Timeline for Coarse moduli spaces of quotient stacks
Current License: CC BY-SA 3.0
7 events
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| Jul 23, 2013 at 13:25 | comment | added | Jason Starr | The OP is correct: Jim Bryan's question has a negative answer. For the action of $\mathbf{G}_m$ on $\mathbb{A}^2\setminus\{(0,0)\}$ via $t*(x,y) = (tx,t^{-1}y)$, the structure sheaf is ample and admits a $\mathbb{G}_m$-linearization. However, the coarse moduli space is the (non-separated) line with doubled origin. So I upvoted this answer. | |
| Jul 23, 2013 at 7:15 | comment | added | Lennart Meier | A source which deals more generally with quotients which are Artin stacks is Alper's Good Moduli Spaces: maths-people.anu.edu.au/~alperj/papers/good_moduli_spaces.pdf But be aware that his notion of a good moduli space is more general than that of a coarse moduli space and applies, for example, also to $[\mathbb{A}^1/\mathbb{G}_m]$. | |
| Jul 23, 2013 at 7:12 | comment | added | Lennart Meier | For $G$ finite, you can just take the set-theoretic quotient (and put an appropriate sheaf on this). See for example Section 3 of Conrad's notes on the Keel-Mori theorem: math.stanford.edu/~conrad/papers/coarsespace.pdf | |
| Jul 23, 2013 at 6:45 | comment | added | stacksgg | @Jim Bryan: Why is that statement true? By $V/G$ you mean either the uniform geometric quotient or the uniform categorical quotient in the sense of Keel-Mori? | |
| Jul 23, 2013 at 0:22 | comment | added | Jim Bryan | Isn't it true that $V/G$ is a quasi-projective scheme if and only if $V$ has a $G$ invariant ample line bundle? Is that what you are looking for? | |
| Jul 22, 2013 at 22:56 | review | First posts | |||
| Jul 22, 2013 at 22:58 | |||||
| Jul 22, 2013 at 22:36 | history | asked | stacksgg | CC BY-SA 3.0 |