1)$ is ample if and only if $E$ is ample on $C$.