# Mgbar

One of the original goals of the Stacks project was to work through most of the “preliminary” material in the paper of Deligne and Mumford. Here I mean the material on algebraic stacks and on moduli stacks of curves, before one actually gets to the “interesting” part, namely, why the moduli stack of curves of a given genus is irreducible. This is now done. Currently the last theorem of the Stacks project is about how the moduli stack Mgbar is a proper and smooth Deligne-Mumford stack over Z for g >= 2.

Enjoy!

PS: I will make an effort to write more frequently here about what is going on with the Stacks project. In particular, I should write about the very successful Stacks project workshop which we just had, about what is next in line to be put in the Stacks project, about the wonderful people who help out with the Stacks project, and about how we’d like more people to help Pieter Belmans to code up parts of the new Stacks project web site!

## 5 thoughts on “Mgbar”

1. Some statistics about the result: it requires 6455 tags (which is 27% of the current toal), in 1493 sections (which is 52% of the sections).

• If I am not mistaken, those 6455 tags do not include definitions (because one usually does not refer to them with \ref{label}). So in reality the ratio is even larger then 27% of the current total, right?

2. Are you planning to include the proof (via deformation theory?) of the fact that the closed complement of $\mathcal{M}_g$ in $\overline{\mathcal{M}}_g$ is the support of a $\mathbf{Z}$-flat relative effective Cartier divisor (giving a most satisfactory sense in which $\mathcal{M}_g$ is “relatively dense” in $\overline{\mathcal{M}}_g$)? Or something about the story of $\mathcal{M}_{g,n}$ (which is useful in a paper I once read about alterations…)?

• All in its own good time, but yes!