Timeline for Example of linearization for GIT
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 22, 2014 at 12:43 | comment | added | Giulio | As varieties, they are both isomorphic to $\mathbb{P}^n$. I think the group acting on $Proj \, Sym \,V$ is $SL(V^{\vee})$ rather than $SL(V)$. | |
| Jul 22, 2014 at 12:04 | comment | added | Sasha | When $\dim V > 2$ projective spaces ${\mathbb{P}}V$ and ${\mathbb{P}}V^\vee$ are not isomorphisc (just because representations $V$ and $V^\vee$ are not isomorphic). | |
| Jul 22, 2014 at 11:28 | comment | added | Giulio | It is a bit confusing. $\mathbb{P}V$ and $\mathbb{P}V^{\vee}$ are isomorphic; even as polarized varieties with polarization $\mathcal{O}(1)$. But they are NOT isomorphic as $SL(V)$-varieties?? (Moreover, $SL(V)$ does not have characters, so the linearization is unique) | |
| Jul 22, 2014 at 11:20 | comment | added | Giulio | So, it depends whether your define $\mathbb{P}V$ as $\Proj Sym(V)$ or $\Proj Sym(V^{\vee})$ ?? | |
| Jul 21, 2014 at 21:35 | comment | added | Sasha | No! In the first case the answer is $V^\vee$, while in the second it is $V$. | |
| Jul 21, 2014 at 20:48 | comment | added | Giulio | I know, but I think the definition of \mathcal{O}(1) changes accordingly. Isn't it?? You should get in both cases the same answer | |
| Jul 21, 2014 at 20:33 | comment | added | Sasha | The answer depends on your convention on what $\mathbb{P}V$ is --- it can be either the moduli space of 1-dimensional subspaces in $V$, or the moduli space of 1-dimensional quotient spaces. Both conventions are used, so it is better to specify what do you mean here. | |
| Jul 21, 2014 at 19:23 | vote | accept | Giulio | ||
| Jul 21, 2014 at 19:23 | vote | accept | Giulio | ||
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| Jul 21, 2014 at 19:23 | vote | accept | Giulio | ||
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| Jul 21, 2014 at 19:23 | vote | accept | Giulio | ||
| Jul 21, 2014 at 19:23 | |||||
| Jul 21, 2014 at 19:23 | vote | accept | Giulio | ||
| Jul 21, 2014 at 19:23 | |||||
| Jul 21, 2014 at 19:07 | answer | added | Pavel Safronov | timeline score: 3 | |
| Jul 21, 2014 at 19:01 | answer | added | Daniel Loughran | timeline score: 1 | |
| Jul 21, 2014 at 17:42 | history | asked | Giulio | CC BY-SA 3.0 |