I'm studying the book "Differential forms of algebraic geometry" of Bott, Tu.
At page 75 there is an exercise about the normal bundle of $CP^1$ in $CP^2$, and there is written that the transition function $g\_{01}$ is $Z\_0/Z\_1$. 
this is wrong for me! i consider the embedding $[X,Y] \mapsto [X,Y,0]$, saying $u=Z\_0/Z\_1$ and $v=Z\_2/Z\_1$, $g\_01$ must be $\partial(v/u)/\partial v = 1/u = Z\_1/Z\_0$. am i doing it wrong?



Normal bundle of $CP^1$ in $CP^2$
With your notations, the normal bundle is spanned over $\{Z\_1\ne 0\}$ by $\partial/\partial v$. Now, over $\{Z\_0\ne 0\}$, take affine coordinates $x=Z\_1/Z\_0$ and $y=Z\_2/Z\_0$, so that, where defined, you have $x=1/u$ and $y=v/u$. On this chart your normal bundle is spanned by $\partial/\partial y$.
The transition function is given by
$$
\partial/\partial v=g\_{01}\cdot\partial/\partial y,
$$ 
so that $g\_{01}=\partial y/\partial v=1/u=Z\_1/Z\_0$.


Your computation is right (after all, these are the transition functions of $\mathcal O(1)$).


