Timeline for When an ample invertible sheaf can be very ample?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 1, 2023 at 11:43 | answer | added | Jason Starr | timeline score: 3 | |
| Apr 1, 2023 at 11:26 | comment | added | Jason Starr | The definition in Hartshorne’s book is “wrong”, I.e., disagrees with the definition in EGA. This is the source of confusion. | |
| Apr 1, 2023 at 1:50 | comment | added | Tabes Bridges | My point is that your hypothesis of $L$ ample on $X$ implies that $L^n$ is very ample, hence induces a closed immersion $X \to \mathbb P_k^n$. If this map is a closed immersion, then it remains a closed immersion when restricted to the fibers of $f$, showing that $L^n$ is very ample relative to $Y$. | |
| Apr 1, 2023 at 1:48 | comment | added | Tabes Bridges | "Very ample on $X$" means "very ample relative to the structure map $X \to \operatorname{Spec}(k)$." | |
| Apr 1, 2023 at 0:12 | comment | added | ZhouQi | Is very ample not a relatitive notion? What is meaning of "being very ample on $X$" without a morphism? @Tabes Bridges | |
| Apr 1, 2023 at 0:09 | comment | added | ZhouQi | Sorry, I can't get your point……You mean this assertion is ture? @Tabes Bridges | |
| Mar 31, 2023 at 23:32 | comment | added | Tabes Bridges | Being very ample on $X$ is a stronger condition than being very ample relative to a morphism, so shouldn't this be true automatically (given that $L$ is ample hence some $L^n$ is very ample)? | |
| S Mar 31, 2023 at 14:20 | review | First questions | |||
| Mar 31, 2023 at 14:41 | |||||
| S Mar 31, 2023 at 14:20 | history | asked | ZhouQi | CC BY-SA 4.0 |