## Picard Groups of Moduli Problems

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First, yes, I've seen Mumford's paper of this title. I'm actually interested in specific ones, and looking for really the most elementary/elegant proof possible.

I'm told that for $g\geq 2$ it is known that the Picard groups of $\mathcal{M}_g$ and $\mathcal{A}_g$ (the moduli spaces of curves of genus $g$ and abelian varieties of dimension $g$) are both isomorphic to $\mathbb{Z}$ (at least, over $\mathbb{C}$). What's the most efficient way to compute this? In fact, for $\mathcal{M}_g$, it's even generated by the Hodge bundle, I'm told. Ideally I want to avoid using stacks (though if stacks give an elegant proof, I'm open to them) and also would like to be able to calculate the degrees of some natural bundles, though I get that that's going to be a bit harder, so I want to focus this question on the computation of the Picard group.

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@Charles: you say you'd like to "avoid stacks" if possible, but that also affects the answer (let alone the methods). Just think of $\mathcal{M}_ 1$: the coarse moduli space is the affine line (trivial Pic), whereas the stack has nontrivial Pic ($\mathbf{Z}/12 \mathbf{Z}$). So specifically what do you mean by $\mathcal{M}_ g$ and $\mathcal{A}_ g$? – BCnrd May 20 2010 at 5:36
I'm also specifically excluding $\mathcak{M}_1$. Assuming that the statements I've made are true, I'll be satisfied with coarse moduli. I'm thinking of $\mathcal{A}_g$ as the Siegel upper half plane moduli the integral symplectic group, and I'm more than willing to take $\mathcal{M}_g$ to be the GIT quotient of the Hilbert scheme if tricanonically embedded curves moduli the projective linear group. However, I'd been under the (possibly mistaken) impression that the stackiness doesn't matter as much for $g\geq 2$, at least for $\mathcal{M}_g$ as automorphisms are finite. – Charles Siegel May 20 2010 at 5:44
I added some words to the question. Hopefully I guessed your intention correctly. – Kevin Lin May 20 2010 at 12:50

## 2 Answers

I'll just talk about the calculation of $\text{Pic}(\mathcal{M}_g)$ as a group (showing that it is generated by the Hodge bundle is then a calculation).

I think the most elementary way to view this problem is to think in terms of orbifolds rather than stacks. Recall that $\mathcal{M}_g$ is the quotient of Tecichmuller space $\mathcal{T}_g$ by the mapping class group $\text{Mod}_g$ (this is the curves analogue of $\mathcal{A}_g$ being the quotient of the Siegel upper half plane by the symplectic group). This action is properly discontinuous but not free (that's why we have an orbifold/stack rather than an honest space). A line bundle on $\mathcal{M}_g$ is then a $\text{Mod}_g$-equivariant line bundle on $\mathcal{T}_g$. There is an equivariant first Chern class homomorphism $c_1 : \text{Pic}(\mathcal{M}_g) \rightarrow H^2(\text{Mod}_g;\mathbb{Z})$. Mumford showed that $H^1(\text{Mod}_g;\mathbb{Z})=0$, so $\text{Pic}(\mathcal{M}_g)$ cannot vary continuously. This implies that $c_1$ is injective. Later, Harer proved that $H^2(\text{Mod}_g;\mathbb{Z}) \cong \mathbb{Z}$ for $g$ large. Since the Hodge bundle is nontrivial, $c_1$ cannot be the zero map, so we conclude that $\text{Pic}(\mathcal{M}_g) \cong \mathbb{Z}$.

Let me now recommend three places that contain more details about the above point of view. First, Hain has a survey entitled "Moduli of Riemann Surfaces, Trancendental Aspects", a large portion of which is devoted to the calculation of the Picard group. He gives many more details of the above sketch. He also shows how to show that the Hodge bundle generates the Picard group. Second, in the first couple of sections of my paper "The Picard Group of the Moduli Space of Curves With Level Structures" I give some extra details about things like Chern classes of orbifold line bundles. Finally, Hain has another survey "Lectures on Moduli Spaces of Elliptic Curves" in which he works all the above out for the moduli space of elliptic curves, where things are a little more concrete.

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Is it profitable to think of these line bundles in terms of central extensions of $\operatorname{Mod}_g$? – S. Carnahan♦ May 20 2010 at 14:22
That an interesting question.  Central extensions by Z of Mod_g correspond to classes in H^2(Mod_g;Z).  The calculations of H^2(Mod_g;Z) that I know of first use topology to give upper bounds on how large H^2(Mod_g;Z) is, and then prove that this bound is realized.  One can either do this using line bundles on moduli space or by using the so-called "Meyer cocycle".  Thus line bundles are more used to study central extensions than the other way around. – Andy Putman May 20 2010 at 14:56
Actually, here's an example of how to use central extensions to prove things about line bundles.  Recently Funar proved that the universal central extension of Mod_g is residually finite (a thm of Deligne shows that the universal central extension of Sp_2g(Z) is not residually finite).  This has a nice implication for the Picard group of moduli space : by passing to high enough finite covers of moduli space, you can make the Hodge bundle divisible by as much as you like.  Deligne's thm implies by passing to finite covers of A_g, the best you can do is make the hodge bundle divisible by 2. – Andy Putman May 20 2010 at 15:07
@Andy, thanks for the detailed answer for the case of curves (and especially the references, going through Hain now).  The case for abelian varieties is fairly similar, with the Symplectic group and $\mathbb{H}_g$ replacing Teichmuller space and and the mapping class group? – Charles Siegel May 27 2010 at 4:15
@Charles : That's right.  The picture for A_g was known long before the work on M_g and was an important inspiration for that work. – Andy Putman May 27 2010 at 4:28

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The fact that the Picard group of the moduli variety (not stack) A(g) is of rank 1, is sketched in a footnote of Mumford's paper on the Kodaira dimension of A(g), in LNM 997. This footnote is elaborated (over Z) in a paper of Smith-Varley: in LNM 1124. Another reference is Freitag's paper in Arch. Math. 40 (1983), pp.255-259. The whole point of Mumford's argument was that it follows from Borel's computation of the rank of the second cohomology group of the symplectic group (i.e. rank one).

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