Timeline for Does a semistable curve descend to a regular base?
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| S Feb 5, 2015 at 5:15 | history | bounty ended | CommunityBot | ||
| S Feb 5, 2015 at 5:15 | history | notice removed | CommunityBot | ||
| Jan 30, 2015 at 14:47 | answer | added | Count Dracula | timeline score: 2 | |
| Jan 29, 2015 at 20:56 | comment | added | user74230 | Sorry, bootstrapping from the locus of smooth curves in $\overline{M}_{g,n}$ by using double and triple fiber powers hits a fatal snag over the issue you raised, so I have deleted my answer via that idea. Without loss of generality $g\ge 2$ (as $g \le 1$ is easy), so over an etale cover where we have an $n$-pointed stable structure we apply Knudsen's "contraction" of enumerated marked points one at a time, getting a stable genus-$g$ curve over $S'$ with the same Pic$^0$; alas, for non-smooth fibers it depends on the ordering (and position) of marked points, so descent is problematic. Hmm! | |
| Jan 29, 2015 at 1:43 | comment | added | Question Mark | @user74230: Could you post this as an answer and explain in a little bit more detail? This would also be useful for whoever would look at this question later (to see that the issue has been resolved). | |
| Jan 28, 2015 at 4:23 | comment | added | Question Mark | @user74230: I think you're looking at the proof of 9.4/4, not of 9.4/1 (which is given on p. 262). In 9.4/4, I think it's better to use GIT (5.1 and 5.3) instead of Deligne-Mumford. | |
| S Jan 28, 2015 at 3:37 | history | bounty started | Question Mark | ||
| S Jan 28, 2015 at 3:37 | history | notice added | Question Mark | Draw attention | |
| Jan 28, 2015 at 3:35 | history | edited | Question Mark | CC BY-SA 3.0 |
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| Jan 28, 2015 at 3:29 | comment | added | Question Mark | @user74230: OK, let's move on, but I will put this on top of the site to see if other people have any ideas for a possible fix. In the smooth genus 1 case, I think though there is a canonical relatively ample line bundle: $\mathrm{Pic}^0_{X/S}$ is then an elliptic curve over $S$, so $\mathscr{O}(0)$ coming from the zero section is canonical and ample. | |
| Jan 28, 2015 at 2:45 | comment | added | Question Mark | @user74230: For the overall fix that we are envisioning, isn't it a problem that the etale-local marking is not canonical? Our overall goal is to build a canonical ample $\mathscr{L}$ after etale localization on $S$, but then all canonicity is lost once we choose markings, no? To be able to descend $\mathscr{L}$ back to over $S$ it seems crucial that the etale-local reduction to the universal case would not depend on noncanonical choices of maps to some moduli stack. | |
| Jan 26, 2015 at 5:35 | comment | added | Question Mark | @user74230: Thanks for your remarks. It seems that for the density needed by BLR, Thm. 2.7 in Knudsen's "The projectivity ... II" paper may suffice, since it describes the substack of singular curves as a normal crossings divisor in $M_{g, n}$. Regarding the readability and correctness of Knudsen, I'll take your word that it all works out in the end, at least until I finish reading BLR. | |
| Jan 26, 2015 at 4:58 | comment | added | user74230 | By the way, Knudsen's paper is not an easy read, or rather the technical details of his study of contraction and clutching morphisms require quite a bit of care. | |
| Jan 26, 2015 at 4:54 | comment | added | user74230 | Knudsen shows that $\overline{M}_{g,n}$ is an iterated curve fibration over the stack built by D-M when $g > 1$, over the D-R stack $\overline{M}_{1,1}$ for $g=1$, and over $\overline{M}_{0,3} = {\rm{Spec}}(\mathbf{Z})$ for $g=0$. From this one reduces to the universal curve $X$ over these "base stacks". The last is $\mathbf{P}^1$. For the others, we want $X$ over the formally smooth deformation ring at a geometric point to have smooth generic fiber. That in turn is clear from how non-smoothness in the universal deformation is encoded in parameters describing the deformation ring. | |
| Jan 26, 2015 at 4:35 | comment | added | Question Mark | Thank you! I will familiarize myself with the work of Knudsen to be able to fix the proof. Do you know, by the way, how do BLR get the locus $S_0 \subset S$ where $X\rightarrow S$ is smooth to be dense, and what inputs from Knudsen would I need to get this density claim to hold? | |
| Jan 26, 2015 at 3:43 | comment | added | user74230 | For your real purposes (not explained in the above question, the aim really being construction of a certain canonical relatively ample line bundle) it is harmless to work etale-locally on the base. But any semistable curve becomes a stable "marked" curve etale-locally on the base, so one can use the $\mathbf{Z}$-smooth Deligne-Mumford stack of stable $n$-pointed genus-$g$ curves for suitable $n$ with $2g-2+n > 0$ (studied in a paper of Knudsen, building on the work of Deligne and Mumford). So this renders the likely erroneous reference to Deligne-Mumford in BLR moot. | |
| Jan 25, 2015 at 23:45 | history | asked | Question Mark | CC BY-SA 3.0 |