Timeline for curve over higher dimensional basis with 0-dimensional locus of bad reduction
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 8, 2015 at 21:51 | comment | added | Ariyan Javanpeykar | My apologies. I now see the point. | |
| Nov 8, 2015 at 21:46 | comment | added | Laurent Moret-Bailly | @AriyanJavanpeykar: But we don't know (yet) that $X\in\mathscr{M}_g(S)$. | |
| Nov 8, 2015 at 19:13 | comment | added | Ariyan Javanpeykar | ...More precicely, for all schemes $S$ and for all $X$ and $Y$ over $S$ (in $\mathcal M_g(S)$), the morphism $\mathrm{Isom}_S(X,Y)\to S$ is proper. (If $g\geq 2$, this morphism is finite. But this is not the case when $g=1$.) The given isomorphism $X_U \cong X^\prime_U$ induces a section of $Isom_S(X,Y)$ over $U$. Since $S$ is regular (noetherian) this "generic section" extends to a section over $S$. See Gabber-Liu-Lorenzini Prop 6.2 math.u-bordeaux1.fr/~qliu/articles/GLL2-Duke.pdf . | |
| Nov 8, 2015 at 19:13 | comment | added | Ariyan Javanpeykar | @TimoKeller In the last part of Moret-Bailly's answer one can also argue as follows. Let $S$ be an integral noetherian regular scheme. Assume $g\geq 1$ and assume that $X_U\cong X_U^\prime$, where $X$ and $X^\prime$ are curves over $S$ in $\mathcal M_g(S)$ and $U\subset S$ is a dense open. The fact that this isomorphism extends to an isomorphism of $X$ and $X^\prime$ over $S$ follows from the separatedness of the stack $\mathcal M_g$ of smooth proper curves of genus $g$... | |
| Nov 7, 2015 at 17:35 | vote | accept | CommunityBot | moved from User.Id=19475 by developer User.Id=69903 | |
| Nov 7, 2015 at 17:35 | comment | added | Laurent Moret-Bailly | @TimoKeller: yes, that's right. | |
| Nov 7, 2015 at 17:09 | comment | added | user19475 | Thank you very much! So you prove that the locus of bad reduction has codimension $\leq 1$? | |
| Nov 7, 2015 at 14:11 | history | answered | Laurent Moret-Bailly | CC BY-SA 3.0 |