Back in 2023, together with Ignacio Barros, I made Mgnbar.info, a website about the geometry of $\overline{\mathrm{M}}_{g,n}$, the moduli space of stable $n$-pointed genus $g$ curves. For a long time it showed a single invariant, the Kodaira dimension. It now has two more layers, which you can pick from the selector above the table, and the Kodaira dimension layer itself has gained some extra information.

The tautological ring. For each $(g,n)$ the table records whether the Chow ring of $\overline{\mathrm{M}}_{g,n}$, and of the open $\mathrm{M}_{g,n}$, is generated by the tautological classes, the natural $\psi$- and $\kappa$-classes and boundary strata, or whether there are genuinely more mysterious cycles.

Point counts. Whether the number of points of $\overline{\mathrm{M}}_{g,n}$ and $\mathrm{M}_{g,n}$ over a finite field $\mathbb{F}_q$ is a polynomial in $q$. The first place where this fails is $\overline{\mathrm{M}}_{1,11}$, where the culprit is the Ramanujan $\tau$-function, the coefficients of the weight-12 cusp form.

Both layers are transcribed from Hannah Larson’s recent survey arXiv:2606.29656, which collects the state of the art on precisely these questions. It is a wonderful read, and any errors in the transcription are of course mine.

More on the Kodaira dimension. Kodaira dimension $-\infty$ is only the coarsest sign that a moduli space is not of general type. The finer notions rational $\Rightarrow$ unirational $\Rightarrow$ rationally connected $\Rightarrow$ uniruled all imply it, and are now shown, as shaded blues, on the cells where the Kodaira dimension is $-\infty$. The baseline is Benzo’s 2014 survey table, extended with the more recent results of Agostini–Barros, Keneshlou–Tanturri, and (for genus 11-15) Verra and Bruno–Verra.

Along the way each layer got its own URL, and clicking a cell now updates the address bar too, so you can link straight to a single entry like mgnbar.info/tautological/#9,4. There is a small about page explaining how to cite the website, and, as with my other websites, Mgnbar.info is now a static site built with Hugo. I’m a big fan of Hugo nowadays, can you tell?

Suggestions and corrections are very welcome, on GitHub or by email.

LLMs were used to build this, which made the whole process much faster.