Timeline for Weights on the linearization
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 10, 2020 at 19:49 | vote | accept | konoa | ||
| Aug 10, 2020 at 16:21 | comment | added | abx | I have written my comment as an answer with some more details, let me know if this not completely clear. | |
| Aug 10, 2020 at 16:20 | answer | added | abx | timeline score: 4 | |
| Aug 10, 2020 at 16:09 | comment | added | konoa | Dear @abx, thanks a lot for the comment! I'm sorry, but I don't follow exactly your passages. You consider $\mathcal{O}(-1)$, and indeed $s_0(\mathbb{P}^2)^c=\mathbb{C}^3\setminus 0$. The you define an action on $\mathbb{C}^3\setminus 0$ with weights $(0,1,2)$, but I don't get the next passage, that is for example $t$ acts precisely as $t^2$ on $L_{e_2}$,I'm sorry The other points are fine, you swap sign since you're working with the dual bundle, and yes, I 'm aware of the last sentence. If you want to write it as an answer I'll surely accept it, thanks again for the patience! | |
| Aug 10, 2020 at 15:53 | comment | added | abx | If your $\mathbb{P}^2$ is $\mathbb{P}(\mathbb{C}^3)$, you can identify the complement of the zero section in $L^{-1}$ with $\mathbb{C}^3\smallsetminus 0$. One possible way to extend your action is to have $t\in\mathbb{C}^*$ acts by $(x,y,z)\mapsto(x,ty,t^2z)$. Then $t$ acts trivially on $L_{e_0}$, as $t$ on $L_{e_1}$ and as $t^2$ on $L_{e_2}$, so that the weights on $L$ are $0,-1,-2$. Note however that you are free to add a fixed integer to these (the linearization is not unique). | |
| Aug 10, 2020 at 15:33 | review | First posts | |||
| Aug 10, 2020 at 15:37 | |||||
| Aug 10, 2020 at 15:26 | history | asked | konoa | CC BY-SA 4.0 |