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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.

2 added 3 characters in body

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.

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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 is false in positive characteristic, though.