Timeline for Principal G-covers with G finite abelian
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 9, 2012 at 11:13 | vote | accept | calc | ||
| Jul 5, 2012 at 15:56 | answer | added | Will Sawin | timeline score: 3 | |
| Jul 5, 2012 at 13:16 | comment | added | Sam Gunningham | @Jason: Ah, thanks I didn't spot that $X$ should be projective. I'll edit my answer. | |
| Jul 5, 2012 at 13:15 | answer | added | Sam Gunningham | timeline score: 3 | |
| Jul 5, 2012 at 13:09 | comment | added | Jason Starr | @Sam: Franz does specify that $X$ is projective, whereas your example is not projective. For an 'etale double cover $f:Y\to X$ of a projective variety $X$, the invertible sheaf $\text{Ker}(\text{Tr}_f:f_*\mathcal{O}_Y \to \mathcal{O}_X)$ is trivial if and only if $h^0(Y,\mathcal{O}_Y)$ equals $2$ if and only if $Y$ is disconnected if and only if $Y$ equals $X \sqcup X$ as a covering of $X$. | |
| Jul 5, 2012 at 12:51 | comment | added | Sam Gunningham | The symbol $\equiv$ was meant to be $\cong$... | |
| Jul 5, 2012 at 12:50 | comment | added | Sam Gunningham | I think what you have written is not quite right as stated. For example etale double covers of $X$ are in correspondence with line bundles $L$ with a specified isomorphism $\mathcal O_X \equiv L^{\otimes 2}$. This is not the same as 2-torsion elements of $Pic^0$. For example, on $\mathbb A^1 - {0}$ there are no non-trivial line bundles, but there is a double cover. | |
| Jul 5, 2012 at 11:03 | history | edited | calc | CC BY-SA 3.0 |
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| Jul 5, 2012 at 10:04 | history | asked | calc | CC BY-SA 3.0 |