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I've been told many times that the Torelli map $J:\mathcal{M}_g\to \mathcal{A}_g$ for ($g\geq 2$, and at least on the level of coarse moduli spaces, over $\mathbb{C}$) gives an isomorphism of Picard groups. On the level of rational picard groups, they're both generated by the determinant of the Hodge bundle, but why is this true on the nose? I'll be more than happy with a proof or with a reference to a detailed proof (actually slightly prefer the latter).

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up vote 9 down vote accepted

I see two ways interpreting "on the nose": 1) At the level of the orbifold Picard group, or that is at stack level, and in my view this is the correct framework, both Pic_{orb}(M_g) and Pic_{orb}(A_g) are generated by the Hodge class. (The Hodge class on M_g is by definition the pull-back of the Hodge class on A_g.) The reference for this result is Arbarello and Cornalba "Picard groups of the moduli space of curves" (1987).

2) At the level of integral Picard groups, the answer is still positive but messier. Pic(M_g) is the sugroup of Pic_{orb}(M_g) generated by the multiple m\lambda of the Hodge class which actually descends to the coarse moduli space. As far as I know, the actual value of this multiple is not known. The bundle L with class m\lambda descends to M_g if every automorphism of a curve C, acts trivially on the fibre of L over C. It is certain to be an even number, for \lambda does not even descend to the smooth part of M_g but 2\lambda does.

The references are Mumford's paper "Stability of proj vareties" (1977) and "Abelian quotients of Teichmüller groups" (1967).

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Very helpful! Is it by any chance known to generalize, say, to a finite cover of the moduli spaces (like level n, (n,2n), or the Prym moduli spaces)? –  Charles Siegel Apr 18 '11 at 16:20

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