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 $\mathcal{M}_g$ 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.

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.