Hello everyone,

I'd like to ask some question about semi-stable reduction of curves.

The Deligne-Mumford theorem tell us "Let $A$ be an Dedekind domain, $K=K(A)$, for any smooth curve $X$ over $K$ and $g(X)>1$, there exist a separable extension $L$ of $K$, such that $X_{L}$ has semi-stable model."

But this theorem does not tell us any information about how to find the extension field $L$, but it seems is very important for arithmetic geometers.

My question is, in arithmetic geometry, why find the good extension field $L$ (for example:tame extension) such that $X_{L}$ has semi-stable model is so important?