Let $M\_g$ be the moduli space of genus $g$ curves, $A\_g$ be the moduli space of principally polarized dimension $g$ abelian varieties. They have dimensions $3g-3,g(g+1)/2$ respectively. The Torelli map is a map $\tau\_g: M\_g \to A\_g$ that takes a curve to it's Jacobian.


For $g=3$, $M\_g$ and $A\_g$ have the same dimension $6$ and I have heard that $\tau\_3$ is a 2-cover that is ramified exactly at the hyperelliptic locus. So there should be an automorphism of $\tau\_3$ that fixes the hyperelliptic locus, right? Can we give an explicit description at the level of genus 3 curves? That is, something like: Take some canonical map $C \to X$ for some variety $X$ and pull back by some involution on $X$?


(I would also appreciate a reference for the fact that $\tau\_3$ is a double cover, it should have something to do with the involution on abelian varieties not extending to a general curve.)



What is the involution on the moduli space of genus 3 curves induced by the Torelli map
The Torelli morphism being a double cover is purely a stacky phenomenon. It is not visible on coarse moduli spaces.


The involution you ask about is supposed to act on the fibers of the Torelli morphism. Here is how this works out. Pick a geometric point of $A\_3$, corresponding to a ppav $(A,\Theta)$. The fiber over this point consists of pairs $(C,\phi)$ where $C$ is a genus three curve and $\phi \colon \mathrm{Jac}(C) \to A$ is an isomorphism of ppav's. The involution on the fiber acts by composing $\phi$ with the inversion map on $A$. The ramification over the hyperelliptic locus arises since it is precisely when $C$ is hyperelliptic that there is an automorphism of $C$ for which the induced automorphism of $\mathrm{Jac}(C)$ is multiplication by $-1$.


