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 $X$? 



Can we bound the minimal degree of a field extension required to obtain semi-stable reduction 
I think the answer to Question 1 is yes. One may use the fact that a curve has semistable reduction iff its Jacobian does and apply Grothendieck's theorem which says that an abelian variety has semistable reduction (over a local field) iff the representation of the intertia group on the Tate module is unipotent (it is always quasi-unipotent). One may ensure the unipotence by requiring the Jacobian variety have level $n$ structure for some $n> 2$; to do this one it suffices to make an extension of the base field of degree at most $|Sp(2g,\mathbb{Z}/n)|$, where $g$ is the dimension of the abelian variety, so the genus of the curve in your case.


(Note that a regular scheme over the ring of integers of a number field remains regular when it is base changed to the ring of integers of its completion at any finite prime, so semistability of a curve is not affected by completion.)


