Timeline for The Sylvester-Gallai theorem over $p$-adic fields
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 13, 2017 at 22:33 | comment | added | Gro-Tsen | The dual configuration appears in this paper by Dolgachev (§5) as well as in the book Geradenkonfigurationen und Algebraische Flächen by Barthel &al (Vieweg 1987) (§2.3) under the name "Ceva configuration". So I guess a logical name for this one would be "Menelaus configuration" since the Ceva and Menelaus theorems are dual. | |
| Dec 31, 2016 at 20:30 | vote | accept | François Brunault | ||
| Dec 31, 2016 at 19:19 | comment | added | David E Speyer | @GilKalai Well, it only applies over fields of characteristic zero, of course: If $\mathbb{F}_p$ is in your field than the points of $\mathbb{P}^n(\mathbb{F}_p)$ form an SG configuration in $\mathbb{P}^n$. But yes, proving it over the complex numbers implies all characteristic zero fields. | |
| Dec 31, 2016 at 19:16 | comment | added | Gil Kalai | Thanks, David. I was aware of Kelly's result but not that it applies over every field... | |
| Dec 31, 2016 at 18:06 | comment | added | Gro-Tsen | @PiotrAchinger Indeed, and when I read OP's question, my immediate reaction was "well, we can certainly take the $n$-torsion of an elliptic curve and it will give a S-G configuration over some field", but this does not work because sometimes you get a line of tangency meeting the curve in only $2$ points; so then when I read David Speyer's answer, my first reaction was, this can't work because the same problem will happen; but it does work. | |
| Dec 31, 2016 at 17:59 | comment | added | Piotr Achinger | This configuration looks a lot like the "torsion points" of the "elliptic curve" given by the cubic equation $xyz=0$. The smooth locus is the disjoint union of three copies of $\mathbb{G}_m$, we take their $n$-torsion points. In that sense the example resembles the configuration of the $3$-torsion of an elliptic curve mentioned by the OP. | |
| Dec 31, 2016 at 17:42 | comment | added | François Brunault | (The case of $\mathbf{Q}_2$ is also still open if I'm not mistaken.) | |
| Dec 31, 2016 at 17:34 | comment | added | David E Speyer | I assume you meant two dimensional, not three dimensional. | |
| Dec 31, 2016 at 17:33 | comment | added | David E Speyer | @GilKalai Yes. "A resolution of the sylvester-gallai problem of J.-P. serre", Kelly, Discrete & Computational Geometry, 1986. See also arxiv.org/abs/math/0403023 for a simpler proof and for results over the quaternions. | |
| Dec 31, 2016 at 17:27 | comment | added | Gil Kalai | Is it always true that a S-G configuration is 3-dimensional? | |
| Dec 31, 2016 at 17:22 | comment | added | Gro-Tsen | Does this configuration have a name? | |
| Dec 31, 2016 at 17:18 | comment | added | François Brunault | Very nice, I didn't expect this! This settles almost completely my question. And it also show SG fails over $\mathbf{Q}_9$, leaving only the case of $\mathbf{Q}_3$... | |
| Dec 31, 2016 at 16:30 | history | answered | David E Speyer | CC BY-SA 3.0 |