Timeline for A log structure on the moduli space of curves
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Mar 26, 2018 at 10:11 | comment | added | Piotr Achinger | I see! What I said means that the sheaf $\bar M_X=\mathbf{N}^n$ is constant, but the extension $0\to \mathcal{O}_X^*\to \mathcal{M}_X^{\rm gp}\to \mathbf{N}^n\to 0$ does not have to be split. The log structure (a'la Deligne-Faltings) in this case corresponds to a map $X\to B\mathbf{G}_m^n$ i.e. the choice of $n$ line bundles on $X$ (use the boundary map $\mathbf{N}^r\to {\rm Pic}(X)$), which here are the pull-backs of the tangent bundle along the $n$ sections of the universal family (or their duals, depending on the conventions). | |
| Mar 25, 2018 at 22:52 | comment | added | Dmitry Vaintrob | @Piotr, I don't think this is true: see my comment to Scott's answer. In fact, the way I want to think about this structure is as a log space underlying the real moduli space of smooth algebraic curves with marked points and real directions at all marked points. Whether or not this Kato-Nakayama realization happens to be locally constant over the base, it certainly does not admit a natural section. | |
| Mar 25, 2018 at 6:22 | vote | accept | Dmitry Vaintrob | ||
| Mar 25, 2018 at 22:44 | |||||
| Mar 25, 2018 at 6:21 | vote | accept | Dmitry Vaintrob | ||
| Mar 25, 2018 at 6:22 | |||||
| Mar 24, 2018 at 22:06 | answer | added | S. Carnahan♦ | timeline score: 5 | |
| Mar 24, 2018 at 19:30 | comment | added | Piotr Achinger | @DmitryVaintrob sorry, after reading Mattia's answer I realized I didn't read your question too carefully. What makes you think that the local system is not constant? Your marked points are ordered, and it seems that the $i$-th copy of the natural numbers corresponds to the $i$-th marked point. So it seems that your log structure is constant, i.e., the pull-back of the log structure on the log point ${\rm Spec}(\mathbf{N}^n\to \mathbf{C})$. | |
| Mar 24, 2018 at 19:22 | answer | added | Mattia Talpo | timeline score: 4 | |
| Mar 24, 2018 at 8:07 | comment | added | Piotr Achinger | F. Kato: ams.org/mathscinet-getitem?mr=1754621 | |
| Mar 24, 2018 at 5:21 | comment | added | David Benjamin Lim | Have you tried looking at this paper of Olsson? | |
| Mar 23, 2018 at 23:52 | comment | added | Dmitry Vaintrob | Right, the log structure will be given by a local system of semigroups locally isomorphic to some power of the natural numbers. I don't think the local system will in general be trivial. | |
| Mar 23, 2018 at 23:44 | comment | added | André Henriques | Could you please explain what it means to pull back a log structure along a morphism of varieties? I seems to me that every time you have an open subscheme U⊂X with D:=X\U a normal crossing divisor, you can take the standard log structure on X associated to D and pull it back to the deepest stratum of D (whatever that means). What do you get in the case when X is affine n-space and D is a bunch of coordinate hyperplanes? | |
| Mar 23, 2018 at 23:02 | history | asked | Dmitry Vaintrob | CC BY-SA 3.0 |