would like to ask you the following question:
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.
