Timeline for Degree-2 étale covers of curves in characteristic 2 vs torsion points on the Jacobian
Current License: CC BY-SA 4.0
12 events
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| Mar 3, 2022 at 0:27 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading
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| Mar 2, 2022 at 14:51 | comment | added | Will Sawin | @Grobber Thanks, added the reference. | |
| Mar 2, 2022 at 14:51 | history | edited | Will Sawin | CC BY-SA 4.0 |
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| Mar 2, 2022 at 14:26 | comment | added | Grobber | @WillSawin Artin--Milne in "Duality in the Flat Cohomology for Curves" proves duality for curves over an algebraically closed field. Unfortunately, I have not looked closely at the paper to be able to give you precise statements. | |
| Feb 22, 2022 at 21:13 | comment | added | Will Sawin | @TCiur $\tilde{C}^n$ admits an action by $(\mathbb Z/2)^n \rtimes S_n$, with $(\mathbb Z/2)^n$ acting by deck transformations of the covering on each factor, and the quotient by this action is $\operatorname{Sym}^n C$, which admits a birational map to the Jacobian if $n=g$. If we instead quotient by $(\mathbb Z/2)^{n-1} \rtimes S_n$ (taking the subspace of $n$-tuples that sum to $0$), we get a double covering of $\operatorname{Sym}^n C$. This double covering descends to the Jacobian if $n=g$ (such descent always happens for birational morphisms between smooth proper varieties). | |
| Feb 22, 2022 at 20:59 | vote | accept | TCiur | ||
| Feb 22, 2022 at 20:59 | comment | added | TCiur | Thank you for your answer! Is there an explicit way to construct $A \to J$ from a given covering $\tilde{C} \to C$? Milne proves these two objects are equivalent but only gives an explicit construction going the other way. | |
| Feb 19, 2022 at 13:17 | comment | added | Will Sawin | @DanielLoughran Yes, it should be fppf. Maybe there isn't one? I added all the details to the concrete proof instead. | |
| Feb 19, 2022 at 13:17 | history | edited | Will Sawin | CC BY-SA 4.0 |
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| Feb 19, 2022 at 12:28 | comment | added | R. van Dobben de Bruyn | @DanielLoughran it is an étale sheaf (even an fppf sheaf); it just doesn't have any nontrivial sections on the small étale site of any reduced scheme... | |
| Feb 19, 2022 at 8:37 | comment | added | Daniel Loughran | I assume the above cohomology is fppf and not etale as $\mu_2$ is not an etale sheaf? In which case what is a good reference for Poincare duality in this setting? | |
| Feb 18, 2022 at 22:07 | history | answered | Will Sawin | CC BY-SA 4.0 |