Together with Ignacio Barros we have created, a website dedicated to the geometry of $\overline{\mathrm{M}}_{g,n}$, the moduli space of stable marked curves.

Kodaira dimension of $\overline{\mathrm{M}}_{g,n}$

Our initial goal was to reproduce a static table that Ignacio had made and which is still available at his homepage, that gives the Kodaira dimension of these moduli spaces.

The reason why this is so interesting is that we've known for more than a hundred years that $\overline{\mathrm{M}}_g$ had Kodaira dimension $-\infty$ (because it is rational) for $g=2,\ldots,10$, by Severi. But then in 1982 Harris and Mumford showed that $\overline{\mathrm{M}}_{25}$ (and $\overline{\mathrm{M}}_g$ for all further odd $g$) are of general type! This was later shown to hold for all $g\geq 24$, and in 2020 for all $g\geq 22$. On the other side we now know that for $g\leq15$ it has Kodaira dimension $-\infty$, and $g=16$ is not of general type.

So there is a drastic change in the complexity of these moduli spaces when you vary $g$. And when you start adding in markings on your curve a similar picture arises, but the change starts happening sooner the more marked points you add.

What's next?

Well, that's for you to decide! Right now it's just a showcase of what is possible, with basic interface for Kodaira dimension.

You can make suggestions on GitHub issues, or send me an email. Please be detailed in what you are suggesting, and include ample references, as I'm no expert.