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Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$.

Does there exist a Néron model $\mathcal X$ for $X$ over $O_K$?

By a Néron model, I mean a smooth model (not necessarily proper) with the "Néron universal property": for any smooth $O_K$-scheme $\mathcal Y$,

$$\mathrm{Hom}(\mathcal Y, \mathcal X) = X(\mathcal Y_K).$$

Is the smooth locus of the minimal regular model of $X$ over $O_K$ a Néron model? Does base change help? That is, does there exist a Néron model for $X_L$ after some suitable base change $L/K$?

Note that if $\mathcal{X}$ is the smooth locus of the minimal regular model of $X$ over $O_K$, we have the "Néron" property $\mathcal{X}(O_K) = X(K)$. (In fact, the image of a section in the minimal regular model lies in the smooth locus.)

This question was asked on stackexchange four months ago:

http://math.stackexchange.com/questions/153369/do-neron-models-of-hyperbolic-curves-exist

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$.

Does there exist a Néron model $\mathcal X$ for $X$ over $O_K$?

By a Néron model, I mean a smooth model (not necessarily proper) with the "Néron universal property": for any smooth $O_K$-scheme $\mathcal Y$,

$$\mathrm{Hom}(\mathcal Y, \mathcal X) = X(\mathcal Y_K).$$

Is the smooth locus of the minimal regular model of $X$ over $O_K$ a Néron model? Does base change help? That is, does there exist a Néron model for $X_L$ after some suitable base change $L/K$?

Note that if $\mathcal{X}$ is the smooth locus of the minimal regular model of $X$ over $O_K$, we have the "Néron" property $\mathcal{X}(O_K) = X(K)$. (In fact, the image of a section in the minimal regular model lies in the smooth locus.)

This question was asked on stackexchange four months ago:

https://math.stackexchange.com/questions/153369/do-neron-models-of-hyperbolic-curves-exist

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$.

Does there exist a Néron model $\mathcal X$ for $X$ over $O_K$?

By a Néron model, I mean a smooth model (not necessarily proper) with the "Néron universal property": for any smooth $O_K$-scheme $\mathcal Y$,

$$\mathrm{Hom}(\mathcal Y, \mathcal X) = X(Y_K).$$

Is the smooth locus of the minimal regular model of $X$ over $O_K$ a Néron model? Does base change help? That is, does there exist a Néron model for $X_L$ after some suitable base change $L/K$?

Note that if $\mathcal{X}$ is the smooth locus of the minimal regular model of $X$ over $O_K$, we have the "Néron" property $\mathcal{X}(O_K) = X(K)$. (In fact, the image of a section in the minimal regular model lies in the smooth locus.)

This question was asked on stackexchange four months ago:

http://math.stackexchange.com/questions/153369/do-neron-models-of-hyperbolic-curves-exist

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$.

Does there exist a Néron model $\mathcal X$ for $X$ over $O_K$?

By a Néron model, I mean a smooth model (not necessarily proper) with the "Néron universal property": for any smooth $O_K$-scheme $\mathcal Y$,

$$\mathrm{Hom}(\mathcal Y, \mathcal X) = X(\mathcal Y_K).$$

Is the smooth locus of the minimal regular model of $X$ over $O_K$ a Néron model? Does base change help? That is, does there exist a Néron model for $X_L$ after some suitable base change $L/K$?

Note that if $\mathcal{X}$ is the smooth locus of the minimal regular model of $X$ over $O_K$, we have the "Néron" property $\mathcal{X}(O_K) = X(K)$. (In fact, the image of a section in the minimal regular model lies in the smooth locus.)

This question was asked on stackexchange four months ago:

http://math.stackexchange.com/questions/153369/do-neron-models-of-hyperbolic-curves-exist

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$.

Does there exist a Néron model $\mathcal X$ for $X$ over $O_K$?

By a Néron model, I mean a smooth model (not necessarily proper) with the "Néron universal property": for any smooth $O_K$-scheme $\mathcal Y$,

$$\mathrm{Hom}(\mathcal Y, \mathcal X) = X(Y_K).$$

Is the smooth locus of the minimal regular model of $X$ over $O_K$ a Néron model? Does base change help? That is, does there exist a Néron model for $X_L$ after some suitable base change $L/K$?

Of course, all of this stuff is clear if $X$ has good reduction over $O_K$.

Note that if $\mathcal{X}$ is the smooth locus of the minimal regular model of $X$ over $O_K$, we have the "Néron" property $\mathcal{X}(O_K) = X(K)$.

This question was asked on stackexchange four months ago:

http://math.stackexchange.com/questions/153369/do-neron-models-of-hyperbolic-curves-exist

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$.

Does there exist a Néron model $\mathcal X$ for $X$ over $O_K$?

By a Néron model, I mean a smooth model (not necessarily proper) with the "Néron universal property": for any smooth $O_K$-scheme $\mathcal Y$,

$$\mathrm{Hom}(\mathcal Y, \mathcal X) = X(Y_K).$$

Is the smooth locus of the minimal regular model of $X$ over $O_K$ a Néron model? Does base change help? That is, does there exist a Néron model for $X_L$ after some suitable base change $L/K$?

Note that if $\mathcal{X}$ is the smooth locus of the minimal regular model of $X$ over $O_K$, we have the "Néron" property $\mathcal{X}(O_K) = X(K)$. (In fact, the image of a section in the minimal regular model lies in the smooth locus.)

This question was asked on stackexchange four months ago:

http://math.stackexchange.com/questions/153369/do-neron-models-of-hyperbolic-curves-exist