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It seems to be a folklore that for any genus $g$, there is a number field $K$ and a curve $X$ over $K$, such that $X$ has good reduction at all the places of $K$. Are any simple proofs of this?

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Perhaps you could just construct them; my guess is that something like $y^2=x^n+1$ has potentially good reduction at every prime. –  Tim Dokchitser Aug 19 '13 at 8:05
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up vote 3 down vote accepted

One way of seeing this is by appealing to Rumely's general local-global principle over $\bar{\mathbb{Z}}$, applied here to the moduli stack: an algebraic scheme over the algebraic integers $\bar{\mathbb{Z}}$ has a solution (point) in $\bar{\mathbb{Z}}$ if and only if it does in all $v$-adic completions $\bar{\mathbb{Z}}_v$. I don't know if this is a simple proof, though - it is probably not what you are looking for.

Here is a link to Rumely's paper:

http://www.math.uga.edu/~rr/ArithAllAlgInt.pdf

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Rumely's paper solves the problem for affine irreducible schemes over the algebraic integers. The extension to Artin stacks is due to Laurent Moret-Bailly, see his paper Problèmes de Skolem sur les champs algébriques, Compositio Math. 125, 1--30, 2001. Moret-Bailly gives applications to moduli stacks of curves; see notably his Exemple 0.9. –  ACL Aug 19 '13 at 7:34
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You can also apply Rumely's theorem directly to the moduli space of tricanonically embedded smooth curves of genus $g\geq2$: this is a scheme (subscheme of some Hilbert scheme), which is quasiprojective, smooth and surjective over $\mathrm{Spec}\:\mathbb{Z}$. Moreover it is a $GL_N$-bundle over the stack $M_g$ for some $N$, hence has geometrically connected fibers over $\mathrm{Spec}\:\mathbb{Z}$ by Deligne-Mumford. So Rumely's theorem applies. –  Laurent Moret-Bailly Aug 19 '13 at 14:52
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