Timeline for Abelian variety
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 25, 2012 at 14:07 | comment | added | meh | As Donu noted, if the surface is smooth the answer is no. If arbitrary singularities are allowed, any surface (including abelian ones) has a birational model of the form $P(x,y,z,w)= 0$. | |
| Oct 24, 2012 at 13:02 | comment | added | user9072 | In case you are not aware, you can edit the question to expand it (you do not have to, and in fact should not ask a new one); while being logged in use the link just below the tag. | |
| Oct 23, 2012 at 22:16 | comment | added | Daniel Loughran | @Jafar: Yes but it depends on the surface. If you have a specific surface that you are interested in, it might be possible to say something. I concur with quid that you should clarify the question. | |
| Oct 23, 2012 at 20:56 | comment | added | Jafar | Dear Dr. Loughan , proving that there is no rational curve in surface (variety) is usually a hard problem. Isn't it? | |
| Oct 23, 2012 at 16:38 | comment | added | user9072 | Could this question, please, be expanded somehow, including a more concise specification of P and an explantion what 'how can we know' should mean in detail. True, one can somehow infer/guess what might be meant, but still I think this will be a much better question with a bit of effort. Finally, a more informative title seems also desirable. | |
| Oct 23, 2012 at 15:59 | comment | added | rita | A necessary condition is that the Albanese map of a smooth model of the surface be birational. | |
| Oct 23, 2012 at 15:56 | comment | added | Daniel Loughran | If your surface contains a rational curve then the answer is again never, as abelian varieties never contain any rational curves. | |
| Oct 23, 2012 at 15:55 | comment | added | Daniel Loughran | @Jafar: Do you have a specific surface in mind? | |
| Oct 23, 2012 at 15:27 | comment | added | Jafar | assume that this is a singular surface in P^3. | |
| Oct 23, 2012 at 15:14 | comment | added | Donu Arapura | If this is a smooth surface in $\mathbb{P}^3$, then the answer is never: by weak Lefschetz, it is simply connected, but a subvariety of an abelian variety is not. | |
| Oct 23, 2012 at 14:53 | history | asked | Jafar | CC BY-SA 3.0 |