Timeline for branch locus of the discriminant map $\overline{\mathcal{H}}_{g',r} \to \overline{\mathcal{M}}_{g,n}$
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 31, 2013 at 16:38 | comment | added | Dan Petersen | You should read a textbook on modular forms! I think there's a nice treatment of all these things about modular curves in the book of Diamond-Shurman. I think I remarked on your other question that Prym=spin in genus one. | |
| Oct 31, 2013 at 16:26 | comment | added | IMeasy | By the way I just remarked that for $\overline{\mathcal{M}}_{1,1}$ if you give no weight to the marked point, prym and spin curve are the same, cause $K=\mathcal{O}$. | |
| Oct 31, 2013 at 10:05 | comment | added | IMeasy | A-ha, I see. In fact those are the points that have $Aut= \mathbb{Z}_4$ and $\mathbb{Z}_6$. Ok, and how does one compute the multiplicity at each point? SHould I expect the mult at 0 to be higher than that at 1728 since there are more automorphisms that do not lift? and (I abuse of your kindness) how does the situation change on the stack? | |
| Oct 31, 2013 at 9:36 | comment | added | Dan Petersen | Oh no, genus one is special. As coarse spaces you have $\overline M_{1,1} \cong \mathbb P^1$ so there's no way you can have only one branch point of a ramified cover!! You can see this by thinking about automorphisms too. Branching always happens in codimension one, but in higher genus the loci of smooth curves with automorphisms have bigger codimension. This is why the only branching comes from the divisor $\delta_1$. For the projection from a modular curve you should expect branching over $\infty$ and over the points with $j$-invariants $0$ and $1728$. | |
| Oct 31, 2013 at 9:29 | comment | added | IMeasy | Ok I see. So it should be the same also in the case of elliptic curves, right? $\overline{\mathcal{M}}_{1,1}^r \to \overline{\mathcal{M}}_{1,1}$ is ramified only over $\delta_{irr}$? and with what multiplicity? | |
| Oct 24, 2013 at 11:47 | comment | added | Dan Petersen | You can see this (informally) by computing degrees: on the level of stacks the map is étale, but then the map to the course space has "degree 1/2" downstairs (that is, on the divisor $\delta_1$) and is generically an isomorphism upstairs so has "degree 1". | |
| Oct 24, 2013 at 11:46 | comment | added | Dan Petersen | You're right, I was thinking of the stack. On the level of course spaces you have to think also about automorphisms. On most divisors the generic curve has no automorphism, so this won't give you any extra ramification, but on $\delta_1$ the generic curve has $2$ automorphisms. If this generically defined automorphism does not lift to the level curve then you get ramification on the level of coarse spaces. | |
| Oct 24, 2013 at 11:31 | comment | added | IMeasy | There seems to be a difference whether you consider the stack or the coarse moduli space. For instance, in the "twin" paper arxiv.org/pdf/1205.0661.pdf (page 13) the authors claim that for the coarse moduli space the discriminant morphism is ramified along some of the divisors I mentioned. Can you explain this? | |
| Oct 24, 2013 at 9:35 | comment | added | Dan Petersen | For $n=0$ the branch locus is $\delta_{irr}$. This is described in many places, for instance here: arxiv.org/abs/1205.0201 | |
| Oct 24, 2013 at 8:22 | history | asked | IMeasy | CC BY-SA 3.0 |