# The injectivity of $\text{Aut}(X) \to \text{Aut} (\text{Pic}^0X)$ for a stable curve

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.)

• @abx Is this a group scheme? If so, I know what it is, but I have no idea about relation between $\text{Pic}$ and it. And I think that this is not the group scheme because in lemma 1.16. the author says an abstract group is isomorphic to $H^1 \otimes \mathbb{G}_m$. ($H^1$ is the cohomology group of a graph.)
– k.j.
Mar 31, 2019 at 13:00
• Of course $\Bbb{G}_m$ is the group scheme, this is a universally adopted notation. See my answer below.
– abx
Mar 31, 2019 at 13:18

Let $$s$$ be a double point of $$X$$, and let $$p,q$$ be the two points of $$\pi ^{-1}(s)$$ (ordering chosen). Let $$\pi :X_s\rightarrow X$$ be the partial normalization of $$X$$ at $$s$$. There is an exact sequence of sheaves $$1\rightarrow \mathcal{O}_{X}^*\rightarrow \pi _*\mathcal{O}_{X_s}^*\xrightarrow{\ \varphi \ } \kappa (s)^*\rightarrow 1$$here $$\kappa (s)^*$$ is the skyscrapper sheaf over $$s$$ with stalk $$k^*$$, and $$\varphi$$ maps a function $$f$$ to $$f(p)/f(q)$$. The coboundary of the associated long exact sequence gives an injective homomorphism of algebraic groups $$h_s:\mathbb{G}_m\rightarrow \operatorname{Pic}^{\mathrm{o}}(X)$$. If $$\operatorname{Sing}(X)=\Sigma$$, the homomorphisms $$h_s \ (s \in \Sigma )$$ define an injective homomorphism $$\mathbb{G}_m^{\Sigma }\rightarrow \operatorname{Pic}^{\mathrm{o}}(X)$$; if $$\phi$$ induces the identity on $$\operatorname{Pic}^{\mathrm{o}}(X)$$, it must therefore preserve $$\Sigma$$ pointwise, and also $$S=\pi ^{-1}(\Sigma )$$ because exchanging $$p$$ and $$q$$ exchanges $$h$$ and $$h^{-1}$$.