Let $K$ be a number field and let $X$ be a smooth projective geometrically connected curve over $K$.

There exists a finite field extension $L/K$ such that $X_L=X\otimes_K L$ has semi-stable reduction, i.e., there exists a semi-stable arithmetic surface $\mathcal{X}$ over the ring of integers $O_L$ with generic fibre $L$-isomorphic to $X_L$. Let $L_m$ be such an extension of minimal degree over $K$.

**Question 1.** Can we bound $[L_m:K]$ in terms of data depending only on $X$?