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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$\begingroup$ I don't think the minimal model program has anything to do with your question. $\endgroup$– Piotr AchingerCommented Feb 11, 2013 at 7:53
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$\begingroup$ 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? $\endgroup$– Rogelio YoyontzinCommented Feb 11, 2013 at 17:56
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$\begingroup$ 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! $\endgroup$– Piotr AchingerCommented Feb 11, 2013 at 19:07
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2$\begingroup$ 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, $\endgroup$– EmertonCommented Feb 12, 2013 at 6:42
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$\begingroup$ Any reference? Thanks! $\endgroup$– Rogelio YoyontzinCommented Feb 26, 2013 at 6:34
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1 Answer
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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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$\begingroup$ What about for Canonical models? $\endgroup$ Commented May 23, 2013 at 20:16
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1$\begingroup$ What do you mean by canonical models? A family of K3 has relative trivial canonical class. $\endgroup$– CYXUCommented May 24, 2013 at 0:24
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