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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]

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    $\begingroup$ 1) The curve in $\mathbb P^2$ given by $xy = p^n z^2$ shows that without the $X$ regular condition, isomorphic special fibers do not imply $S$-isomorphisms. 2) A nice such $Y$ exists when $X$ and $X'$ are birational - take the closure of a birational map - which you can use the classification of genus $0$ curves over fields to investigate. $\endgroup$
    – Will Sawin
    Commented Jan 18, 2021 at 2:18
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    $\begingroup$ Assuming the first, third and fourth conditions (for both $X$ and $X'$, it does seem likely that if $X_k$ is isomorphic to $X_k'$ then $X$ is isomorphic to $X'$. You can see this if $k$ is algebraically closed by iteratively blowing down $X$ and $X'$ at at the curves corresponding to degree one vertices of the dual graph of the special fibre to get to the case where the special fibre is irreducible. Over a more general field, one can blow down Galois orbits of such curves as long as they are disjoint, so one can reduce to some simple minimal cases which could be analyzed further. $\endgroup$
    – naf
    Commented Jan 18, 2021 at 8:36
  • $\begingroup$ @WillSawin Ah yes, you're right. Thank you very much. I had forgotten about that example. I think I may need to modify the question to include regularity. By birational, you mean as schemes, so, as long as the generic fibers are isomorphic? Also, do you have a favorite reference for the classification of genus 0 curves? $\endgroup$
    – PrimeRibeyeDeal
    Commented Jan 18, 2021 at 17:57
  • $\begingroup$ @naf Thank you for the response. That does seem like a promising strategy, but I'm afraid I don't see exactly how you're using the conditions 1, 3, 4. As for the minimal cases, I'm not sure how to proceed even in those cases. What if we know special fiber is irreducible? Or even suppose special and generic fibers are all $\cong \mathbb{P}^1$? How to show isomorphism then? $\endgroup$
    – PrimeRibeyeDeal
    Commented Jan 18, 2021 at 18:02
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    $\begingroup$ Regularity implies that the vertices of the dual graph are always (-1)-curves so they can be blown down to get something that is still regular. I also assumed implicitly that $X_k$ is reduced. If both fibres are $\mathbb{P}^1$ then the class of a section in the Picard group of $X$ is a relatively very ample bundle which gives a map to $\mathbb{P}^1_S$ which is an isomorphism. $\endgroup$
    – naf
    Commented Jan 19, 2021 at 0:56

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