# Good reduction or not, that is the question

Ok, admittedly this title is not the best one that I can come up with. I just read (small part of) this paper. The authors constructed a family of degenerating Calabi–Yau 3-folds (over a punctured disc), with trivial monodromy action and have a (potentially) semistable reduction which is not a smooth filling.

If I understand it correctly, the authors believe that this family has no (potentially) good reduction. Namely, however we pull this family back along maps from another punctured disc to the current one given by “$z^n$“, it won’t admit a smooth filling.

Now the (philosophical) question is, how can we tell if a smooth variety over the fraction field of a DVR does NOT have a smooth integral model over that DVR?

I know this (kind of fake) example: if we take a non-archimedean Hopf surface, then its étale cohomology groups are unramified (if you use l-adic coefficients) or Crystalline (if you use p-adic coefficients). But it doesn’t have a smooth reduction. Why? Because the weight is wrong. Ok, so this example is lame.

So I think there is a task I can give myself: to find a smooth projective variety over a non-archimedean field such that (1) its étale cohomology groups are “good reduction” meaning unramified/Crystalline of the correct weight and, (2) it does not admit any potential good reduction. I assume it would be easy to find a potential example, like the example discussed in the above link, but it perhaps is quite difficult to prove that such a thing is an actual example.

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Update on Dec. 8th: Bhargav reminded me that a curve of compact type is one such example. Meanwhile Daniel said that one can detect (this example) by looking at the monodromy action on the (algebraic) fundamental group of this curve and decided it doesn’t have (potential) good reduction. On the other hand, Dingxin pointed the existence of this paper of Robert Friedman. This paper exhibits an example of degenerating family of quintic surfaces over a punctured disk, which doesn’t have a (potential) good reduction, with the property that a finite power of the monodromy action is homotopic to identity!

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In memory of Michael Zhao.