Timeline for Potentially good, semi-stable reduction => good reduction ?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 19, 2013 at 9:26 | comment | added | Matthieu Romagny | Dear Matthew, sure, this is how I thought about these things. Thanks. | |
| Jun 19, 2013 at 2:34 | comment | added | Emerton | $d+1$-dimensional varieties, and so to think about the former carefully you have to understand the latter. Best wishes, Matthew | |
| Jun 19, 2013 at 0:53 | comment | added | Emerton | Dear Jason, Thanks for this example. Dear Matthieu, You should probably take my comments as just a broad outline of how to think about these kinds of question. The total space of a degenerating family of surfaces (as in Jason's example) is three-dimensional, and so I am guessing that his example is related to phenomena in the minimal model program for three-folds (something that I don't know much about, but that is a deeply researched area that others know a huge amount about!). The theory of reduction of $d$-dimensional varieties is closely related to the theory of minimal models for ... | |
| Jun 17, 2013 at 16:41 | comment | added | Jason Starr | In response to Emerton: there are also examples of this behavior with degenerating families of K3 surfaces, e.g., a family over $\text{Spec} \mathbb{C}[[t]]$ whose total space has an $A_1$-singularity. After an etale base change, there are two small modifications (related to each other by a flop), so that the family has potentially good reduction. For the original family, you can blow up the singular point, resolving the singularity at the expense of adding an additional irreducible component to the central fiber. | |
| Jun 17, 2013 at 7:22 | vote | accept | Matthieu Romagny | ||
| Jun 17, 2013 at 7:19 | comment | added | Matthieu Romagny | Dear Matthew, OK. In fact I wondered whether I'd add a comment to my question, to mention that the situation is somehow under control for curves and abelian varieties essentially due to the uniqueness of smooth models. I'm glad to hear that although not much is known in general, one expects the phenomenon to be quite typical. Thanks! | |
| Jun 17, 2013 at 3:04 | comment | added | Emerton | ... surfaces over a field, and minimal models are unique except in the case of rational and ruled surfaces. So e.g. for curves of genus $\geq 1$, I think that the answer to your question will be yes. In higher dimension, I'm not sure what is known; the theory of semistable models is less well-developed, because one doesn't have resolution of singularities and related tools in mixed characteristic. Anyway, it is not coincidence that Will Savin's counterexample has genus $0$. Regards, | |
| Jun 17, 2013 at 3:01 | comment | added | Emerton | Dear Matthieu, I think the answer should typically be yes, despite Will Savin's counterexample. One way to think about it is as follows: take your semistable model over $K$, base-change to $K'$ (where it is no longer semistable: $T_1 \cdots T_r = \pi$ turns into $T_1 \cdots T = (\pi')^e$), blow-up to make it semi-stable again. Now your assumption is that there is also a good reduction model over $K$'. So when minimal models are unique, you will (I think) get a contradiction. (You have to think a bit about minimality of models.) Now models for curves over DVRs are like models for ... | |
| Jun 16, 2013 at 1:03 | answer | added | Will Sawin | timeline score: 9 | |
| Jun 15, 2013 at 20:45 | history | asked | Matthieu Romagny | CC BY-SA 3.0 |