# What is the involution on the moduli space of genus 3 curves induced by the Torelli map

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.)

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 $$-1$$.
• To be complete, let me mention that the Torelli map $\tau _g:M_g\rightarrow A_g$ for ordinary moduli spaces is an embedding for $g\geq 2$ (and in char. zero), a result of Oort and Steenbrink.