Timeline for Morphism attached to a big and globally generated line bundle
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 7, 2022 at 11:40 | vote | accept | Maqui | ||
| Jun 7, 2022 at 10:02 | comment | added | Ennio Mori cone | You can also see it because if $\phi_L$ maps to something lower-dimensional, then $0=f^*(O(1))^n$, so $L^n=0$. However, since $L$ is globally generated (hence nef), it is big if and only if $L^n>0$. | |
| Jun 6, 2022 at 19:26 | answer | added | Karl Schwede | timeline score: 1 | |
| Jun 5, 2022 at 20:37 | comment | added | Maqui | @KarlSchwede Thanks! if you post this as an answer I'd be happy to accept it. | |
| Jun 5, 2022 at 3:38 | comment | added | Karl Schwede | I don't see it in Positivity, but one way to see it is this, let $Z = \phi_L(X)$, let $\eta$ be the generic point of $Z$ and let $f : X_{\eta} \to \eta$ be the base change. Note $L|_{X_{\eta}}$ is still big and globally generated. Furthermore $X_{\eta}$ is a projective variety over the field $k(\eta)$ and $L|_{X_{\eta}} = \phi_L^* O(1)|_{X_{\eta}} = f^* k(\eta) = O_{X_{\eta}}$. Now the only time the structure sheaf of a projective variety can be big is if the variety is 0-dimensional, so $X_{\eta}$ is zero dimensional, thus $X \to Z$ is generically finite. | |
| Jun 5, 2022 at 1:28 | comment | added | Maqui | @JasonStarr Thanks for the prompt reply! Do you happen to know a reference? | |
| Jun 5, 2022 at 1:25 | comment | added | Jason Starr | Welcome new contributor. Yes, the morphism $\phi_L$ is generically finite to its image. | |
| S Jun 5, 2022 at 1:11 | review | First questions | |||
| Jun 5, 2022 at 2:03 | |||||
| S Jun 5, 2022 at 1:11 | history | asked | Maqui | CC BY-SA 4.0 |