Closely related is [this question][1].

Suppose $R$ is a DVR with fraction field $K$ and residue field $k$ (say finite), and $S = \mathrm{Spec}(R)$.

I am interested in proper, flat schemes $X \to S$ where the generic fiber $X_K$ is a curve of *genus zero*. In many ways these are simple objects, but I've been unable to find good answers or references, one reason being that positive genus is often assumed.

My main questions are: given such $X, X'$

1) Does isomorphic special fibers $X_k \cong X_k'$ imply $S$-isomorphism $X \cong X'$?
2) Is there always some $Y$ with maps to each ($X\leftarrow Y \to X'$), or is the category connected?

I'm interested in the answers in general, but especially when one or more of these conditions hold:

 - $X_K \cong \mathbb{P}^1_K$ (there is a rational point)
 - $R = \mathbb{Z}_p$
 - $X_k$ is a normal crossings divisor
 - $X$ is regular
 - $X \to S$ is smooth

Any references would be very much appreciated as well. My first though was Qing Liu's book, but I haven't found my answers there.

Thank you.

  [1]: https://mathoverflow.net/questions/309949/are-relative-curves-x-to-s-determined-by-their-fibers