Timeline for Submersion implies many rational points in image?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 7, 2018 at 20:14 | vote | accept | darko | ||
| Oct 2, 2018 at 9:22 | answer | added | Daniel Loughran | timeline score: 5 | |
| Oct 1, 2018 at 18:01 | comment | added | darko | @DanielLoughran: It would certainly suffice if we can conclude that the number of $\mathbb{F}_p$-points is $O(p^{\dim Y})$, so a sensible bound on the log-size of the image would be fine. You don`t think it is of any use that the map $\alpha$ is essentially linear and so its fibers are linear intersections with $X$? By the way, I would be satisfied with resolving this even just for large enough primes $p$. | |
| Oct 1, 2018 at 15:49 | comment | added | Daniel Loughran | I'm still not sure what you are after and I think that the submersion condition you are imposing is not as important as you think it is. For example the map $x \to x^2$ on the affine line is a smooth morphism away from the origin, hence a submersion on points over these fields, but is clearly not surjective on $\mathbb{F}_p$-points. The image contains $(p-1)/2$ many points; is this a "good enough" lower bound for you? | |
| Oct 1, 2018 at 15:42 | comment | added | darko | @aginensky: I really want to have a rational map $\mathbb{P}(V) \to \mathbb{P}(W)$ coming from a surjective linear map. | |
| Oct 1, 2018 at 15:41 | comment | added | darko | @DanielLoughran: Indeed, as an analytic function. I have edited the relevant part in the post. Would smoothness be enough to give an answer to the question? How is smoothness over $\mathbb{F}_p$ related to that over $\mathbb{R}/\mathbb{Q}_p/\mathbb{C}$? | |
| Oct 1, 2018 at 15:35 | history | edited | darko | CC BY-SA 4.0 |
added 43 characters in body
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| Oct 1, 2018 at 15:09 | comment | added | meh | Usually $A \colon V \to W$ surjective is considered as inducing an immersion , $ \alpha \colon \mathbb{P}(W) \hookrightarrow \mathbb{P}(V)$. Is that a typo or do you want to do it that way. | |
| Oct 1, 2018 at 15:08 | comment | added | Daniel Loughran | I think it would be good to clarify more the question. By "local submersion", do you mean on the level of $\mathbb{R}/ \mathbb{Q}_p/\mathbb{C}$-points? The analogue of submersion in algebraic geometry is a smooth morphism, so are you asking that $X \to Y$ is a smooth morphism of schemes? This in itself is a very mild assumption, as by generic smoothness it always holds over some open subset of the base (in char $0$). And when counting $\mathbb{F}_p$-points, one is often happy to remove closed subvarieties as these usually contribute negligibly to the count. | |
| Oct 1, 2018 at 14:32 | history | asked | darko | CC BY-SA 4.0 |