Timeline for Properly elliptic surface with no multiple fibers and without a section
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 13, 2018 at 14:35 | vote | accept | user564401 | ||
| Jul 13, 2018 at 11:39 | answer | added | Jason Starr | timeline score: 6 | |
| Jul 12, 2018 at 16:54 | comment | added | Jason Starr | I meant to write $\mathbb{P}^2\times \mathbb{P}^1$, not $\mathbb{P}^2\times \mathbb{P}^2$. Sorry about that. | |
| Jul 12, 2018 at 16:16 | comment | added | user564401 | @JasonStarr Can you point me toward a reference? Also, it's not clear to me how this would get one closer to an elliptic surface, but maybe I'm just not seeing it. The only way I can think of is to start with an elliptic surface without a section and with multiple fibers, then do a log transformation. But then of course, determining what one ends up with is nontrivial... Curious to see if anyone can point me toward a specific example of this. | |
| Jul 12, 2018 at 0:14 | comment | added | Jason Starr | Possibly you can use the infinitesimal approach to Noether-Lefschetz theory to prove that a very general hypersurface in $\mathbb{P}^2\times \mathbb{P}^2$ of bidegree $(3,d)$, $d\gg 0$, has no section (obviously it has no multiple fibers, since that is codimension $>2$ in the parameter space of plane cubic curves). | |
| Jul 11, 2018 at 21:19 | comment | added | user564401 | I do need it to be projective. Edited accordingly. | |
| S Jul 11, 2018 at 21:09 | history | suggested | user564401 | CC BY-SA 4.0 |
clarification of the questions
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| Jul 11, 2018 at 18:54 | review | Suggested edits | |||
| S Jul 11, 2018 at 21:09 | |||||
| Jul 11, 2018 at 17:10 | comment | added | Jason Starr | Do you require the surface to be projective? (Maybe that is part of "properly".) If Kaehler, non-projective surfaces are acceptable, then you can take the base change of a non-projective elliptic K3 by a high degree morphism from $\mathbb{P}^1$ to $\mathbb{P}^1$. | |
| Jul 11, 2018 at 16:58 | review | First posts | |||
| Jul 11, 2018 at 17:05 | |||||
| Jul 11, 2018 at 16:56 | history | asked | user564401 | CC BY-SA 4.0 |