Let $K$ be a number field and let $g$ be a positive integer. Does there exist a smooth projective geometrically connected curve $X/K$ of genus $g$ such that $X$ does not have semi-stable reduction over $K$?
I can write down curves over certain number fields without semi-stable reduction, but I can't do it for a general number field.
Yes. Take $f\in O_K$ a uniformizing element of some prime $\mathfrak p$. Consider the hyperelliptic curve defined by the equation $$y^2=x^{2g+1}+f.$$ Then this curve doesn't have semi-stable reduction at $\mathfrak p$. In fact, this equation defines a proper regular model of the curve over the localization $O_{K, \mathfrak p}$ and this model is minimal because its closed fiber is irreducible (defined by $y^2=x^{2g+1}$), and it is not semi-stable. If the curve had semi-stable reduction, the minimal regular model would also be semi-stable. Further, this curve is far from being semi-stable because its Jacobian has purely additive reduction at $\mathfrak p$.
Add a more elementary explanation on why the curve doesn't have semi-stable reduction at $\mathfrak p$. Suppose for simplicity that $\mathfrak p$ is prime to $2(2g+1)$. As saw in the comments, the curve has potentially good reduction above $\mathfrak p$. So if it had already semi-stable reduction over $K$, then it would already have good reduction over $K$. This implies (as for elliptic curves) that after a suitable homographic transformation on $x$, we will get a new polynomial with discriminant invertible in $O_{K, \mathfrak p}$. But such transformation changes the valuation of the discriminant by a multiple of $2(2g+1)$ while the initial discrimiant has valuation $2g$. Impossible.
