Mgnbar.info: two new layers

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…