Timeline for Semistable curves of genus $g\geq 2$ form an Artin algebraic stack in the etale topology?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 6, 2022 at 15:39 | comment | added | Johan | OK, I think you can make a semistable genus 2 example by gluing 2 elliptic tails to the genus $0$ Example 2.3 in Fulghesu's "The stack of rational curves". It is a bit delicate. Good luck! Also Jason meant "etale local" and not "zariski locally on the target" above. | |
| Sep 5, 2022 at 23:10 | comment | added | S.D. | @Jason Starr It would be great if you explain the elliptic curve example? | |
| Sep 5, 2022 at 23:09 | comment | added | S.D. | @Jason Starr Thank you for the references. Though I want to learn the proof (if it is correct) of it being an Artin stack in the Etale topology. By emphasising on "Etale topology" I mean to say that the families of semi-stable curves of genus greater than equal to $2$ glue in Etale topology under suitable gluing conditions. I understand the proof in the case where we also allow algebraic spaces as Johan mentioned. | |
| Sep 5, 2022 at 1:03 | comment | added | Jason Starr | You are correct; there are elliptic curve examples. I was thinking of the result that every relative curve over a scheme is (Zariski locally on the target) also projective. So the projection morphism over the Artin stack is, etale locally on the target, a schematic morphism. As you say, the morphism is not schematic. | |
| Sep 5, 2022 at 0:26 | comment | added | Johan | Well... that is the thing that is wrong for the prestable curves: there exists a scheme $T$ and a prestable curve $X$ over $T$ such that $X$ is not a scheme. So the morphism from the universal curve to the moduli stack is not representable by schemes (for the prestable case). | |
| Sep 5, 2022 at 0:11 | comment | added | Jason Starr | @Johan I think the sentence is ambiguous. Perhaps the OP wants the morphism from the universal curve to the Artin stack to be schematic (of course that morphism is schematic). | |
| Sep 4, 2022 at 22:51 | comment | added | Johan | OK, the last parenthetical sentence of the question suggests that @S.D. wants to only work with schemes. Then in the prestable case it doesn't work. I don't know about the semistable case. | |
| Sep 4, 2022 at 21:10 | comment | added | Jason Starr | Of course my article with Johan de Jong and Xuhua He proves that the stack of all curves is an Artin stack. (This is a folk result; we just wrote down one proof.) | |
| Sep 4, 2022 at 21:08 | comment | added | Jason Starr | I am not sure what you mean by "Artin algebraic stack in the Etale topology." Anyway, I recommend that you look at "Gromov-Witten invariants in algebraic geometry" by Kai Behrend: personal.math.ubc.ca/~behrend/gwag.pdf (published in Invent. math.). Although that Artin stack is not the main point of the article, Behrend does work with that stack in the article. | |
| Sep 4, 2022 at 11:03 | history | asked | S.D. | CC BY-SA 4.0 |