Timeline for Is there an analogy of Sumihiro's equivariant Chow's lemma for DM stack?
Current License: CC BY-SA 3.0
7 events
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| Nov 28, 2017 at 13:23 | comment | added | Ariyan Javanpeykar | You're welcome. I will come back and write down an answer once I'm sure that what I say is really correct. Or maybe somebody else will find another (easier?) example. | |
| Nov 28, 2017 at 13:17 | comment | added | keaton | Thank you for your kind explanation. Your comments help me very much. Also results in the reference you mentioned are exactly what I was finding these days. | |
| Nov 27, 2017 at 17:00 | comment | added | Ariyan Javanpeykar | More precisely, write $Y:=\mathbb{P}^1$ and $X = \mathbb{P}^1$ and consider $X$ with the natural action of $\mathbb{G}_a$ (via the inclusion $\mathbb{A}^1\subset \mathbb{P}^1$). My previous comment shows that the question now becomes whether there is a $\mathbb{G}_a$-action on $Y$ and a $\mathbb{G}_a$-equivariant morphism $Y\to X$ which ramifies at infinity. If this data does not exist, then the answer to your question is negative. | |
| Nov 27, 2017 at 12:27 | comment | added | Ariyan Javanpeykar | Let $G=\mathbb{G}_a$ and consider the action of $G$ on the weighted projective line $\mathcal{P}(1,n)$ with $n\geq 2$ over $\mathbb{C}$. Note that $\mathcal{P}(1,n)$ is a smooth proper DM-curve. Let $Y\to X$ be a generically finite, proper surjective morphism with $Y$ a normal integral scheme. Then $Y$ is a smooth projective connected curve. If the $\mathbb{G}_a$-action on $X:=\mathcal{P}(1,n)$ lifts to $Y$, then $g(Y) =0$, so that $Y =\mathbb{P}^1$. Now, the question becomes whether there is a $\mathbb{G}_a$-equivariant morphism $f:\mathbb{P}^1\to \mathbb{P}^1$ which ramifies at infinity. | |
| Nov 27, 2017 at 12:24 | comment | added | Ariyan Javanpeykar | If $G$ is a torus and you are ok with working etale locally, you can find an answer to your question in Alper-Hall-Rydh's Section 2.3 of sites.math.washington.edu/~jarod/papers/luna.pdf | |
| Nov 26, 2017 at 19:09 | history | edited | keaton | CC BY-SA 3.0 |
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| Nov 26, 2017 at 3:28 | history | asked | keaton | CC BY-SA 3.0 |