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In my algebraic geometry class we had a theorem that for a scheme $X$ over an affine scheme $\operatorname{Spec} A$ giving a morphism $X \to \mathbb{P}^n_A$ is the same as giving an invertible $\mathcal{O}_X$-sheaf $\mathcal{L}$ and $n + 1$ global sections $s_0, \dotsc , s_n \in \Gamma(X, \mathcal{L})$ which generate $\mathcal{L}$. In the proof we then defined $X_{s_i} = \{ x \in X \mid (s_i)_x \notin \mathfrak{m}_x \mathcal{L}_x\}$ and morphisms $$ f_i \colon X_{s_i} \to \mathbb{P}_A^n $$ by \begin{align*} A\Bigl[ \frac{x_0}{x_i}, \dotsc ,\frac{x_n}{x_i} \Bigr] &\to \Gamma(X_{s_i}, \mathcal{O}_{X_{s_i}}) \,, \\[0.5em] \frac{x_j}{x_i} &\mapsto s_j\vert_{X_{s_i}} (s_i\vert_{X_{s_i}})^{-1} \,. \end{align*} Here I am confused, the $s_j$ are global sections of $\mathcal{L}$ but why are then the $s_j\vert_{X_{s_i}} (s_i\vert_{X_{s_i}})^{-1} \in \Gamma(X_{s_i}, \mathcal{O}_{X_{s_i}})$? Im sorry if any of the notation is unclear, please ask and I will explain what is meant.

Jendrik Stelzner
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Glasenapp
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  • The ration of two sections of a line bundle is a rational function. – Sasha Dec 15 '22 at 19:27
  • You find a more "scheme theoretic" construction of the Segre embedding here which may be of help: https://math.stackexchange.com/questions/4117064/stacks-project-construction-of-segre-embedding-a-clarification/4123144#4123144 – hm2020 Dec 16 '22 at 13:38
  • @hm2020 ok thank you this helped me quite a bit, for others in Hartshorne II 7.1 , goes a tiny bit more into detailed, and I found the post https://math.stackexchange.com/questions/659200/hartshorne-page-150-theorem-7-1?rq=1 which disusses exactly the same problem. – Glasenapp Dec 16 '22 at 15:22

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