Timeline for $p$-divisibility of Picard groups
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Aug 22, 2023 at 8:36 | vote | accept | Boaz Moerman | ||
| Aug 19, 2023 at 4:20 | answer | added | naf | timeline score: 3 | |
| Aug 18, 2023 at 5:23 | comment | added | naf | If $\widetilde{C}$ is smooth, it suffices to know that $\widetilde{C} \setminus C$ supports a divisor of degree prime to $p$, which always holds by your assumption on $k$. This implies that $\mathrm{Jac}(\widetilde{C})$ is $p$-divisible (see the answer by R. van Dobben de Bruyn below). It follows that the Picard group is also $p$-divisible, since if $L$ is a line bundle whose class in the Jacobian is divisible by $p$, then the obstruction to finding a $p$-th root in the Picard group is an element of $\mathrm{Br}(k)[p]$, but $\mathrm{Br}(k)[p]=0$ (by the same cohomological argument). | |
| Aug 18, 2023 at 0:14 | answer | added | R. van Dobben de Bruyn | timeline score: 1 | |
| Aug 17, 2023 at 19:55 | history | edited | Boaz Moerman | CC BY-SA 4.0 |
Added brackets in exponent for clarity (otherwise l^p might look like p if zoomed out)
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| Aug 17, 2023 at 17:39 | comment | added | Piotr Achinger | I thought you meant $k$ is algebraically closed in the last sentence, sorry. In fact I misunderstood your question and thought $k$ has characteristic $p$, and anyway your assumption on $k$ disallows the removal of a point of degree $p$ from $\tilde C$. | |
| Aug 17, 2023 at 15:33 | history | edited | Boaz Moerman | CC BY-SA 4.0 |
Added examples of such fields $k$.
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| Aug 17, 2023 at 15:04 | comment | added | Boaz Moerman | Could you explain why $A(k)$ is $p$-divisible? I know that it is true for separably closed fields, but it is not clear to me why it should hold for the fields I described. | |
| Aug 17, 2023 at 14:18 | history | edited | Boaz Moerman | CC BY-SA 4.0 |
added 22 characters in body
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| Aug 17, 2023 at 13:34 | comment | added | Piotr Achinger | Yes, it’s true that $A(k)$ is always divisible. What worries me is the situation when you’re removing a point of degree $p$ from the compact curve. | |
| Aug 17, 2023 at 10:40 | history | asked | Boaz Moerman | CC BY-SA 4.0 |