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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?

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closed as too localized by Ryan Budney, Deane Yang, Igor Rivin, Dan Petersen, Andreas Blass Jan 10 '12 at 22:25

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up vote 0 down vote accepted

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 $\mathcal O(1)$).

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In my Bott-Tu edition is says $g_{01}=z_1/z_0$ which is the right answer.

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