Timeline for Equivalence between $\bar{\mathcal{M}}_{g,n}$ and ${\mathcal{M}}_{g,n}^{logbas}$
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Jun 13, 2023 at 20:35 | comment | added | Jason Starr | The locally closed (disjoint) strata in your stratification are themselves disjoint unions of open and closed subsets indexed by the "marked dual graph", i.e., the dual graph with a nonnegative integer attached to each vertex indexing the geometric genus of the normalization of the corresponding irreducible component, and with "flags" = "half-edges" indexing marked points. This combinatorics is in the article of Behrend and Manin on "motivic" Gromov-Witten invariants. | |
| Jun 12, 2023 at 13:30 | comment | added | Matthias | Is it possible to construct another stratification, where the strata correspond to the combinatorical types of the parametrized curves? | |
| Jun 12, 2023 at 8:43 | comment | added | Piotr Achinger | In your example, I think the map $S\to \mathcal{M}$ will land in the stratum if and only if the images of $\pi_i$ in $\Gamma(S, \mathcal{O}_S)$ are all nilpotent. So you should take formal completion, ${\rm Spf}(P\to k[[P]])$, whose underlying reduced scheme is ${\rm Spec}(k)$. | |
| Jun 12, 2023 at 7:10 | history | asked | Matthias | CC BY-SA 4.0 |