Timeline for Synthetic projective lines
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| Dec 3, 2015 at 20:20 | comment | added | Mike Shulman | I finally got a copy of the Buekenhout-Cohen book, which finally contains an actual definition of what he means by "perspectivity", which is not what I thought: he means the restriction to the line in question of a central collineation of the ambient projective space (an automorphism that fixes some hyperplane pointwise and all lines through some point setwise). There's still an exercise I have to do to convince myself that this works, but at least it seems more plausible. | |
| Nov 20, 2015 at 19:23 | comment | added | Mike Shulman | Excellent, that makes sense. Thanks again! | |
| Nov 20, 2015 at 14:46 | comment | added | Tim Penttila | The paper has five citations:Hirschfeld's book, the Buekenhout-Cohen book, J.Tits, Twin buildings and groups of Kac-Moody type 1992,PM Johnson Semiquadratic sets...1999 H Van Maldeghem Moufang lines 2007. None of them really take the idea much further. The last one is essntially studying the translation line in the Luneburg plane from this perspective. | |
| Nov 20, 2015 at 14:32 | comment | added | Tim Penttila | (continued) This is for the restricted definition where there's only the hypothesis when the two points are equal. Buekenhout has replaced Desargues (little Desargues in the restricted sense) by the symmetry it induces on a line, and a general line in a non-Desarguesian plane won't have that symmetry. | |
| Nov 20, 2015 at 14:22 | comment | added | Tim Penttila | @MikeShulman A line in a projective plane is a projective line in this sense if the plane is a translation plane with respect to this line. Most lines in non-Desarguesian projective planes are not projective lines in this sense. | |
| Nov 19, 2015 at 17:14 | comment | added | Felipe Voloch | @MikeShulman Re: obscure journal. "one used to glace breathlessly through the table of contents of each new number of the Hamb, Abhand. with the hope, seldom disappointed, of finding Artin's name there". A. Weil "Review of the Collected Papers of Emil Artin". The review itself was published in some obscure journal but you can find it in Weil's collected works. | |
| Nov 18, 2015 at 19:39 | comment | added | Mike Shulman | My question is how one can prove that a line in a nonDesarguesian projective space is a "projective line" in this sense? | |
| Nov 18, 2015 at 18:31 | comment | added | Max Horn |
@MikeShulman So, it is an open problem which projective lines are integrable or realizable (i.e. come from projective planes). And as I suggested earlier, I believe this is a very difficult problem, given that any really "useful" solution probably would also settle the question of whether there are projective planes of non-prime-power order or not. With "useful solution" I mean that one would want an intrinsic characterization of integrable projective lines which is powerful enough to decide for a concrete projective line whether it is integrable or not.
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| Nov 18, 2015 at 18:26 | comment | added | Max Horn | @MikeShulman My comment was misleading: The definition of projective line in the book is general. And lines in any thick projective plane "are" projective lines in this sense. But in fact any set of cardinality at least three can be made into a projective line, by assigning the trivial group to each point. Since there are no projective planes of order 6, nor 10, this proves that there are projective lines which do not come from projective planes. In that direction, the only relevant result in the book seems to be a classiciation of projective lines coming from Desarguesian projective planes. | |
| Nov 18, 2015 at 16:06 | comment | added | Mike Shulman | @MaxHorn Interesting! (ILL actually got me the paper faster than the book, since it can be delivered electronically.) The paper seems to be clear that he intends it to work in the non-Desarguesian case, e.g. he says "Let S be... a projective plane... and let L be a line of S ... L becomes a projective line in the sense of the introduction ... If S is Desarguesian then it is also clear that L is a Desarguesian projective line." But maybe by the time he wrote the book he realized that was wrong? | |
| Nov 18, 2015 at 7:37 | comment | added | Max Horn | The book by Buekenhout and Cohen is much more easy to obtain. I just had a look at the copy in my shelf; section 6.2 quite cleary states that it is about classifying projective lines in thick projective spaces, which are Desarguesian. | |
| Nov 18, 2015 at 5:41 | comment | added | Mike Shulman | Hmm, actually, now I am confused again. If that's what he means by "perspectivity" then I don't see how to prove his axiom (1) for a line in a projective plane without Desargues's theorem, or how to prove his axiom (4) without Pappus's theorem (the problem in both cases being closure of the putative groups under composition). | |
| Nov 18, 2015 at 4:20 | history | edited | Mike Shulman | CC BY-SA 3.0 |
fix spelling of Buekenhout
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| Nov 18, 2015 at 4:10 | comment | added | Mike Shulman | Found it. It took me a little while to figure out that by "perspectivity" he means what most projective geometry books I've read would call "a projectivity from a line to itself that is the composite of two perspectivities", but after realizing that, his definition makes sense. Do you know whether this has been carried further by anyone? Without an MSN entry I can't search for citations. | |
| Nov 18, 2015 at 3:52 | vote | accept | Mike Shulman | ||
| Nov 17, 2015 at 18:23 | comment | added | Mike Shulman | Wow, thanks! This looks like a really obscure journal; MSN doesn't even know that it had an issue 43 (it lists issues 1-15, 23, 31, 39, 42, and 52-77). Off to ILL! | |
| Nov 15, 2015 at 23:38 | review | Late answers | |||
| Nov 15, 2015 at 23:43 | |||||
| Nov 15, 2015 at 23:23 | review | First posts | |||
| Nov 15, 2015 at 23:29 | |||||
| Nov 15, 2015 at 23:21 | history | answered | Tim Penttila | CC BY-SA 3.0 |