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Let be X a reduced and irreducible curve over a field $L_0$. Let $L$ an extension of $L_0$ and set

\begin{gather*} \overline{X}=L \otimes X. \end{gather*}

Assume $\overline{X}$ also irreducible. Now, let $P_0 \in X$ be a regular closed point and set $P=L \otimes P_0$. If $\mathscr{O}_{\overline{X}}(P)$ is the ideal sheaf of $P$, how can I calculate $\chi(\mathscr{O}_{\overline{X}}(P))$? I do not really know much of sheaf cohomology, so any comment will be really helpful to me.

Thanks!

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    $\begingroup$ That is only well defined if $X$ is proper. In that case, it equals $\chi(X,\mathcal{O}_X)$ minus the degree of the field extension of the residue field of $P$ over $L_0$. $\endgroup$
    – Jason Starr
    Commented Jun 25 at 19:34
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    $\begingroup$ To clarify how to derive the result given in Jason Starr's comment, write down the additivity of $\chi$ for the short exact sequence $0 \to \mathcal{O}_{\bar X}(P) \to \mathcal{O}_{\bar X} \to \mathcal{O}_P \to 0$ and $\mathcal{O}_P$ is a skyscraper sheaf. $\endgroup$
    – Gro-Tsen
    Commented Jun 25 at 19:56
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    $\begingroup$ By the way, the standard notation is $\mathscr{O}_{\bar{X}}(-P)$. $\endgroup$
    – abx
    Commented Jun 26 at 4:30

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