Timeline for Endomorphism of globally generated sheaves on curves
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 17, 2017 at 14:49 | vote | accept | user43198 | ||
| Apr 17, 2017 at 6:10 | comment | added | R. van Dobben de Bruyn | Also, usually $(-)_x$ denotes the stalk, not the fibre. For this type of question it doesn't matter, because by Nakayama's lemma surjecting onto the stalk is equivalent to surjecting onto the fibre. | |
| Apr 17, 2017 at 5:55 | answer | added | R. van Dobben de Bruyn | timeline score: 6 | |
| Apr 17, 2017 at 1:56 | comment | added | R. van Dobben de Bruyn | Another comment: global generation of $\mathcal E$ is completely irrelevant. Indeed, $\mathcal E \otimes \mathcal L$ is globally generated for a sufficiently ample line bundle $\mathcal L$, and $\mathcal End(\mathcal E \otimes \mathcal L) = \mathcal End(\mathcal E)$. | |
| Apr 17, 2017 at 1:53 | comment | added | R. van Dobben de Bruyn | First counterexample: consider $\mathcal E = \mathcal O \oplus \mathcal O(1)$ on $\mathbb P^1$. Then $\mathcal End(\mathcal E) = \mathcal O(-1) \oplus \mathcal O \oplus \mathcal O \oplus \mathcal O(1)$, so this is not globally generated (even though $\mathcal E$ is). Indeed, global sections do not surject onto the stalk for any point, because the $\mathcal O(-1)$ factor does not have a nonzero global section. Of course, similar examples exist on curves of higher genera. | |
| Apr 17, 2017 at 0:52 | history | asked | user43198 | CC BY-SA 3.0 |