Timeline for Does there exist a holomorphic fibration of genus two over $\mathbb{P}^{1}$ with $7$ nodal singularities?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Sep 12, 2015 at 13:03 | answer | added | David E Speyer | timeline score: 2 | |
| Sep 12, 2015 at 12:45 | comment | added | David E Speyer | I'm sorry, I just redid the RH computation, and now I get $20$. So we have to work harder, to exclude the ramification profile $(2,2,2)^6$, $(2,2,1,1)$. This sounds like a question for Mike Zieve ... However, as I explain below, it is easy to find solutions by letting $R$ acquire nodes. | |
| Sep 11, 2015 at 19:01 | comment | added | David E Speyer | At most three of those points of $R$ can map to any given point in $\mathbb{P}^1$. So we get ramification over at least $22/3 > 7$ points if we follow this route. The question is about whether we can find a way, by less obvious choices of the parameters, to get down to $7$. | |
| Sep 11, 2015 at 19:00 | comment | added | David E Speyer | I figured I'd explain where the $7$ is presumably coming from. As Yusuf says, $X$ should be a double cover of a $\mathbb{P}^1$ bundle over $\mathbb{P}^1$. The simplest case is $\mathbb{P}^1 \times \mathbb{P}^1$. And $X$ should be ramified over some $R$ of degree $(6,n)$ in $\mathbb{P}^1 \times \mathbb{P}^1$. We must have $n$ even in order for the double cover to exist, so the simplest choice is $(6,2)$. Then, if $R$ is smooth, it is genus $5$. So Riemann-Hurwitz shows that $R \to \mathbb{P}^1$ is ramified at $22$ points of $R$. (continued) | |
| Sep 11, 2015 at 7:36 | answer | added | Remke Kloosterman | timeline score: 4 | |
| Sep 10, 2015 at 18:36 | answer | added | Yusuf Mustopa | timeline score: 0 | |
| Sep 10, 2015 at 1:10 | comment | added | guest | Yes, I want 7 nodal fibers. | |
| Sep 10, 2015 at 0:58 | comment | added | Daniel Litt | Just checking--do you want 7 nodal fibers? Or actually seven nodes on the surface? | |
| S Sep 10, 2015 at 0:31 | history | suggested | Henry.L | CC BY-SA 3.0 |
It should be a complex manifold and I edited the title. I added a sentence in the beginning and reformat it.
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| Sep 10, 2015 at 0:28 | review | Suggested edits | |||
| S Sep 10, 2015 at 0:31 | |||||
| Sep 9, 2015 at 23:23 | review | First posts | |||
| Sep 9, 2015 at 23:42 | |||||
| Sep 9, 2015 at 23:19 | history | asked | guest | CC BY-SA 3.0 |