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 ?
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.
The answer is affirmative when $X_K$ is smooth and geometrically connected with genus $\ge 2$ (which I am guessing are implicit hypotheses), without restriction on the residual characteristic nor on the generic characteristic. The key ingredients are theorems of Grothendieck concerning semistable reduction and the link between stable reduction of curves and semistable reduction of Jacobians (proved by Deligne and Mumford).
By increasing $K$ if necessary, the "minimality" hypothesis on $L$ reduces the problem to showing that if the ${\rm{Gal}}(L/K)$-action on the special fiber $X_k$ of $X_{R_L}$ is trivial then $X$ has stable reduction over $R$. By Deligne-Mumford, it is equivalent to deduce that the Jacobian $J$ of $X_K$ has semistable reduction. By Grothendieck's inertial criterion for semistable reduction, this in turn is equivalent to deducing that the action of $G_K = {\rm{Gal}}(K_s/K)$ on $V_{\ell}(J)$ is unipotent for some (equivalently, any) prime $\ell \ne {\rm{char}}(k)$.
By a theorem of Raynaud, $P := {\rm{Pic}}^0_{X_{R_L}/R_L}$ is a scheme, and more specifically is semi-abelian with generic fiber $J_L$. The subgroup $G_L = {\rm{Gal}}(K_s/L)$ of $G_K$ acts unipotently on $V_{\ell}(J)$, and Grothendieck's orthogonality theorem (applied over $R_L$) gives much more: $V_{\ell}(J)^{G_L} = V_{\ell}(P_k)$ with $V_{\ell}(J)/V_{\ell}(J)^{G_L}$ canonically dual to $V_{\ell}(T)$ where $T$ is the maximal torus of $P_k$ (here using the auto-duality of $J$ over $K$). This "canonicity" includes $G_K$-equivariance, so to prove the unipotence of the $G_K$-action on $V_{\ell}(J)$ it suffices to prove the triviality of the ${\rm{Gal}}(L/K)$-action on $V_{\ell}(P_k)$. But this latter Galois action is just the composition of functoriality of $V_{\ell}$ applied to the ${\rm{Gal}}(L/K)$-action on $P_k = {\rm{Pic}}^0_{X_k/k}$ arising from the ${\rm{Gal}}(L/K)$-action on $X_k$. (Various implicit compatibility verifications are left as an exercise.) This latter action is assumed to be trivial, so we are done.
This is correct in characteristic $0$. Let $G$ be the kernel of the action; then one has to show that $X_{R_L}/G$ is a stable curve over $R_L^G$. This is Lemma 3.2 in my paper with Dan Abramovich Complete moduli for fibered surfaces, http://arxiv.org/abs/math/9804097.
I am pretty sure that it is false in positive characteristic, though.
[Edit:] I guess I was wrong about positive characteristic.
