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 $$0\rightarrow N\rightarrow \pi^\ast E^\ast \rightarrow O\_h(1)\rightarrow 0,$$ 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 $O\_h(1)\cong O\_l(1)\otimes \pi^\ast(\wedge^2E^\ast)$, 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!



Are these two definitions of $\mathcal{O}(1)$ over a ruled surface closely related?
If $E$ is a rank 2 bundle then $E^\* \cong E \otimes \Lambda^2E^\*$. This gives the required relation for $O(1)$ bundles.


