Timeline for How should I think about this scheme constructed from a line bundle
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 25, 2013 at 1:15 | vote | accept | Daniel Barter | ||
| Nov 24, 2013 at 4:29 | answer | added | S. Carnahan♦ | timeline score: 3 | |
| Nov 24, 2013 at 2:57 | comment | added | Julian Rosen | @Marty $\oplus_n \mathscr{L}^{\otimes n}$ is canonically isomorphic to $\oplus_n (\mathscr{L}^\vee)^n$ (by a map that negates degree), so we can identify the total space of a line bundle (minus zero section) with the total space of its dual (minus zero section). Concretely, a nowhere vanishing section $\gamma$ of $\mathscr{L}$ over some open $U\subset X$ determines a nowhere vanishing section $\tilde{\gamma}$ of $\mathscr{L}^\vee$ over $U$ by the condition $\tilde{\gamma}(\gamma)=1$. | |
| Nov 24, 2013 at 2:33 | comment | added | Marty | What Jason said below, and Julian said above... except that it should be the total space of the dual line bundle (minus the zero section). | |
| Nov 24, 2013 at 1:47 | comment | added | Julian Rosen | I think $L$ can be viewed as the total space of the line bundle $\mathscr{L}$, minus the image of the zero section. | |
| Nov 24, 2013 at 1:27 | answer | added | Jason Starr | timeline score: 6 | |
| Nov 24, 2013 at 0:41 | comment | added | Daniel Barter | I should add that I understand the obvious properties of $L$, e.g the pushforward of its structure sheaf is $ \oplus_n \mathscr{L}^n$ | |
| Nov 24, 2013 at 0:36 | history | asked | Daniel Barter | CC BY-SA 3.0 |