Can we bound the minimal degree of a field extension required to obtain semi-stable reduction  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 $X$? 
 A: For an abelian variety $A/K$, Serre-Tate says that you get semistable reduction if you adjoin enough torsion to $K$. For example, adjoining all of the 3-torsion will suffice. It seems plausible (but I don't know for certain) that if the Jacobian $J$  of your curve $X$ has semistable reduction, then so does $X$. If that's the case, then you can take $L$ to be $K(J[3])$, whose degree is bounded by a function of $\dim(J)=\text{genus}(X)$.
A: 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.)
