This is lemma 1.15. of Deligne, Mumford's paper. Let $X$ be an irreducible stable curve over an algebraically closed field, $\phi$ an automorphism on $X$ which induces the identity on $\text{Pic}^0X$. Then is $\phi$ the identity on $X$?

Let $X'$ be the normalization and $\phi'$ the action of $\phi$ on $X'$. (induced by the universal property of the normalization) The author says that choosing a singular point of $X$ and an prdering of its inverse image in $X'$ defines a distinct morphism from $\mathbb{G}_m$ to $\text{Pic}^0X$, and therefore the inverse image $S$ of the singular locus of $X$ is pointwise fixed by $\phi'$. But I can't understand this at all. (And I don't know what $\mathbb{G}_m$ is.)