Let $X\_{K}$ be a curve over a complete DVR $R$, $R/m:=k$ an algebraically closed field. We suppose the minimal field extension $L$ of $K$ such that $X\_{L}$ has stable model $X\_{R\_{L}}$, and the special fiber is $X\_{k}$. We obtain an action of $Gal(L/K)$ on $X\_{k}$.


My question is:


Does $Gal(L/K) \longrightarrow Aut(X\_{k})$ is an injection ?



Galois action on special fiber of a stable model
This appears in the literature, see e.g. Theorem 2.2 in Lehr and Matignon, Wild monodromy and automorphisms of curves, Duke Math. 2006. It is attributed there to Deligne and Mumford.


