Comment by Gro-Tsen on How can I calculate $\chi(\mathscr{O}(P))$

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.