Ok, I got it, that was a bit subtle.

may that my terminology "natural isomorphism" confused you. By natural isomorphism I meant a natural transformation which is an isomorphism.

Thanks a lot Mat, this is a very nice example. With this example in mind it also gives the motivation why one should base change to a larger field in order to acquire semi-stable reduction.

Hi Mat, could you elaborate a little bit on the sentence "it is not true in general that having semistable etale cohomology implies having semistable reduction". I'd like to understand your comment. And what is the meaning of your comment on the geometrical side?

Thanks Francois. So is there some kind of relationship between Hironaka's desingularisation over a field $K=Frac(R)$ of characteristic 0 (the existence of a smooth compactification $X'$ of $X$ which is isomorphic to $X$ away from a normal crossing divisor) and the conjecture which predicts that every scheme of $R$ is potentially semi-stable.