Timeline for Is there a complex surface into which every Riemann surface embeds?
Current License: CC BY-SA 3.0
16 events
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| Jan 15, 2016 at 1:18 | comment | added | S. Li | Or maybe we don't even know whether $M_g$ is of general type in char p? Sorry I know very little about moduli of curves... | |
| Jan 14, 2016 at 17:14 | comment | added | Olivier Benoist | @S.Li : indeed, it does not seem that the argument goes through (at the place where the Kodaira dimension argument is used) and I do not know if there is a way to make it work. | |
| Jan 13, 2016 at 17:14 | comment | added | S. Li | Your answer is really nice! But it seems to me that it would not be generalized in characteristic p? | |
| Nov 3, 2015 at 17:16 | history | edited | Olivier Benoist | CC BY-SA 3.0 |
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| Oct 30, 2015 at 19:17 | comment | added | Michael Albanese | @OlivierBenoist: Would you mind including this argument in the answer when you get a chance? I'd really like to know the solution in the general case. | |
| Oct 30, 2015 at 10:20 | comment | added | Olivier Benoist | @MichaelAlbanese : You're right : I have written the argument for algebraic surfaces. On the other hand, it seems to me that on non-algebraic surfaces, there will not be high-dimensional linear systems of smooth curves (otherwise, you would get, after blow-up, a generically finite morphism to projective space). This should show that the space of smooth curves in $S$ is a countable union of spaces of bouded dimension. This cannot fill $M_g$ if $g$ is big. I do not have time now to write it down: does it sound sensible ? | |
| Oct 30, 2015 at 1:18 | comment | added | Michael Albanese | @OlivierBenoist: Thanks for your answer. Just to check, does your argument hold for any complex surface, or only algebraic surfaces? | |
| Oct 29, 2015 at 12:37 | vote | accept | Michael Albanese | ||
| Oct 28, 2015 at 0:26 | comment | added | Olivier Benoist | @WalterNeff Oh: I was indeed assuming that $S$ is smooth, sorry ! If $S$ is singular, you are right that one can consider a desingularization $S'$ of $S$. The hypothesis on $S$ implies that every smooth projective curve may be embedded in $S'$, with the exception of at most a finite number of isomorphism classes of curves.The argument applies as well in this situation. | |
| Oct 27, 2015 at 23:59 | comment | added | Walter Neff | What is the natural morphism $H_i\to \operatorname{Pic}(S)$? Have you desingularized $S$? | |
| Oct 27, 2015 at 23:42 | history | edited | Olivier Benoist | CC BY-SA 3.0 |
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| Oct 27, 2015 at 20:36 | comment | added | Olivier Benoist | @July Haha: I had not noticed that :-) | |
| Oct 27, 2015 at 20:32 | history | edited | Olivier Benoist | CC BY-SA 3.0 |
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| Oct 27, 2015 at 20:15 | history | edited | Olivier Benoist | CC BY-SA 3.0 |
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| Oct 27, 2015 at 20:12 | comment | added | M.G. | +1 for using Albanese against himself :-P | |
| Oct 27, 2015 at 20:08 | history | answered | Olivier Benoist | CC BY-SA 3.0 |