All varieties are assumed over $\mathbb{C}$. Consider a geometrically ruled surface $X$ over a curve $C$, it is known that $X$ can be realized as the projective bundle associated to a rank $2$ vector bundle $E$ over $C$, that is $X=\mathbb{P}(E)$.
But there are two ways to associate a projective space $\mathbb{P}(V)$ to a vector space $V$: $\mathbb{P}_l(V)$ consists of lines in $V$; $\mathbb{P}_h(V)$ consists of hyperplanes in $V$ (all passing through $0$). These two projective spaces are dual to each other but not isomorphic naturally. (This is the problem!)
Hartshorne uses $\mathbb{P}_h(E)$, since he defined it by $\textrm{Proj }S(E)$; while Beauville uses $\mathbb{P}_l(E)$. It is obviously true that $\mathbb{P}_h(E)=\mathbb{P}_l(E^*)$, we denote it by $X$. Let $\pi:X\rightarrow C$ be the projection map.
The difference comes from the definitions of $O_X(1)$ over $X$. Hartshorne defines it using $\textrm{Proj}$ construction, so $O_l(1)$ is a natural quotient of $\pi^*E$; while Beauville defines it using the exact sequence $0\rightarrow N\rightarrow \pi^*E^*\rightarrow O_h(1)\rightarrow 0$, here $N$ is the natural line bundle over $X=\mathbb{P}_l(E^*)$.
Question: Are these two definitions closely related?
When $E$ is decomposable, I prove that $O_h(1)\cong O_l(1)\otimes \pi^*(\wedge^2E^*)$, but I cannot prove it in general.
I'm sorry that the description is a little bit long, but I'm really confused by the subtle difference between the two definitions of the projective bundle associate to $E$. Thanks for your help!