Following is quoted from -Nakayama, On Weierstrass models-, 


" Let $S$ be a complex surface, and $L$ a line bundle on it. consider $P=\mathbb{P}(\mathcal{O}\_S\oplus L^2 \oplus L^3)$. Let $a$ and $b$ be arbitrary sections of $L^{-4}$ and $L^{-6}$ and let $(x,y,z)$ be the canonical sections of $\mathcal{O}\_P(1)\otimes L^{-2}$, $\mathcal{O}\_P(1)\otimes L^{-3}$ and $\mathcal{O}\_P(1)$ respectively which correspond to the natural injections of $L^2$, $L^3$ and $\mathcal{O}\_S$ into $\mathcal{O}\_S\oplus L^2 \oplus L^3$. Then the Weierstrass model is given by equation $y^2z=x^3+axz^2+bz^3$ in $P$ and is an elliptic fibration over $S$ ..."


My question: When I do calculations for my self, I see that the embeddings given above should correspond to canonical sections of $\mathcal{O}\_P(1)\otimes L^{2}$, $\mathcal{O}\_P(1)\otimes L^{3}$ and $\mathcal{O}\_P(1)$, not the one he is saying and therefore we should consider $(a,b)$ as sections in dual of what he has said and at the end we should get an equation in $\mathcal{O}(3)\otimes L^6$ not in $\mathcal{O}(3)\otimes L^{-6}$ ? How does he get $x$ (and similarly $y$) from embedding $ L^2 \rightarrow \mathcal{O}\_S \oplus L^2 \oplus L^3$??


My calculation:
We have the exact sequnce 
$$0\rightarrow \mathcal{O}\_P(-1) \rightarrow \mathcal{O}\oplus L^2 \oplus L^3 \rightarrow Q \rightarrow 0$$ 


over $P$, from which we get 


$$0 \rightarrow Hom(\mathcal{O}(-1),\mathcal{O}(-1)) \rightarrow Hom(\mathcal{O}(-1),\mathcal{O}\oplus L^{2} \oplus L^{3}) \rightarrow T\_{W/S} \rightarrow 0$$


Here the last object is the relative tangent bundle and the first map is given by three sections $(z,x,y) \in Hom(\mathcal{O}(-1),\mathcal{O}\oplus L^{2} \oplus L^{3})\cong \Gamma(\mathcal{O}(1)) \oplus \Gamma(\mathcal{O}(1)\otimes L^{2}) \oplus \Gamma(\mathcal{O}(1)\otimes L^{3})$ as above. These are analogue of $(x,y,z)$ coordinate on projective space.



Question on a projective bundle associated to a vector bundle
There are two competing definitions for $\mathbb P(\mathcal E)$, one classifies subbundles of $\mathcal E$ of rank 1, the other classifies quotients. With the latter (as used in EGA or Hartshorne), you have a canonical quotient $p^\*\mathcal E\to\mathcal O(1)$ instead of a canonical subbundle $\mathcal O(-1)\to p^\*\mathcal E$. As long as $\mathcal E$ is locally free of finite rank, there is no big difference, you can just dualize $\mathcal E$ to pass from one to the other.


