Timeline for Comparison of two monodromies
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 18, 2021 at 9:30 | comment | added | Francesco Polizzi | Maybe one can use in some way (at least when $\gamma$ has order $2$) the fact that the elementary abelian group $V=G/\langle \gamma \rangle$ acts freely on the quotient varieties $\Sigma_g/\langle \gamma \rangle$ and $X /\langle \gamma \rangle$ | |
| Apr 18, 2021 at 7:38 | comment | added | Francesco Polizzi | Ah, ok. In my situation, since I am assuming actual ramification, then $\gamma$ is non-trivial (and moreover I have that $G/\langle \gamma \rangle$ is elementary abelian). | |
| Apr 17, 2021 at 22:47 | comment | added | Will Sawin | @FrancescoPolizzi I mean, if you let $\gamma$ be trivial, the abelian case is OK. | |
| Apr 17, 2021 at 22:32 | comment | added | Francesco Polizzi | Abelian groups cannot occur, since I want ramification on the diagonal, hence the image of the element $\gamma$, that is a non-trivial commutator in $\pi_1(\Sigma_g \times \Sigma_g - \Delta)$, must give a non-trivial commutator in $G$. For extra-special $p$-groups, this means that the image of $\gamma$ must lie in the center $Z(G) \simeq \mathbb{Z}_p$. Maybe, for extra-special $2$-groups one can say something more precise about the monodromies, since in that case the image of $\gamma$ is the unique generator of the center $Z(G) \simeq \mathbb{Z}_2$. | |
| Apr 17, 2021 at 22:26 | comment | added | Will Sawin | @FrancescoPolizzi I currently can't see how to do any case except for abelian groups, which perhaps suggests that extra-special groups are not far off, but I don't see how to extend it yet. | |
| Apr 17, 2021 at 22:11 | comment | added | Francesco Polizzi | Thank you for the nice answer. Can you see any condition on the finite group $G$ implying equality? For instance, in my case $G$ is extra-special. | |
| Apr 17, 2021 at 22:06 | history | answered | Will Sawin | CC BY-SA 4.0 |