Timeline for Section of a Ruled surfaces
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 13, 2010 at 20:10 | vote | accept | Rex | ||
| Dec 13, 2010 at 19:09 | answer | added | Donu Arapura | timeline score: 6 | |
| Dec 13, 2010 at 18:20 | comment | added | Mohan | The point is Tsen's theorem implies one needs to know only that the fibers over geometric (closed) points are projective lines. This is because, as Donu says, then the generic fiber is a conic (smooth and genus zero, so canonical bundle is of degree -2, its negative giving an embedding in the plane). Now Tsen's theorem implies such conics have a rational point since the base field is the function field in one variable over an algebraically closed field and hence $C^1$. | |
| Dec 13, 2010 at 12:47 | comment | added | Donu Arapura | Keerthi, thanks for adding the clarification. | |
| Dec 13, 2010 at 3:56 | comment | added | Keerthi Madapusi | Since Donu's comment ended so tragically, let me just add a tiny bit of clarification: you do require the generic fiber as well to be a projective line (at least geometrically, which, as Donu shows, is equivalent to its being $\mathbb{P}^1$ on the nose). The problem with Hartshorne of course is that he insists on working over algebraically closed fields, which makes this sort of thing seem a little suspect. | |
| Dec 13, 2010 at 0:54 | comment | added | Donu Arapura | I don't have Hartshorne with me at the moment, so I can't say with any assurance what he intended. There are two possible definitions: (1) a ruled surface means that fibres over geometric points are $\mathbb{P}^1$'s, or (2) a ruled surface means that fibres over all points are $\mathbb{P}^1$'s. The good news is that these are in fact by Tsen's theorem. Assuming (2), you can realize the generic fibre as a plane conic in $\mathbb{P}^2$, and conclude that it has a rational point over $k(C)$ by the aforementioned theorem. Conics with points are isomorphic to $\mathbb{P}^1$. I think I'm outof spac | |
| Dec 13, 2010 at 0:19 | history | asked | Rex | CC BY-SA 2.5 |