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?



curves with good reduction everywhere
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>


