I am following Vakil's notes on Foundations of algebraic geometry. We have just introduced twisting sheaves ${\cal O}(n)$ on $\mathbb P^1_k$ by gluing the structure sheaves on standard open subsets with a non-standard map.

Later in the text, we take a rational function $s \in {\cal O}_X(U)$ and associate to it a rational section of ${\cal O}(n)$, but the procedure on how this is done is not explained. It is not clear to me how to get a section of $s$, since we constructed ${\cal O}(n)$ by gluing and there seems to be some choice related to which chart of ${\cal O}(n)$ one chooses to interpret the function $s$. As ${\cal O}(n)$ is non-trivial, this choice will produce different rational functions.

Example: How can we interpret $1/x_0$ as a section of ${\cal O}(1)$? Or of ${\cal O}(-1)$? We can think of $1/x_0$ as defined on an open subset of $D(x_0)$ as well as an open subset of $D(x_1)$. If we start with $D(x_0)$, then it gives a regular function $1/x_0$ there, and this function corresponds to $x_1^2$ on $D(x_1)$ if we are working with ${\cal O}(1)$, and to $1$ on $D(x_1)$ if we are working with ${\cal O}(-1)$.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.