Timeline for Does ampleness descend along finite maps?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 16, 2018 at 5:50 | vote | accept | jacob | ||
| Sep 15, 2018 at 1:15 | answer | added | darx | timeline score: 5 | |
| Sep 14, 2018 at 15:25 | comment | added | Mohan | @R.vanDobbendeBruyn You are right, it is a mistake. | |
| Sep 14, 2018 at 14:21 | comment | added | R. van Dobben de Bruyn | @Mohan: your criterion is not true for non-proper varieties. For example, consider $\mathscr L = \mathcal O_X$ on $X = \mathbb A^2 \setminus \{0\}$, which is ample by Tag 01QE (and take $\mathscr F = \mathcal O_X$ as well). | |
| Sep 14, 2018 at 13:24 | comment | added | Mohan | The best proof (not necessarily the easiest) would be to show that with your definition, a line bundle $L$ is ample if and only for any coherent sheaf $F$, $H^i(F\otimes L^n)=0$ for all large $n$ and all $i>0$. This property transfers nicely under finite maps. | |
| Sep 14, 2018 at 13:21 | history | edited | jacob | CC BY-SA 4.0 |
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| Sep 14, 2018 at 13:20 | comment | added | jacob | Could you please explain why? | |
| Sep 14, 2018 at 13:19 | comment | added | Mohan | With your definition, a line bundle is ample if and only if it is when you pull it back under a finite map. | |
| Sep 14, 2018 at 13:03 | history | asked | jacob | CC BY-SA 4.0 |