Timeline for Curves with negative self intersection in the product of two curves
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Dec 2, 2009 at 12:13 | vote | accept | Dmitri Panov | ||
| Dec 2, 2009 at 12:13 | |||||
| Oct 24, 2009 at 19:23 | history | edited | naf | CC BY-SA 2.5 |
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| Oct 24, 2009 at 14:02 | comment | added | Dmitri Panov | Thanks again! I belive this is correct, but I see it will require me some time to understand, anyway, I had this question in mind for a couple of years, so I don't mind to spend time to work out the details. | |
| Oct 24, 2009 at 13:59 | vote | accept | Dmitri Panov | ||
| Dec 2, 2009 at 12:12 | |||||
| Oct 24, 2009 at 12:45 | comment | added | naf | The only thing which is not elementary is the fact that Shimura curves are compact; the Hecke cycles can be defined very explcitly using some simple group theory and the properties I mentioned are immediate from the definitions. It might be easier to first think about the case of classical modular curves (associated to congruence subgroups of SL(2,Z)) and the Hecke cycles on self products of these; the case of Shimura curves works in essentially the same way but because they are compact one can deduce the negativity of the self-intersection without any calculations at all. | |
| Oct 24, 2009 at 11:19 | comment | added | Dmitri Panov | Thanks for the refference, I'll try to study it. I do belive that this can work but I just wonder, can you estimate how many pages would take an honest proof of this fact? At least what will be the steps? Also I am curious if these curves will be geodesic with respect to the product metric? Thanks again. | |
| Oct 24, 2009 at 11:14 | history | edited | naf | CC BY-SA 2.5 |
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| Oct 24, 2009 at 10:53 | comment | added | naf | @Dmitri. Yes, Shimura curves are compact quotients of H^2 but the main point is that these correspond to very specific discrete subgroups of PSL(2,R). By a search on google I found the book "Quaternion Orders, Quadratic forms and Shimura curves" by Alsina and Bayer; it looks like a good reference. I am not sure about what happens when the curve gets larger; I will add a comment later if I have anything sensible to say about it. | |
| Oct 24, 2009 at 9:43 | comment | added | Ilya Nikokoshev | I wonder if we know that these graphs are reduced and irreducible? | |
| Oct 24, 2009 at 9:27 | comment | added | Dmitri Panov | Thanks a lot for this comment! This seems very promissing, indeed all curves will etale projections will have negative self intersection. Unfortunatelly I don't know anything about Shimura curves. Are they compact quotinets of H^2? Where can I learn a bit on this? I also wonder what will happen if we let the curve get larger and larger --- will it converge to some lamination? | |
| Oct 24, 2009 at 6:16 | history | answered | naf | CC BY-SA 2.5 |