Timeline for $p$-divisibility of Picard groups
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 22, 2023 at 10:58 | comment | added | naf | One does not need geometric integrality, or even geometric irreducibility (as I had written). The sequence is very general, see Prop 4 on p. 204 of the book "Neron models" by Bosch, Raynaud and Lutkebohmert. If the curve is not geometrically integral then the (geometric) components of the Picard scheme would not be parametrized by $\mathbb{Z}$, so a little more care needs to be taken to get the last equation. Also, note that one can replace the base field $k$ by the field of constants of $\widetilde{C}$ to simplify this problem and also for the Brauer group issue. | |
| Aug 22, 2023 at 9:42 | comment | added | Boaz Moerman | $\newcommand{\mr}{\mathrm}$ It seems the only place where geometric integrality is used (which is an assumption I prefer to avoid, though integrality is fine), is potentially in the exact sequence $$0\rightarrow \mr{Pic}^0(C)\rightarrow J(k)\rightarrow \mr{Br}(k)$$ (which I assumed you were referring to with the reference to the Brauer group). Is this the right sequence, and do you know a good reference for it? It is mentioned by Milne in jmilne.org/math/xnotes/JVs.pdf, but he does not provide a reference. | |
| Aug 22, 2023 at 8:41 | comment | added | Boaz Moerman | Thank you for the update! Your answer is exactly what I was looking for and is a nice proof of the result. | |
| Aug 22, 2023 at 8:36 | vote | accept | Boaz Moerman | ||
| Aug 22, 2023 at 2:26 | history | edited | naf | CC BY-SA 4.0 |
added 515 characters in body
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| Aug 21, 2023 at 15:44 | comment | added | naf | @BoazMoerman: You are right, probably the proof works in general. I will edit the answer soon. | |
| Aug 21, 2023 at 6:37 | comment | added | Boaz Moerman | Thank you for your answer! For the counterexample, you say that a form of the additive group is killed by $p$, but I do not see why this is the case, as I assume $p$ is different from the characteristic. Am I misunderstanding something here? | |
| Aug 19, 2023 at 4:43 | history | made wiki | Post Made Community Wiki by naf | ||
| Aug 19, 2023 at 4:20 | history | answered | naf | CC BY-SA 4.0 |