Timeline for Do varieties without rational curves contain sub-polynomially many rational points?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 6, 2017 at 17:04 | answer | added | jacob | timeline score: 2 | |
| Aug 1, 2017 at 0:36 | comment | added | Noam D. Elkies | Surely not a cubic threefold because once you have a rational point you have a hyperplane section that's a cubic surface with a rational point and then @DanielLoughran's observation applies. | |
| Jul 31, 2017 at 20:21 | comment | added | Will Sawin | @NoamD.Elkies A cubic threefold might provide an interesting case to consider. The class of a curve in the Chow group should define a point in the intermediate Jacobian. If the intermediate Jacobian has few rational points then there should not be so many curves, but I don't know if you can get it down to none. | |
| Jul 31, 2017 at 8:14 | comment | added | Daniel Loughran | I don't think any Fano variety, or more generally any variety which is rationally connected over $\bar{\mathbb{Q}}$ will work. I expect that such varieties contain many rational curves over $\mathbb{Q}$ as soon they contain a rational point $x \in X(\mathbb{Q})$. But admittedly I don't know how to prove this; probably it is very difficult in general. | |
| Jul 31, 2017 at 7:44 | comment | added | Noam D. Elkies | Hm, true. Might a degree $d$ hypersurface in ${\bf P}^d$ still work for some $d>3$? | |
| Jul 31, 2017 at 7:22 | comment | added | Daniel Loughran | @Noam D.Elkies: Any smooth cubic surface over $\mathbb{Q}$ with a rational point contains many rational curves: just take the tangent hyperplane to a general rational point. | |
| Jul 30, 2017 at 22:40 | comment | added | Noam D. Elkies | "Does not contain any rational curves" over $\bf Q$ or over $\overline{\bf Q}$? If you mean the former, there may already be many examples among cubic surfaces, though it's probably intractable to prove that there is even one such surface with a rational point and no rational curves. | |
| Jul 30, 2017 at 21:51 | answer | added | Daniel Loughran | timeline score: 6 | |
| Jul 30, 2017 at 19:36 | comment | added | Will Sawin | One case to consider for this problem is that of Calabi-Yau varieties without rational curves. These are at least conjecturally the furthest you can get from general type without containing a rational curve. Their existence is discussed here mathoverflow.net/questions/69716/… The examples mentioned are quotients of abelian varieties, so maybe there is hope of verifying this for these, although the statement for the quotient does not follow from the known statement for the variety. | |
| Jul 30, 2017 at 13:34 | comment | added | Will Sawin | I take it your height is not logarithmic, i.e. the height of the point $n$ on $\mathbb P^1$ is $n$ and not $\log n$? | |
| Jul 30, 2017 at 11:50 | history | asked | jacob | CC BY-SA 3.0 |