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Let $A$ be a DVR with an algebraically closed residue field $k$. Consider a morphism $f: X \to Y$ of arithmetic surfaces (regular integer projective and flat schemes over $A$ of dimension 2), such that $f$ induces a finite morphism on generic fibers. Assume that $X$ and $Y$ are semistable, the generic fibers of $X$ and $Y$ are smooth, and $Y$ is relatively minimal, i.e. does not contain any exceptional divisor.

Q: Is there any chance that $f$ factors through a birational map $X \to X'$, where $X'$ is a relatively minimal surface?

As far I understand, the answer lies in the consideration of an irreducible component $C$ of the closed fiber $X_k$ of $X$, which is isomorphic to $\mathbb P^1_k$ and intersects the other components of $X_k$ in exactly one point. With respect to the morphism $X \to X'$ the component $C$ is then contracted. If $f$ contracts $C$, so $f$ can possibly factor through $X'$. But in general (it seems to me) it could happen, that $f(C)$ is irreducible and intersects the other components of $Y_k$ in at least two points.

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up vote 3 down vote accepted

I'll generalize the set-up a bit. The answer to your question should be negative.

Let $S$ be a Dedekind scheme with function field $K$.

Let $\pi_K:X_K \to Y_K$ be a finite morphism of smooth projective geometrically connected curves over $K$. Let $Y$ be a (regular projective) model for $Y_K$ over $S$. Let $\pi:(Y,X_K)\to Y$ be the normalization of $Y$ in the function field of $X_K$. You are asking whether the minimal resolution of singularities of $N(Y,X_K)$ is relatively minimal. Let me first try to convince you this is what you are asking for.

Explanation. Your morphism $X\to Y$ factors as $X\to N(Y,X_K) \to Y$ with $X\to N(Y,X_K)$ birational projective surjective. Asking whether $X\to Y$ factors through a morphism $X\to X_m$, where $X_m$ is the minimal regular model of $X_K$ over $O_K$, is equivalent to asking whether the minimal resolution of singularities of $X^\prime\to X$ coincides with $X_m$.

You can give examples of when this will not happen. But, it could happen if you modify $Y$.

Let me just finish by mentioning three relevant references.

Besides Qing Liu's book, you could take a look at the article

http://www.math.u-bordeaux1.fr/~qliu/articles/Compositio.pdf

Moreover, Liu-Lorenzini's article seems also relevant

http://www.math.u-bordeaux1.fr/~qliu/articles/modcove.pdf

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This helps a lot, thank you. –  finite Feb 8 '13 at 10:39
    
Hi Ariyan, welcome to MSE ! Is there a reason to write Lorenzini-Liu instead of the alphabetical order ? –  user18119 Feb 8 '13 at 22:26
    
Thank you! No reason at all to write Lorenzini-Liu. I make such silly mistakes all the time! –  Ari Feb 8 '13 at 22:43
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On a more mathematical note: the answer is not complete in my opinion. I haven't had the time to write down a simple "counterexample". One way one can do it I think is to take a genus $g\geq 1$ curve $Y_K$ over $K$ with a smooth projective model $Y$ over $O_K$ and a free action of a non-trivial finite group $G$. This gives a morphism $Y\to Y/G$ with $Y/G$ semi-stable. I think one can write down an example where $Y/G$ is not regular. If $X\to Y/G$ denotes the minimal resolution of singularities, it's clear that the base-change of $Y\to Y/G$ via $X\to Y/G$ can not factorize through $Y\to X$. –  Ari Feb 8 '13 at 22:52
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@Ariyan: in the second paper you quoted, Prop. 6.6 gives an example of $X_K\to Y_K$ étale cyclic, with the special of $X$ having a singular point $x_0$ fixed by the Galois group. This forces $X/G$ to be singular, so $X$ can't dominate $Y$. You can't find an example with $X$ smooth by op. cit. 4.10. –  user18119 Feb 9 '13 at 13:08

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