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Let $f:\mathbb{P}^1 \rightarrow X$ be a morphism with $X$ a smooth projective algebraic variety of dimension $n$, then by Grothendieck $\mathcal{N}_{f}=\bigoplus_{i=1}^{n-1}\mathcal{O}_{\mathbb{P}^1}(a_i)$, where $\mathcal{N}_{f}$ is the normal sheaf associated to $f$.

Is there some relation between those $a_{is}$ and the properties of the morphism $f$? and what is the geometric interpretation?.

..for example if $X$ has dimension $2$ then $\mathcal{N}_{f}=\mathcal{O}_{\mathbb{P}^1}(a_1)$ thus $a_{1}=c_{1}(\mathcal{N}_{f})$ (the first Chern class of the normal sheaf) and it is the autointersection of $f(\mathbb{P}^1)\subset X$ , but in higher cases $c_{1}(\mathcal{N}_{f})=\sum_{i=1}^{n-1}a_i$ and then I don't know what is the relation between them... I wanna find a geometric interpretation of those $a_is$ in $f$. Anyone could help me with this ?

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    $\begingroup$ There is a huge literature about this. You might start with Kollár's Rational curves on algebraic varieties. $\endgroup$
    – abx
    Commented Apr 18, 2020 at 18:24
  • $\begingroup$ Yes i am reading this book and Debarre , but i didnt find some about this coefficients $\endgroup$
    – alberth
    Commented Apr 18, 2020 at 18:27
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    $\begingroup$ Come on! The coefficients tell you how your map deforms, a fundamental tool in algebraic geometry. Read better. $\endgroup$
    – abx
    Commented Apr 18, 2020 at 18:29
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    $\begingroup$ @abx, I don't think the "Come on!" and "Read better" are appropriate or needed in your comment. $\endgroup$
    – Deane Yang
    Commented Apr 18, 2020 at 18:48
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    $\begingroup$ Looking at the proof of the result should answer this. On th eother hand, I hope we get an enlightening answer from a member of the community :) $\endgroup$
    – user347489
    Commented Apr 18, 2020 at 18:54

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