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I wonder if a semistalbe K3 surface over a $p$-adic field has a minimal semistable model. I guess yes but I do not find any reference.

Also, if we have a semistable K3 surface with a log structure, there exist a minimal log semistable model?

Thanks.

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I don't think the minimal model program has anything to do with your question. –  Piotr Achinger Feb 11 '13 at 7:53
    
No? I really do not know a lot about it, but look this paper: arxiv.org/pdf/1010.2577v2.pdf You are probably right and I am looking then on a wrong sobject. Do you thing so? –  Yoyontzin Feb 11 '13 at 17:56
    
I think you're right. I never heard of MMP in such context (I thought MMP dealt with birational classification of varieties, not with models over DVRs, but I guess they are related somehow). Sorry for confusion! –  Piotr Achinger Feb 11 '13 at 19:07
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Dear Piotr, There is a close relationship between the theory of minimal models and the theory of semistable reduction. To see this, you could think about the relationship between the birational classification of surfaces and the theory of good models of curves over DVRs. Regards, –  Emerton Feb 12 '13 at 6:42
    
Any reference? Thanks! –  Yoyontzin Feb 26 '13 at 6:34
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1 Answer 1

up vote 3 down vote accepted

The answer is yes when p>3. Look at Kawamata's paper

Semistable minimal models of threefolds in positive or mixed characteristic. J. Algebraic Geom. 3 (1994), no. 3, 463–491.

and a correction in

Index 1 covers of log terminal surface singularities. J. Algebraic Geom. 8 (1999), no. 3, 519–527.

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Thanks so much for the reference! I will look at it! –  Yoyontzin May 23 '13 at 20:16
    
What about for Canonical models? –  Yoyontzin May 23 '13 at 20:16
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What do you mean by canonical models? A family of K3 has relative trivial canonical class. –  CYXU May 24 '13 at 0:24
    
Your are right. –  Yoyontzin Aug 5 '13 at 1:06
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