I know that the Torelli morphism $t_g:\mathcal{M}_g\rightarrow \mathcal{A}_g$ between the stacks of smooth curves of genus $g$ and principally polarized abelian varieties of dimension $g$ is of order 2 ramified on the hyperelliptic locus. I also know that the natural map $\sigma:Aut(C)\rightarrow Aut (J(C),\Theta)$ which to an automorphism of the smooth curve $C$ associates the induced automorphism on $(J(C),\Theta)$ is an isomorphism for $C$ hyperelliptic and is such that $Aut(J(C),\Theta)=<Aut(C),-1>$ otherwise.

My question is: how are the two notions linked? I know this is a trivial question, but i can't see how, for a non-hyperelliptic curve $C$, $Aut(J(C),\Theta)=<Aut(C),-1>$ should imply that $t_g^{-1}(J(C),\Theta)$ consists of two points in the stack $\mathcal{M}_g$..

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    $\begingroup$ Your question has already been answer but let me give a really lowbrow comment. It's sometimes useful to think of a point with isotropy group of order $\vert G \vert$ as "$1/\vert G \vert$th of a point". So for instance if you have point of $M_g$ corresponding to a curve with no automorphisms, and it maps to its jacobian in $A_g$ which has the inversion automorphism, then this corresponds to "a whole point" mapping to "half of a point". This makes the map 2-to-1 in the sense of stacks. $\endgroup$ Oct 20, 2014 at 9:15

2 Answers 2


Look at the definition of fiber product in the category of stacks: For a scheme $S$, the $S$-points of the fiber product $X \times_Z Y$ are triples $(x, y, \sigma)$ where $x \in X(S), y \in Y(S)$and $\sigma$ is a (iso)morphism between the images of $x$ and $y$ in the groupoid $Z(S)$.

Let $c:Spec\ k \to \mathcal{A}_g$ be the morphism corresponding to the point $(J(C), \Theta)$. The $Spec\ k$ valued points of $\mathcal{M}_g \times_{t_g, \mathcal{A}_g, c} Spec\ k$ are triples $(Spec\ k \to \mathcal{M}_g, Spec \ k \to Spec \ k, \sigma)$ where $\sigma$ is an isomorphism between images in $\mathcal{A}_g$. So if the map $Spec\ k \to \mathcal{M}_g$ corresponds to $C$ (which it must, for a $\sigma$ to exist, by Torelli theorem), then composing with $t_g$ corresponds to $(J(C), \Theta)$ in $\mathcal{A}_g$, and we can choose $\sigma$ to be an automorphism of $(J(C), \Theta)$.


It's not so much that it's two points of the stack, it's that the point of $\mathcal{A}_g$ has inertia, that is, the map restricted to a point (corresponding to a general curve) looks like $*\mapsto */\mathbb{Z}/2\cong B\mathbb{Z}/2$.

The key is that a point of a moduli stack "looks like" $*/Aut$ where $Aut$ is the automorphism group of the object corresponding to the moduli point. So, for any point, we have $*/Aut(C)\mapsto */<Aut(C),-1>$ which is degree 2, with general point being like I described above.

You can think of it as just being that if you have a set and quotient by $\mathbb{Z}/2$ it should be a double cover if the action is free, and here it's just not free, so we also have a double cover, because the target stops being just a set, but is rather a groupoid.


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