Timeline for Betti numbers of moduli spaces of smooth Riemann surfaces
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Jul 24 at 0:40 | history | edited | Harry Richman | CC BY-SA 4.0 |
add hyperlinks to articles
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| Aug 24, 2022 at 12:17 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question was bumped anyway)
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| Feb 9, 2012 at 11:33 | comment | added | algori | domenico -- thanks for the reference. | |
| Feb 9, 2012 at 6:36 | comment | added | domenico fiorenza | The paper by Riccardo Murri has now appeared: arxiv.org/abs/1202.1820 | |
| Sep 20, 2010 at 15:09 | comment | added | algori | Torsten -- indeed, if we fix 3 points, we get the complement of a collection of hyperplanes, which we only have to quotient by $S_3$, but this comes at a price: we lose track of the action of $S_n$ on the moduli space. (More precisely, it's not clear to me how to recover it.) To the best of my knowledge, the integral cohomology of the non-ordered configuration space of the sphere is not known in general. It is torsion in degrees $\neq 0,3$ and it is conjectured by F. Napolitano to stabilize in each degree $\neq 2$ as the number of the points goes to $\infty$. | |
| Sep 20, 2010 at 13:29 | comment | added | Torsten Ekedahl | My point was that the case of the configuration space in the current case ($\mathbb P¹$ minus three points) is much simpler than the general case and unless I am mistaken it is torsion free. In any case it is a hyperplane complement which has been studied for a long time. However, I am taking about ordered points. The unordered case is trickier if you want integral cohomology, I think that Arnold did the unordered onfiguration space för $\mathbb R²$ and there definitely is torsion. | |
| Sep 19, 2010 at 21:30 | comment | added | algori | Torsten -- rationally the answer should be classical (I've basically sketched a proof above). The integral case seems tricker since some shortcuts that are there rationally get closed. E.g. it is not clear to me how to compute the cohomology ring of the configuration space: in Totaro's approach Hodge theory is used in an essential way. Taking the quotient may also be problematic. However even integrally this does not look like a hopelessly complicated problem and I was wondering if someone has done it. | |
| Sep 19, 2010 at 19:13 | comment | added | Torsten Ekedahl | The $g=0$ is a hyperplane complement and its cohomology is very classical even integrally (my recollection was that that was the starting point for Keel's computation but I may misremember). | |
| Sep 19, 2010 at 18:11 | history | edited | algori | CC BY-SA 2.5 |
typo in formula corrected
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| Sep 19, 2010 at 18:10 | comment | added | algori | domenico -- thanks, will correct this. | |
| Sep 19, 2010 at 8:29 | comment | added | domenico fiorenza | algori - thanks, that's a great formula for the Betti numbers of $\mathcal{M}_{0,n}$! just for sake of completeness, it seems to me the Poincare' polynomial of $\mathcal{M}_{0,n}$ should end with the factor $(1+(n−2)t)$ rather than with $(1+(n−1)t)$. tom - thanks a lot for the reference! | |
| Sep 18, 2010 at 4:39 | comment | added | Tom Church | At the cost of self-promotion, let me mention a relevant theorem: for an appropriate labeling of the irreducible representations of $S_n$, the decomposition of $H^i(\mathcal{M}_{0,n})$ as an $S_n$--representation is independent of $n$ once $n\geq 4i$. In particular, you can reduce this recipe to a finite computation which implies the decomposition for all $n$. This is proved for $F(\mathbb{C},n)$ as Theorem 4.1 in Church-Farb, "Representation theory and homological stability", front.math.ucdavis.edu/1008.1368. The closely related case of $F(\mathbb{P}^1,n)$ will appear soon. | |
| Sep 18, 2010 at 4:03 | history | edited | algori | CC BY-SA 2.5 |
typo
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| Sep 18, 2010 at 3:39 | history | edited | algori | CC BY-SA 2.5 |
a typo and a small correction
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| Sep 18, 2010 at 3:32 | history | edited | algori | CC BY-SA 2.5 |
improved phrasing
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| Sep 18, 2010 at 2:30 | comment | added | algori | domenico -- I think you are right, the result you are interested in does not seem to be there. My fault. To try and make up for it I've added a description of the cohomology of the open part. | |
| Sep 18, 2010 at 2:29 | history | edited | algori | CC BY-SA 2.5 |
added a description in genus 0; deleted 7 characters in body
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| Sep 17, 2010 at 10:11 | comment | added | domenico fiorenza | apparently (i.e., if I'm not misunderstanding) Keel deals with the moduli space `$\overline{\mathcal{M}}_{0,n}$ of genus zero $n$-marked stable curves, rather than with the open stratum $\mathcal{M}_{0,n}$ of smooth curves | |
| Sep 16, 2010 at 21:02 | comment | added | domenico fiorenza | yes, it is a colleague of mine (Riccardo Murri) who has written a program implementing it (or, better, the version considered by Kontsevich), and it is precisely this program we are now going to test against known results (clearly, the only fault can be in the implementation of the algorithm). thanks a lot for the references. a paper containing the implementation of the algorithm and the computed results will be available on the arxiv in a month or so. I will announce it here as it appears | |
| Sep 16, 2010 at 18:30 | history | answered | algori | CC BY-SA 2.5 |