Closely related is this question.

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

I am interested in regular, 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
• $X \to S$ 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.

[Edit: added regularity assumption]

Closely related is this question.

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

I am interested in regular, 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)
• $R = \mathbb{Z}_p \text{ or } \mathbb{F}_p[[t]]$
• $X_k$ is a normal crossings divisor
• $X \to S$ 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.

[Edit: added regularity assumption]

Closely related is this question.

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.

Closely related is this question.

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

I am interested in regular, 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
• $X \to S$ 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.

[Edit: added regularity assumption]