Timeline for Number of $\mathbb F_p$ points constant mod $p$?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 27, 2015 at 10:16 | comment | added | Mikhail Bondarko | @Jason Starr Thank you! I didn't know that (being not an expert in conditions of this sort). | |
| Jun 26, 2015 at 14:15 | comment | added | Will Sawin | I think the statement you want doesn't follow from standard motivic conjectures. You need a special conjecture just to prove statements of this type. See my question mathoverflow.net/questions/209667/… However, this should cause no problem for Allen. | |
| Jun 26, 2015 at 14:06 | comment | added | Jason Starr | @MikhailBondarko. Regarding the impression that testing rational connectedness is hard, that depends very much on the variety. There is a conjecture of Mumford that, if true, implies that testing rational connectedness is "computable" (if not particularly efficient). In many cases, it really is easy to test rational connectedness: for smooth projective varieties in characteristic $0$, it is a matter of finding a single morphism from $\mathbb{P}^1$ that pulls back the tangent bundle to be ample. | |
| Jun 26, 2015 at 12:54 | comment | added | Mikhail Bondarko | Yet my impression is that the notions rationall connectivity/unirationality/etc. are mostly adapted to smooth projective varieties. Still you certainly may look at "nice compactifications" of your $X$ (and ask whether there exists a compactification with prescribed properties). | |
| Jun 26, 2015 at 12:38 | comment | added | Mikhail Bondarko | Yes; this seems to be a very reasonable idea! Yet I have an impressing that testing rational connectivity may be rather hard. So I would be glad to read any your further questions in this direction.:) | |
| Jun 26, 2015 at 12:33 | comment | added | Allen Knutson | Your #4 sounds like "what does it imply?" rather than "what does it suggest?" (what would imply it, that one should be on the lookout for?). I have a certain class of varieties; I don't know what properties they have, but now I'm going to try to prove rational connectivity directly for them (not actually imply it from the point-counting fact). | |
| Jun 26, 2015 at 11:32 | history | edited | Mikhail Bondarko | CC BY-SA 3.0 |
added 730 characters in body
|
| Jun 26, 2015 at 11:00 | history | edited | Mikhail Bondarko | CC BY-SA 3.0 |
added 152 characters in body
|
| Jun 26, 2015 at 9:27 | comment | added | Mikhail Bondarko | Yet my "general" pessimism does not mean that one cannot hope to obtain certain information in special cases (i.e., if $X belongs to a "simple" class of varieties; smooth and affine does not seem to be sufficient here). | |
| Jun 26, 2015 at 9:09 | history | answered | Mikhail Bondarko | CC BY-SA 3.0 |