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Does a smooth proper variety having semi-stable reduction as well as potentially good reduction have good reduction ?

Note that over a $p$-adic field, this is true for the Galois representations in the $p$-adic étale cohomology of $X$.

(With a bit more details: fix a field $K$ complete for a discrete valuation, with ring of integers $\mathcal{O}_K$, and a smooth proper $K$-scheme $X$. We say that $X$ has good reduction if it is the generic fibre of a smooth proper $\mathcal{O}_K$-scheme $\mathcal{X}$. We say that $X$ has semi-stable reduction if it is the generic fibre of a flat proper $\mathcal{O}_K$-scheme $\mathcal{X}$ such that étale-locally on $\mathcal{X}$ there exists a smooth morphism $\mathcal{X}\to \text{Spec}(\mathcal{O}_K[T_1,\dots,T_r]/(T_1\dots T_r-\pi))$ where $\pi$ is a prime element of $\mathcal{O}_K$. We say that $X$ has potentially one of the two properties above if it has it after a finite extension $K'/K$.)

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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 ... – Emerton Jun 17 '13 at 3:01
... 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, – Emerton Jun 17 '13 at 3:04
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! – Matthieu Romagny Jun 17 '13 at 7:19
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. – Jason Starr Jun 17 '13 at 16:41
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 ... – Emerton Jun 19 '13 at 0:53
up vote 9 down vote accepted

No. Take $K=\mathbb Q_3$. Consider the projective genus $0$ curve $x^2+y^2+3z^2$. This has bad reduction, since it has no rational points. It is semistable, since after adjoining $i$ it has exactly that form. It has potentially good reduction, since all genus $0$ curves do.

So you at least need to say it has good reduction etale locally.

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Maybe I'm being really dumb, but why does no rational points imply bad reduction? I know the standard argument for genus $1$, but it involves Lang's theorem. – Matt Jun 16 '13 at 2:31
If it had good reduction you could lift a point over the residue field using Hensel's lemma, and all genus $0$ curves over finite fields have points. Alternately, by a well-known fact there are two genus $0$ curves over a local field, one with good reduction and rational points and one with bad reduction and without rational points. – Will Sawin Jun 16 '13 at 2:58
Oops. Of course. In the one-dimensional case the set of torsors is identified with the Brauer group of the field under the standard connecting homomorphism, which in this case is $C_1$. Sorry. The other part of the argument is the same. I was trying to remember why genus $0$ curves over finite fields always have points. – Matt Jun 16 '13 at 4:26
Thank you Will. Now I'm a bit confused with the Galois representation counterpart, e.g. like in Matthew Emerton's example answering this MO question. Namely, in that example Matt Emerton claims that $P$ and $E$ have the same étale cohomology but I would have thought that the Galois action on the cohomologies match only after restriction to a finite extension $K'/K$. Am I wrong? – Matthieu Romagny Jun 16 '13 at 15:31
What are P and E? – Will Sawin Jun 16 '13 at 17:04

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