Timeline for Why does this vector bundle on the surface sit in this exact sequence?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 20, 2015 at 16:40 | vote | accept | gradstudent | ||
| Apr 20, 2015 at 14:48 | comment | added | Sasha | Correct. If $codim Z = 1$ then $I_Z$ is invertible and $N \otimes I_Z = N'$ with $N'$ a line bundle. | |
| Apr 20, 2015 at 12:28 | comment | added | gradstudent | We have an injective map $E/E_1\longrightarrow {E/E_1}^{\vee\vee}=F$. Now $F$ is a line bundle, therefore, tensoring by $F^{\vee}$, we get an injective map $E/E_1\otimes F^{\vee}\longrightarrow F\otimes F^{\vee}=\mathcal{O}_X$. Therefore, $E/E_1\otimes F^{\vee}$ is an ideal sheaf of a closed subscheme say $Z$. So $E/E_1 = F\otimes I_Z$. So I get what you said. Why is the codimension of $Z$ two? | |
| Apr 19, 2015 at 19:15 | comment | added | Sasha | These are basic properties, try looking into Huybrechts--Lehn. | |
| Apr 19, 2015 at 19:04 | comment | added | gradstudent | Thanks @Sasha! Can I know some reference for this? | |
| Apr 19, 2015 at 18:34 | history | answered | Sasha | CC BY-SA 3.0 |