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Let $X_K$ be a smooth proper curve over a field $K$, and let $S$ be a Dedekind scheme with function field $K$.

Let $X$ be the proper regular minimal model of $X_K$ over $S$.

Let $Y_K$ be another smooth proper curve over $K$, equipped with a finite map $Y_K\rightarrow X_K$.

Let $Y$ be the normalization of $X$ inside the function field of $Y_K$.

Must $Y$ be the proper regular minimal model of $Y_K$?

If not, what goes wrong, and are there reasonable situations where this is true? (perhaps if we assume $Y_K/X_K$ etale?)

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    $\begingroup$ The natural map $f:X_1(11) \rightarrow X_0(11)$ is an isogeny between elliptic curves over $\mathbf{Q}$ (upon using compatible $\mathbf{Q}$-points for the origin of each), and its dual $f^*$ is a counterexample over $R=\mathbf{Z}_{(11)}$. Indeed, these have semistable reduction at 11, so by the link between minimal regular proper models and Neron models for elliptic curves we see via computing $-{\rm{ord}}_{11}(j)$'s that the minimal regular proper model $\mathscr{X}_0(11)$ has irreducible special fiber whereas $\mathscr{X}_1(11)$ has reducible special fiber (so $f^*$ doesn't extend over $R$). $\endgroup$
    – nfdc23
    Oct 1, 2016 at 6:59
  • $\begingroup$ @oxeimon You will find more on your question in arxiv.org/abs/math/0412075 and/or math.u-bordeaux.fr/~qliu/articles/modcove.pdf If I remember correctly, you can even find examples where $Y$ (hence $X$ if $g(X) >0$) has good reduction over $S$ such that $Y\to X$ does not extend to a morphism of minimal regular models. $\endgroup$
    – Ariyan Javanpeykar
    Oct 1, 2016 at 7:44
  • $\begingroup$ @AriyanJavanpeykar: Suppose $Y$ and $X$ are smooth proper over local $S$ with geometrically connected $K$-fiber of genus $>0$. Note that $P_X:={\rm{Pic}}_{X/S}^1$ is a torsor for $J_X:={\rm{Pic}}^0_{X/R}$, and likewise for $Y$. For finite $f:Y_K\rightarrow X_K$, the Albanese map $(P_Y)_K=P_{Y_K}\rightarrow P_{Y_K}=(P_X)_K$ of torsors uniquely extends to an $S$-map $A:P_Y\rightarrow P_X$ over $J_Y\rightarrow J_X$ (work etale-locally on $S$ to get $S$-points). But $Y$ is the closure in $P_Y$ of the canonical $Y_K\hookrightarrow P_{Y_K}$, and likewise for $X$, so $A:Y\rightarrow X$ extending $f$. $\endgroup$
    – nfdc23
    Oct 1, 2016 at 20:59
  • $\begingroup$ @nfdc23 Nice argument. Is the obtained morphism $Y\to X$ necessarily finite? (That's what I should have written in the last sentence of my comment, and I think I remember seeing an example where $Y\to X$ is not finite. Am I wrong?) Note that, more generally, if $S$ is a Dedekind scheme with function field $K$, and $X$ and $Y$ are smooth proper curves over $K$ with genus $>0$, then any morphism $Y\to X$ extends to a morphism $\mathcal Y^{sm}\to \mathcal X^{sm}$, where $\mathcal Y$ (resp. $\mathcal X$) is the minimal regular model of $Y$ (resp. $X$) over $S$; see arxiv.org/abs/1312.4822 $\endgroup$
    – Ariyan Javanpeykar
    Oct 2, 2016 at 7:48
  • $\begingroup$ @AriyanJavanpeykar: Yes, it must be finite. By Stein factorization considerations, the smooth proper special fiber is geometrically connected and hence irreducible. By properness and flatness over $S$ and surjectivity over $K$, the map must be surjective. Thus, on special fibers it is a surjection between irreducible curves, so quasi-finite, so the map over $S$ is quasi-finite and therefore (by properness) is finite. $\endgroup$
    – nfdc23
    Oct 2, 2016 at 15:48

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There is no reason for $Y$ to be minimal, even if $Y_K\rightarrow X_K$ is étale. Take an elliptic curve $Y_K$ over $K=\mathbb{C}(t)$ with a rational point $y$ of order 2. Translation by $y$ gives an involution of $Y_K$, which extends to the minimal model $Y_0$. This involution will have some fixed points; it extends to the surface $Y$ obtained by blowing up the fixed points, and the quotient $X$ is smooth over $\mathbb{C}$, hence regular. $Y$ is the normalization of $X$ in $\mathbb{C}(Y)$ but is not minimal.

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