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I am trying to understand what happens to the Chern Classes of an invertible sheaf $F$ over a complete intersection reduced curve of genus $g$ and degree $d$, when viewed as a invertible sheaf of $\mathbb{P}^3$ by the natural inclusion map $i : C \to \mathbb{P}^3$, if the sheaf $F$ has degree $g-1$.

My first idea was to look into the pushfowards formula that defines the Chern classes, but the problem is that on the paper I am following the authors says that this sheaf $i_{*}F$ has non zero second Chern class, and I can not see why.

Thank you for any help.

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    $\begingroup$ The pushforward of an invertible sheaf is rarely an invertible sheaf, and definitely isn't one in this case; it's supported on $C$, not on all of $\mathbb P^3$. To talk about Chern classes, you'd need to resolve this sheaf using vector bundles on $\mathbb P^3$, and it's easy to believe that they might need dimension $>1$ (I haven't actually computed a resolution). $\endgroup$
    – Allen Knutson
    May 19, 2016 at 23:31
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    $\begingroup$ The Grothendieck-Riemann-Roch Theorem computes the Chern character (and hence Chern classes) of the derived pushforward of a sheaf. If the morphism is a closed embedding, the derived pushforward coincides with the usual pushforward, so GRR gives what you want. $\endgroup$
    – Sasha
    May 20, 2016 at 9:36

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