Timeline for Synthetic projective lines
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 20, 2015 at 19:22 | comment | added | Mike Shulman | Thanks! I can certainly hope for a more geometric description though. (-: And Buekenhout's, when it works, is geometric. | |
| Nov 20, 2015 at 15:07 | comment | added | Tim Penttila | Yes, it is the Buekenhout definition, when the two points are equal (and no hypothesis with the points distinct). The Moufang set people tend not to cite people who worked on the topic before Tits, probably because they didn't use the terminology of Tits. (So, for instance, John Faulkner's early work is also not often cited.) | |
| Nov 19, 2015 at 12:51 | comment | added | Tom De Medts | I added some more details now. I haven't included a general "synthetic" construction. In general, one can start from one root group $U$ and a permutation of the non-trivial elements of $U$ (satisfying certain conditions) to construct any possible Moufang set. I'm not sure whether that's what you're after; it's quite far away from a "geometric" construction but probably the best one can hope for for 1-dimensional geometric objects... | |
| Nov 19, 2015 at 12:48 | history | edited | Tom De Medts | CC BY-SA 3.0 |
Added more details, in particular about the example PSL_2(k)
|
| Nov 18, 2015 at 19:43 | comment | added | Mike Shulman | I was just saying I think the answer above would be enriched by a concise description of the construction, so a reader doesn't have to make their way through 20 pages of notes to piece together the definition. And octonion division algebras are one particular nonDesarguesian situation, but is there a synthetic construction that works in general? | |
| Nov 18, 2015 at 18:17 | comment | added | Max Horn | This is described in section 5 of Tom's course notes, in particular section 5.2 for skew fields and octonion division algebras (which are non-associative). | |
| Nov 18, 2015 at 16:12 | comment | added | Mike Shulman | I don't suppose you could add a short description of what the groups $U_x$ are in the case of $\mathbb{P}^1(k)$, and ideally a synthetic description if $X$ is a line in a synthetic projective plane? This looks a lot like the definition in Buekenhout's paper, where I think the groups are supposed to be "central collineations", except that Buekenhout slices up the collineations according to their axis as well as center, so I'm wondering if the structures are the same. Does this work for projective lines over division rings, or in non-Desarguesian planes? | |
| Nov 18, 2015 at 11:41 | comment | added | Max Horn | Of course Moufang sets are not that helpful if one also wants to capture projective lines contained in projective planes with small or even trivial automorphism groups. But catching those sounds like a very difficult problem to me, given that it is not known if there are projective planes of non-prime-power order... So if one could characterize projective lines intrinsically via axioms, deciding whether the finite ones can have non-prime-power order ought to be a very difficult problem. | |
| Nov 18, 2015 at 11:37 | comment | added | Max Horn | Well, I also recommend those course notes, and I am not a co-author, so I think I am allowed to do that ;-) | |
| Nov 18, 2015 at 11:24 | history | edited | Tom De Medts | CC BY-SA 3.0 |
fixed typo
|
| Nov 18, 2015 at 8:40 | history | edited | Tom De Medts | CC BY-SA 3.0 |
added 12 characters in body
|
| Nov 18, 2015 at 8:34 | history | answered | Tom De Medts | CC BY-SA 3.0 |