Timeline for An abelian variety has good reduction $\iff$ the Neron model is proper
Current License: CC BY-SA 4.0
5 events
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| Oct 6, 2018 at 19:29 | comment | added | gdb | @k.j. Yes, it does. Moreover, there is exactly one such model (up to a canonical isomorphism). | |
| Oct 6, 2018 at 19:23 | comment | added | k.j. | Thank you very much! So, does every smooth proper model of an abelian variety have a group scheme structure? | |
| Oct 6, 2018 at 19:13 | comment | added | gdb | $\mathfrak A$ is smooth over a dvr => its generic fibre is dense in $\mathfrak A$ (also $\mathfrak A$ is irreducible since you can check irreducibility on an open dense subset). I think that this is sufficient for your purposes. But actually you can show more, any smooth proper model of an abelian variety is isomorphic to its Neron model. This is explained in Proposition 1.4/2 in the book "Neron models". | |
| Oct 6, 2018 at 19:05 | comment | added | msteve | The proof of of this direction requires some work: you must show that the abelian $R$-scheme $\mathfrak{X}$ satisfies the Neron mapping property. This is carefully explained in Proposition 2.2.4 of these notes. | |
| Oct 6, 2018 at 18:47 | history | asked | k.j. | CC BY-SA 4.0 |