Timeline for $Pic$ of the stack of elliptic curves vs. $Pic$ of the coarse space
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Nov 10, 2013 at 14:03 | history | bounty ended | IMeasy | ||
| S Nov 10, 2013 at 14:03 | history | notice removed | IMeasy | ||
| Nov 9, 2013 at 18:26 | vote | accept | IMeasy | ||
| Nov 7, 2013 at 11:34 | comment | added | naf | In the tame case, my claim follows from Lemma 2.3 of the Fulton--Olsson paper. | |
| Nov 7, 2013 at 4:13 | answer | added | Jonathan Wise | timeline score: 7 | |
| Nov 6, 2013 at 15:51 | vote | accept | IMeasy | ||
| Nov 6, 2013 at 15:51 | |||||
| Nov 5, 2013 at 0:33 | answer | added | Puzzled | timeline score: 2 | |
| Nov 4, 2013 at 20:34 | comment | added | IMeasy | I really would love to see a proof. Do you have a reference? | |
| Nov 4, 2013 at 15:17 | comment | added | Lennart Meier | The general statement is, I think, the following: Let $f:\mathcal{X} \to X$ be the map from a DM-stack to its coarse moduli stack, $L$ a line bundle on $\mathcal{X}$ and $n$ the lowest common multiple of the orders of all automorphism groups. Then $f_*L^{\otimes n}$ is a line bundle and $L^{\otimes n} \to f^*f_*L^{\otimes n}$ an isomorphism. At least, I know a proof in the case that $\mathcal{X}$ is separated and of finite type over a noetherian base scheme. | |
| S Nov 4, 2013 at 12:42 | history | bounty started | IMeasy | ||
| S Nov 4, 2013 at 12:42 | history | notice added | IMeasy | Draw attention | |
| Nov 2, 2013 at 11:09 | comment | added | naf | It is easy to see that the 12'th tensor power of the Hodge bundle--and no smaller tensor power--descends to the coarse moduli space: consider the possible automorphism groups of elliptic curves and their actions on the tangent space at the identity. | |
| Nov 2, 2013 at 0:30 | comment | added | S. Carnahan♦ | You can also consider the generator $\Delta$ of Pic on the coarse space, and lift to the degree 24 cover $X(3) \to \overline{\mathcal{M}_{1,1}}$. I think degree of $\omega_P$ for a representable elliptic moduli problem $P$ is computed somewhere in Katz-Mazur. | |
| Nov 1, 2013 at 23:14 | comment | added | Lennart Meier | If I remember correctly, Mumford only does it over a field not of characteristic 2 or 3. The general reference is Fulton and Olsson: arxiv.org/abs/0704.2214 | |
| Nov 1, 2013 at 20:36 | comment | added | IMeasy | $Pic(\mathcal{M}_{1,1})$ is $\mathbb{Z}/12\mathbb{Z}$ (see Mumford). It is $Pic(\overline{\mathcal{M}}_{1,1})$ that is $\mathbb{Z}$. This is pretty standard, see for instance sect.6 of arxiv.org/abs/0812.1803v2 | |
| Nov 1, 2013 at 18:11 | history | edited | IMeasy | CC BY-SA 3.0 |
added 2 characters in body
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| Nov 1, 2013 at 17:06 | comment | added | Joël | That seems reasonable. But how do you know that $Pic \mathcal M_{1,1} = \mathbb Z$? You have a proof or reference? | |
| Nov 1, 2013 at 14:15 | history | asked | IMeasy | CC BY-SA 3.0 |