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Suppose that $K$ is a $p$-adic field, that is a field of characteristic $0$ whose ring of integers is a complete discrete valuation ring $\mathcal O_K$ and with residue field $k$ (algebraic closed) of characteristic $p$.

Let $X\rightarrow \mathop{Spec}(\mathcal{O}_K)$ be a projective threefold with generic fibre $X_K$ a (smooth) $K3$-surface. If $p>5$ we can assume that $X$ is a semi-stable minimal model of $X_K$.

Is it true that the special fibre $X_0 = X \otimes{k}$ is a combinatorial $K3$-surface?

I know that it is true for the complex numbers, that is, if $X\rightarrow \Delta$ is a projective semi-stable degeneration of $K3$-surfaces over the open complex disk, then the central fibre $X_0$ is combinatorial (in the sense of Firedman-Kulikov). (1. Persson, U.; Pinkham, H. Degeneration of surfaces with trivial canonical bundle. Ann. of Math. (2) 1981, 113, 45–66.) and (Kulikov).

Moreover Friedman studied the converse situation over the complex numbers and proved that any combinatorial $K3$-surface over the complex numbers is the central fibre of a family of $K3$-surfaces. 1. Persson, U.; Pinkham, H. Degeneration of surfaces with trivial canonical bundle. Ann. of Math. (2) 1981, 113, 45–66. 2. Friedman, R. Global smoothings of varieties with normal crossings. The Annals of Mathematics 1983, 118, 75–114.

Latter Nakkajima prove it for combinatorial $K3$ surfaces that admit a log-strtucture family of $K3$-surface. 1. Nakkajima, Y. Liftings of simple normal crossing log $K3$ and log Enriques surfaces in mixed characteristics. J. Algebraic Geom. 2000, 9, 355–393.

My question is then to know if it is know that for a semi-stable $K3$ surface over a field $K$ of mixed characteristic with minimal semi-stable model, the generic fibre is combinatorial.

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  • $\begingroup$ Some of your references seem to have ended up in the middle of sentences. Also, "desecrate" should be "discrete". $\endgroup$
    – user5117
    Commented Nov 18, 2013 at 17:10
  • $\begingroup$ ups… that is right! $\endgroup$
    – Rogelio Yoyontzin
    Commented Nov 18, 2013 at 17:12
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    $\begingroup$ you should have a look at Maulik's paper "Supersingular K3 surfaces for large primes" (arxiv.org/abs/1203.2889), and in particular to Section 4 $\endgroup$
    – Christian Liedtke
    Commented Nov 19, 2013 at 10:47
  • $\begingroup$ Yes, I have seen that paper. He works on characteristic $p$ and I am on mixed characteristic. Looking at Maulik's arguments on section 4, he claims that having a semi-stable minimal model, then indeed the special fibre is combinatorial, however there is something that I am missing: He said that this follows by Nakkajima, but for Nakkajima we need to have that the special fibre to be a simple normal crossing log K3-surface. Why is it true in Maulik's paper? $\endgroup$
    – Rogelio Yoyontzin
    Commented Nov 20, 2013 at 3:25
  • $\begingroup$ Is is true that if we have a semi-stable minimal model $X\rightarrow \mathop{spec}(R)$ with $R$ a complete discreat valuation ring, with smooth generic fibre a $K3$-surface, then the special fibre $X_0$ (with the induced log structure) is a simple normal crossing Log $K3$-surface? Even if we assume that the special fibre is reduced and that its components are geometrically irreducible why is it $d$-semistable and why $H^1(X_0,\mathcal O_{X_0}) = 0$ and $\Omega_{X_0}^2(log) = \mathcal{O}_{X_0}$? $\endgroup$
    – Rogelio Yoyontzin
    Commented Nov 20, 2013 at 3:38

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