Let $C$ be a smooth quasiprojective curve over a field $k$ of characteristic $p>0$, and let $X\rightarrow C$ be a flat morphism, smooth over $C\setminus 0$ for some $k$-point $0$, with generic fiber a $K3$ surface.
I have two questions on the reduction of $X$ at $0$. -Is it true that after taking a finite cover of $X$, $X$ has semi-stable reduction at $0$ ? -Assuming that the generic fiber of $X$ is ordinary, to what extent does the Kulikov classification of degenerations of $K3$ in characteristic zero extend to the case of positive characteristic ?
By the semistable reduction theorem for abelian varieties and resolution of singularities for surfaces, I would expect the first question to have a positive answer, using the Kuga-Satake family associated to $X$. However, I can't seem to make it work...