Timeline for How many points determine a line?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jan 25, 2012 at 8:34 | comment | added | Gunnar Þór Magnússon | I think Edison said he needed a hundred tries before coming up with one good idea. The same seems to be true for learning anything, and this process now seems to be public. | |
| Jan 25, 2012 at 8:33 | vote | accept | Gunnar Þór Magnússon | ||
| Jan 24, 2012 at 18:52 | answer | added | Sándor Kovács | timeline score: 9 | |
| Jan 24, 2012 at 18:49 | comment | added | user5117 | Anyway, all that is just to say that the question in your last paragraph has a rather trivial answer, for either interpretation of the term "projective line". On the other hand, the problem of enumerating the rational curves passing through certain sets of points (even in projective space) is a very famous and actively-researched one. I guess the keywords to search for are Gromov-Witten invariants and Kontsevich moduli spaces. | |
| Jan 24, 2012 at 18:41 | comment | added | user5117 | The issue now is that your original motivation is no longer true: in projective space there can be lots of rational curves through a given set of points. For instance in P^2, there is a 3-dimensional family of conics through any pair of points, and any smooth member of that family is isomorphic to P^1. | |
| Jan 24, 2012 at 18:32 | history | edited | Gunnar Þór Magnússon | CC BY-SA 3.0 |
added 17 characters in body
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| Jan 24, 2012 at 18:31 | comment | added | Gunnar Þór Magnússon | @Artie: That's my fault. I had a feeling that just "rational curve" left the question too open for an answer, and may have tried to compensate with less than perfect results given my knowledge of the subject. Feel free to answer the question with just "rational curve" and ignore the "degree 1" business. | |
| Jan 24, 2012 at 18:23 | comment | added | user5117 | The questions seems difficult to make sense of, because of the phrase "rational curves of degree 1". The degree of a subvariety is defined with respect to a given projective embedding of your surface (or complex manifold, or whatever). But given such an embedding, the fact that there is a unique line in projective space through two given points implies the same thing for the surface. | |
| Jan 24, 2012 at 18:19 | comment | added | Gerhard Paseman | If by line you mean geodesic, there are likely to be very bumpy manifolds where several geodesics share the same $\kappa$-many points for appropriate cardinals $\kappa$. Even with fewer bumps, considering hyperbolic planes and similar geometries may suggest examples to you. Gerhard "Ask Me About System Design" Paseman, 2012.01.24 | |
| Jan 24, 2012 at 18:11 | history | asked | Gunnar Þór Magnússon | CC BY-SA 3.0 |